On Light, Vision, Appearance, Color and Imaging

On Light, Vision, Appearance, Color and Imaging

This is a topic close to my heart. My masters thesis research was on color prediction and modeling. My research work was published by IPST Atlanta as Technical paper no 469. Those who work in Paper and Printing Industry know IPST very well. It used to be called Institute of Paper Chemistry and was based in Appleton, Wisconsin.

Key Terms

  • Color Science
  • Human Vision
  • Color Models
  • Industrial Color
  • Measurement
  • Color Physics
  • Color Chemistry
  • Color Perception
  • Color Psychology
  • Instruments
  • Light
  • Color Technology
  • Light Absorption
  • Light Scattering
  • Psycho-Physics of Color
  • Reflectance
  • Refraction
  • Gloss
  • Texture
  • Colorimeter
  • Spectrophotometer
  • Color Dyes and Pigments
  • Paint, Plastics, Paper, Textiles
  • Digital Color
  • Device Independent color
  • Computer Monitors
  • Color Theory
  • Color Physics
  • Kubelka Munk Theory
  • Munsell Colors
  • Pantone Colors
  • RIT
  • Newton’s Optics
  • Goethe Color Theory
  • Four Color Problem
  • Primary Colors
  • CIE LAB color
  • CIE LCH color
  • Visual Match
  • Instrument Match
  • Radiative Transfer Theory
  • Two Flux vs Multi Flux Models
  • CIE
  • ICC
  • Optical Society of America
  • Inter-Society Color Council ISCC
  • Color and Appearance
  • Whiteness
  • Yellowness
  • Color Profiles
  • Color Scales
  • RGB
  • CMYK
  • Rods and Cones

Human Vision

Retina of Human eye has two kind of cells responsible for color vision

Rod Cells. Rod Cells are used for motion and lightness

Cone Cells. Cone Cells are responsible for color vision in the eye retina.

Image Source: Basics of Color Imaging/Yao Wang

Image Source: Basics of Color Imaging/Yao Wang

Image Source: Clarkvision

In the references, I have included many links to articles and papers on the following importatnt topics of color. Many companies who meaure and do testing of color provide excellent tutorials on color. See References.

  • Light and Visible Spectrum
  • What is Color?
  • Color Perception in Human
  • Color Models for Visual Perception
  • Color Physics
  • Dyes and Pigments
  • Color Chemistry
  • Optical Properties of Materials

Color Standards

  • ICC International Color Consortium
  • CIE
  • ISCC Inter Society Color Council

Coloring of Materials

  • Paper
  • Textiles
  • Paints
  • Plastics

Color Meaurement in Industry

  • Colorimeters
  • Spectrophotometers

Color Measurement Companies

  • Xrite
  • Datacolor
  • Konica Minolta
  • Hunterlab
  • Technidyne

Color Prediction and Control

  • Prediction in Lab
  • Online Prediction and Control

Kubelka Munk Theory (KM)

It was developed using Radiative Transfer Theory to measure Light Absorption and Light Scattering by objects. Reflection and Transmission.

Limitations of KM Theory

  • Only Two Flux
  • Errors in measuring Strong Absorption and Weak Scattering
  • Correlation between K and S. As K goes up S goes down
  • Use of Single Constant Vs Two Constant KM Theory
  • Can not measure effects of Fluroscent Dyes FWA OBA

Several efforts have been made since early 1990s, to revise, modify KM theory or develop other multiflux models to improve prediction better than KM model.

Key Recent Researchers

  • Li Yang
  • Per Edstrom
  • L G Coppel
  • Tarja Shakespeare
  • H Granberg

My related posts

Some of my earlier published papers

Sounds True: Speech, Language, and Communication

Myth of Invariance: Sound, Music, and Recurrent Events and Structures

Understanding Rasa: Yoga of Nine Emotions

Key sources of Research

Measurement and Control of the Optical Properties of Paper




Optical paper properties and their influence on colour reproduction and perceived print quality


Ivana Jurič

Igor Karlovits

Ivana Tomić

Dragoljub Novaković


An assessment of Saunderson corrections to the diffuse reflectance of paint films

A García-Valenzuela et al 

2011 J. Phys.: Conf. Ser. 274 012125


Review: Optical properties of paper: theory and practice.

R. Farnood.

In Advances in Pulp and Paper Research, Oxford 2009, 

Trans. of the XIVth Fund. Res. Symp. Oxford, 2009,
(S.J. I’Anson, ed.), pp 273–352, FRC, Manchester, 2018

Diffuse Reflectance Spectroscopy; Applications, Standards, and Calibration (With Special Reference to Chromatography)

R. W. Frei

Analytical Research and Development, Pharmaceutical Department Sandoz Ltd., 4002 Basel, Switzerland

(May 26, 1976)

Optical models for colored materials

Mathieu Hébert
Institut d’Optique Graduate School, Saint-Etienne. mathieu.hebert@institutoptique.fr

The Use of Reflectance Measurements in the Determination of Fixation of Reactive Dyes to Cotton

N. Ahmed, D. P. Oulton, J. A. Taylor*

Textile sand Paper Group, School of Materials, University of Manchester, P.O. Box 88, Sackville Street, Manchester M60 1QD, United Kingdom

Received 3 January 2005; accepted 9 August 2005

Two-flux and multiflux matrix models for colored surfaces

Mathieu Hébert

Université de Lyon, Université Jean Monnet de Saint-Etienne, CNRS UMR 5516 Laboratoire Hubert Curien, F-42000, Saint-Etienne, France.

Patrick Emmel
14 rue de Münchendorf, 68220 Folgensbourg, France.



USDA 1967


Optical Response from Paper

Doctoral Thesis

H Granberg 2003



Martin A. Hubbe, Joel J. Pawlak and Alexander A. Koukoulas

2008 BioResources online Journal

Examination of the revised Kubelka–Munk theory: considerations of modeling strategies

Per Edström

Department of Engineering, Physics and Mathematics, Mid Sweden University, SE-87188 Härnösand, Sweden

Received April 5, 2006; revised July 3, 2006; accepted July 18, 2006; posted September 11, 2006 (Doc. ID 70185); published January 10, 2007

Revised KubelkaMunk theory. I. Theory and application

Li Yang and Bjo ̈rn Kruse

Campus Norrko ̈ ping (ITN), Linko ̈ ping University, S-601 74, Norrko ̈ ping, Sweden

Received November 20, 2003; revised manuscript received May 3, 2004; accepted May 5, 2004

Revised Kubelka–Munk theory II Unified framework for homogeneous and inhomogeneous optical media

Article in Journal of the Optical Society of America A · November 2004

Li Yang, Bjo ̈rn Kruse, and Stanley J. Miklavcic

Campus Norrko ̈ ping (ITN), Linko ̈ ping University, S-601 74, Norrko ̈ ping, Sweden

Revised Kubelka–Munk theory. III. A general theory of light propagation in scattering and absorptive media

Li Yang

Graphical Technology/Package Printing Group, Department of Chemical Engineering, Karlstad University, S-651 88 Karlstad, Sweden

Stanley J. Miklavcic

Center for Creative Media Technology, Department of Science and Technology, Linköping University, S-601 74 Norrköping, Sweden

Received January 18, 2005; accepted March 9, 2005

Qualifying the arguments used in the derivation of the revised Kubelka-Munk theory: reply

Yang, Li 

Linköping University, The Institute of Technology. Linköping University, Department of Science and Technology.2007 (English)


A novel method for studying ink penetration of a print.

Yang L, Fogden A, Pauler N, Sävborg Ö, Kruse B.

Nordic Pulp & Paper Research Journal. 2005;20(4):423-429.


A study in ink-jet color reproduction

L i Y a n g 2003

Department of Science and Technology Link ̈oping University

SE-601 74 Norrko ̈ping Sweden

Color Prediction and Separation Models in Printing

-Minimizing the Colorimetric and Spectral Differences employing Multiple Characterization Curves

Yuanyuan Qu

Department of Science and Technology Linköping University, SE-601 74 Norrköping, Sweden Norrköping 2013

Deriving Kubelka–Munk theory from radiative transport 

Christopher Sandoval and Arnold D. Kim*

Applied Mathematics Unit, School of Natural Sciences, University of California, Merced, 5200 North Lake Road, Merced, California 95343, USA
*Corresponding author: adkim@ucmerced.edu

Received November 12, 2013; accepted January 6, 2014;
posted January 16, 2014 (Doc. ID 201164); published February 21, 2014


Vesna Džimbeg-Malčić, Željka Barbarić-Mikočević, Katarina Itrić


  • Source: Tehnicki vjesnik / Technical Gazette . Jan-Mar2012, Vol. 19 Issue 1, p191-196. 6p. 
  • Author(s): Džimbeg-Malčić, Vesna; Barbarić-Mikočević, Željka; Itrić, Katarina

Applicability conditions of the Kubelka–Munk theory

William E. Vargas and Gunnar A. Niklasson

Extension of the Kubelka–Munk theory of light propagation in intensely scattering materials to fluorescent media 

Leonid Fukshansky and Nina Kazarinova

  • Journal of the Optical Society of America
  • Vol. 70,
  • Issue 9,
  • pp. 1101-1111
  • (1980)

What Has Been Overlooked in Kubelka-Munk Theory?

Author: Yang, Li

Source: NIP & Digital Fabrication Conference, 2005 International Conference on Digital Printing Technologies. Pages 332-679., pp. 376-379(4)

Publisher: Society for Imaging Science and Technology


Kubelka Munk Theory for Efficient Spectral Printer Modeling

Mekides Assefa


On Measurements of Effective Residual Ink Concentration (ERIC) of Deinked Papers Using Kubelka-Munk Theory

D.W. Vahey, J.Y. Zhu and C.J. Houtman

Single‐constant simplification of Kubelka‐Munk turbid‐media theory for paint systems—A review

Roy S. Berns Mahnaz Mohammadi

First published: 25 April 2007

Color Research and Application J


Spectrophotometric color prediction of mineral pigments with relatively large particle size by single- and two-constant Kubelka-Munk theory

Authors: Li, JunfengWan, Xiaoxia

Source: Color and Imaging Conference, Volume 2017, Number 25, September 2017, pp. 324-329(6)

Publisher: Society for Imaging Science and Technology


Theory of light propagation incorporating scattering and absorption in turbid media

Li Yang and Stanley J. Miklavcic

Department of Science and Technology, Link ̈oping University, S-601 74, Norrko ̈ping, Sweden

Article in Optics Letters · May 2005

Kubelka-Munk Model for Imperfectly Diffuse Light Distribution in Paper

Li Yang􏰀
Holmen Paper Development Center (HPD), Holmen AB, Sweden E-mail: li.yang@holmenpaper.com

R. D. Hersch

Ecole Polytechnique Fédérale de Lausanne (EPFL), School of Computer and Communication Sciences, Lausanne 1015, Switzerland

Quantification of the Intrinsic Error of the Kubelka–Munk Model Caused by Strong Light Absorption



Anisotropic reflectance from turbid media. I. Theory

Neuman, Magnus 

Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.(Pappersoptik och färg)

Edström, Per 

Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.(Pappersoptik och färg)

2010 (English)

In: Journal of the Optical Society of America A, ISSN 0740-3232, Vol. 27, no 5, p. 1032-1039

Mathematical modeling and numerical tools for simulation and design of light scattering in paper and print

Edström, Per 

Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.(FSCN – Fibre Science and Communication Network)ORCID iD: 0000-0002-0529-1009

Mid Sweden University

2007 (English)

Theoretical Investigation of Bioactive Papers Using the Kubelka-Munk Theory

Elina Levi Gendler

Masters of Applied Science Chemical Engineering and Applied Chemistry University of Toronto

The color prediction model of fluorescent prints

Na DongYixin ZhangGuoyun Shi

Proceedings Volume 7241, Color Imaging XIV: Displaying, Processing, Hardcopy, and Applications;72411N (2009) 
Event: IS&T/SPIE Electronic Imaging, 2009, San Jose, California, United States


State of the art on macroscopic models for the determination of thin films optical properties

G. Saridakis, D. Kolokotsa

Technological Educational Institute of Crete, Greece

M. Santamouris

Radiative properties of optically thick fluorescent turbid media

Alexander A. Kokhanovsky

Journal of the Optical Society of America AVol. 26,Issue 8,pp. 1896-1900(2009)


Radiative properties of optically thick fluorescent turbid media: errata 

A. A. Kokhanovsky  

  • Journal of the Optical Society of America
  • Vol. 27,
  • Issue 9,
  • pp. 2084-2084
  • (2010)


Spectral Reflectance Model of a Single Sheet of Blank Paper*

Yongchi XU** and Shisheng ZHOU** **

Faculty of Printing and Packaging Engineering, Xi’an University of Technology, Xi’an, 710048 China


Next Generation Simulation Tools for Optical Properties in Paper and Print

Per Edström
Mid Sweden University, TFM, SE‐87188 Härnösand, Sweden,

Improving the performance of computer color matching procedures 

  • Journal of the Optical Society of America A
  • Vol. 25,
  • Issue 9,
  • pp. 2251-2262
  • (2008)


A two-phase parameter estimation method for radiative transfer problems in paper industry applications

Per Edstro ̈ m*

Department of Engineering, Physics and Mathematics, Mid Sweden University, Ha ̈rno ̈sand, Sweden

Inverse Problems in Science and Engineering

Vol. 16, No. 7, October 2008, 927–951

A Guide to Understanding Color


Understanding CIE *L*a*b Colour Space


Understanding Color

Giordino Beretta 2008 and 2010

HP Labs



Konica Minolta

The basics of Color Perception and Measurement


The Color Guide and Glossary


Color Differences & Tolerances Commercial Color Acceptability


Defining and Communicating Color: The CIELAB System


Color Science Course




Using Color Effectively in Computer Graphics

Lindsay W. MacDonald

University of Derby, UK

Color Management Fundamentals

Color realism and color science

Alex Byrne

Department of Linguistics and Philosophy, Massachusetts Institute of Technology, Cambridge, MA 02139
abyrne@mit.edu mit.edu/abyrne/www

David R. Hilbert

Department of Philosophy and Laboratory of Integrative Neuroscience, University of Illinois at Chicago, Chicago, IL 60607
hilbert@uic.edu http://www.uic.edu/~hilbert/


Introduction to Color Models


Color Appearance Models

Second Edition

Mark D. Fairchild

Munsell Color Science Laboratory Rochester Institute of Technology, USA


By George R. Rossman



Light and Color

Colour physics and colour measurement: state-of-the-art and challenges

S Westland


Mil􏰀osz Michalski

Institute of Physics Nicolaus Copernicus University

July 3, 2012

Lecture 26: Color and Light

On the Kubelka-Munk Single-Constant/Two-Constant Theories

Ning Pan and others


Kubelka-Munk Prediction for Dark Mixtures

  • December 2013
  • Conference: 5th International Color and Coatings Congress (ICCC 2013)
  • At: Isfahan, Iran


Farhad Moghareh Abed

Roy S. Berns


Colour Measurement and Analysis in Fresh and Processed Foods: A Review

Pankaj B Pathare

Umezuruike Linus Opara

Fahad Al-Julanda Al-Said


Extending Kubelka-Munk’s Theory with Lateral Light Scattering

Safer Mourad *, Patrick Emmel **, Klaus Simon and Roger David Hersch **

IS&T’s NIP17: International Conference on Digital Printing Technologies

Kubelka Munk Model in Paper Optics: Successes, Limitations and Improvements

L. Yang

Page 81


Gülen Bayhan



Dibakar Raj Pant.

University of Joensuu Department of Computer Science Pro gradu
April, 2006


Next Generation Simulation Tools for Optical Properties in Paper and Print

Per Edström
Mid Sweden University, TFM, SE‐87188 Härnösand, Sweden, per.edstrom@miun.se

The optical properties of bleached kraft pulp

Steven R. Middleton and Anthony M. Scallan, 

Pulp and Paper Research Institute of Canada, Pointe Claire, Canada

Light Scattering in Fibrous Sheets

Edwin W Arnold

IPC PhD Thesis 1962




Department of Information and Graphic Arts Technology, Faculty of Natural Sciences and Engineering, University of Ljubljana, Snežniška 5, SI-1000 Ljubljana, Slovenia

Determining optical properties of mechanical pulps 

Anette Karlsson, Sofia Enberg, Mats Rundlöf, Magnus Paulsson and Per Edström

Nordic Pulp and Paper Research Journal Vol 27 no.3/2012

Color iQC and Color iMatch Multi Flux Matching Guide

Version 8.0 | July 2012


Industrial Color Physics 

By Georg A. Klein



MARCH 1993

IPST Technical Paper Series 469

Click to access tps-469.pdf

Application of Kubelka-Munk Theory in Device-independent Color Space Error Diffusion

Shilin Guo and Guo Li

Hewlett-Packard Company, San Diego Site

The Practical Guide To Color Theory For Photographers

In-Depth Guide on How to Measure Color in Plastics


A Guide to Understanding Color Communication

Tintometer Group

A partial explanation of the dependence between light scattering and light absorption in the Kubelka-Munk model 

M. Neuman, L. G. Coppel and P. Edström

Nordic Pulp and Paper Research Journal Vol 27 no.2/2012

Limitations of the efficiency of fluorescent whitening agents in uncoated paper

Gustafsson Coppel, Ludovic 

Andersson, Mattias 

Edström, Per 

Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.

Kinnunen, Jussi 

Univ Eastern Finland, Dept Math & Phys, FI-80101 Joensuu, Finland.

2011 (English) In: Nordic Pulp & Paper Research Journal, ISSN 0283-2631, E-ISSN 2000-0669, Vol. 26, no 3, p. 319-328

Determination of light scattering coefficient of dark and heavy sheets

KNOX, J.M., and WAHREN, D.

Tappi J 1984

Whiteness and Fluorescence in Layered Paper and Boards

Perception and Optical Modelling

L G Coppel

PhD Thesis

Mid Sweden University

Extension of the Stokes equation for layered constructions to fluorescent turbid media

Ludovic G. Coppel,1,2 Magnus Neuman,2 and Per Edström2,*
1Innventia AB, Box 5604, SE-11486 Stockholm, Sweden
2Department of Natural Sciences, Engineering and Mathematics, Mid Sweden University, SE-87188 Härnösand, Sweden 

*Corresponding author: per.edstrom@miun.se

Received January 3, 2012; accepted January 20, 2012;
posted January 24, 2012 (Doc. ID 160521); published March 22, 2012

Determination of quantum efficiency in fluorescing turbid media.

Coppel LG,  Andersson M,  Edström P

Applied Optics, 31 May 2011, 50(17):2784-2792

Extension of the Kubelka–Munk theory of light propagation in intensely scattering materials to fluorescent media 

Leonid Fukshansky and Nina Kazarinova

  • Journal of the Optical Society of America
  • Vol. 70,
  • Issue 9,
  • pp. 1101-1111
  • (1980)


Revised Optical Properties of Turbid Media on a Base of Generally Improved Two-Flux Kubelka-Munk Approach

D. A. Rogatkin1, and V. V. Tchernyi2

Understanding Color Communication


Correspondences between the Kubelka-Munk and the Stokes model of strongly light-scattering materials. II: Implications

[1] North Carolina state univ., dep. wood & paper sci., Raleigh NC 27695-8005, United StatesSource

Tappi journal1989, Vol 72, Num 7, pp 159-163

Precise Color Communication

Konica Minolta

The Color Guide and Glossary




Principles of Color Technology for Color Imaging Scientists and Engineers




Using Color Effectively in Computer Graphics

Lindsay W. MacDonald

University of Derby, UK

Color Management Fundamentals Wide Format Series

Introduction to Color Models

Anisotropic reflectance from turbid media. I. Theory

Neuman, Magnus 

Edström, Per 

Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.(Pappersoptik och färg)

2010 (English)In: Journal of the Optical Society of America A, ISSN 0740-3232, Vol. 27, no 5, p. 1032-1039

Anisotropic reflectance from turbid media. II. Measurements 

Magnus Neuman and Per Edström

  • Journal of the Optical Society of America A
  • Vol. 27,
  • Issue 5,
  • pp. 1040-1045
  • (2010)


Angular dependence of fluorescence from turbid media

Ludovic G. Coppel,1,∗ Niklas Johansson,and Magnus Neuman2


Limitations in the efficiency of fluorescent whitening agents in uncoated paper

Ludovic G. Coppel, Mattias Andersson, Per Edström and Jussi Kinnunen

Fluorescence model for multi-layer papers using conventional spectrophotometers

 L. G. Coppel, M. Andersson, M. Neuman and P. Edström

Nordic Pulp & Paper Research Journal | Volume 27: Issue 2

Whiteness Assessment: A Primer Concepts, Determination and Control of Perceived Whiteness

September 2006

Claudio Puebla



Dr. Tarja Shakespeare1, Dr. John Shakespeare2



Tarja Shakespeare, John Shakespeare 

US Patent

A fluorescent extension to the Kubelka–Munk model

Tarja Shakespeare

John Shakespeare


Radiative properties of optically thick fluorescent turbid media

Alexander A Kokhanovsky 1


A New Look at Fundamentals of the Photometric Light Transport and Scattering Theory. Part 2: One-Dimensional Scattering with Absorption

Authors: Persheyev S.Rogatkin D.A.Published: 22.11.2017 
Published in issue: #6(75)/2017 
DOI: 10.18698/1812-3368-2017-6-65-78


Spectral prediction model for color prints on paper with fluorescent additives.

Hersch RD1

Applied Optics, 30 Nov 2008, 47(36):6710-6722


Relationship between the Kubelka-Munk scattering and radiative transfer coefficients

Suresh N Thennadil 1


Effect of strong absorption on the Kubelka-Munk scattering coefficient

A. KoukoulasB. Jordan

Published 1997


A note concerning the interaction between light scattering and light absorption in the application of the Kubelka-Munk equations

Mats RundlöfJ. A. Bristow

Published 1997


Color Measurements on Prints Containing Fluorescent Whitening Agents

Mattias Andersson and Ole Norberg

Digital Printing Center, Mid Sweden University, 89118 Örnsköldsvik, Sweden


Colorant modelling for on-line paper coloring: evaluations of models and an extension to Kubelka-Munk model

Shakespeare, T. (2000)

Tampere University of Technology


Fluorescent White Dyes: Calculation of Fluorescence from Reflectivity Values 

Eugene Allen

1964 OSAJ


Extension of the Kubelka–Munk theory for fluorescent turbid media to a nonopaque layer on a background

Article in Journal of the Optical Society of America A · July 2011


Tutorial on Fluorescence and Fluorescent Instrumentation

Colour measurement in practice 

Contemporary wool dyeing and finishing

Dr Rex Brady Deakin University

Separation of the Spectral Radiance Factor Curve of Fluorescent Substances into Reflected and Fluoresced Components 

Eugene Allen

  • Applied Optics
  • Vol. 12,
  • Issue 2,
  • pp. 289-293
  • (1973)


Fluorescence and kubelka‐munk theory

James S. Bonham

First published: Autumn (Fall) 1986

Color Research and Application J


Spectrophotometry of fluorescent pigments

R Donaldson1


British Journal of Applied PhysicsVolume 5Number 6



He Guoxin (Department of Textile Technology)



Problems in colour measurement of fluorescent paper grades

Tarja Shakespeare John Shakespeare11

Analytica Chimica Acta

Volume 380, Issues 2–3, 2 February 1999, Pages 227-242


Spectral Colour Prediction Model for a Transparent Fluorescent Ink on Paper*

Patrick Emmel, Roger David Hersch

Laboratoire de Systèmes Périphériques

Ecole Polytechnique Fédérale de Lausanne (EPFL),

The extended Kubelka–Munk theory and its application to spectroscopy



The color prediction model of fluorescent prints

Na DongYixin ZhangGuoyun Shi

Author Affiliations +Proceedings Volume 7241, Color Imaging XIV: Displaying, Processing, Hardcopy, and Applications;72411N (2009) 
Event: IS&T/SPIE Electronic Imaging, 2009, San Jose, California, United States



He Shi,a Hongbin Liu,a,b,* Yonghao Ni,a,c Zhirun Yuan,d Xuejun Zou,d and Yajun Zhou

The Kubelka-Munk Theory for Color Image Invariant Properties

Jan-Mark Geusebroek, Theo Gevers, Arnold W.M. Smeulders Intelligent Sensory Information Systems, University of Amsterdam

Kruislaan 403, 1098 SJ Amsterdam, The Netherlands

Determination of quantum efficiency in fluorescing turbid media 

Ludovic Gustafsson Coppel, Mattias Andersson, and Per Edström

Applied OpticsVol. 50,Issue 17,pp. 2784-2792(2011)


4.2 Colour Science


Quantification of the Intrinsic Error of the Kubelka–Munk Model Caused by Strong Light Absorption


Effect of Moisture on Paper Color


page 85

Basic equations used in computer color matching, II. Tristimulus match, two-constant theory 

Eugene Allen

Journal of the Optical Society of AmericaVol. 64,Issue 7,pp. 991-993(1974)

Spectrophotometric color formulation based on two-constant Kubelka-Munk theory

Eric Walowit

(1985). Thesis. Rochester Institute of Technology

Basic Equations Used in Computer Color Matching 

Eugene Allen

Journal of the Optical Society of AmericaVol. 56,Issue 9,pp. 1256-1259(1966)


Computer-Aided Color Formulation (How to Formulate Color)

Posted March 02, 2017 by Mike Huda



COMIC: An Analog Computer in the Colorant Industry

July-Sept. 2014, pp. 4-18, vol. 36



An investigation of the optical scattering and absorption coefficients of dyed handsheets and the application of the ICI system of color specification to these handsheets

Foote, William J. (William John)

1938 PhD Thesis IPC




FEBRUARY, 1987 IPC Technical Paper 223

Mathematical Modelling of
Light Scattering in Paper and Print

Per Edström

PhD Thesis Mid Sweden University Sweden 2004

A Comparison Between the Coefficients of the Kubelka-Munk and DORT2002 Models

Per Edström
Mid Sweden University 2003

Simulation and modeling of light scattering in paper and print applications

Edström P. (2010)

In: Kokhanovsky A. (eds) Light Scattering Reviews 5. Springer Praxis Books. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10336-0_10


Measuring and Modelling Light Scattering in Paper

Niklas Johansson

Department of Natural Sciences Mid Sweden University

Doctoral Thesis No. 224 O ̈ rnsko ̈ ldsvik, Sweden 2015

Does the photon-diffusion coefficient depend on absorption?

T. Durduran and A. G. Yodh

B. Chance

D. A. Boas

J. Opt. Soc. Am. A/Vol. 14, No. 12/December 1997

Seyyed Hossein Nasr: On Islamic Philosophy

Seyyed Hossein Nasr: On Islamic Philosophy

Key Terms and People

  • Perennial Philosophy
  • Sanatan Dharma
  • Traditional Studies
  • Islamic Studies
  • Islamic Philosophy
  • Islamic Cosmology
  • Frithjof Schuon
  • Titus Burckhardt
  • Marco Pallis
  • Martin Lings
  • Rene Guenon
  • Ananda K. Coomaraswamy
  • Seyyed Hossein Nasr


I first came to know of Professor Nasr through my interactions with Ernest G. McClain, a famed Archeo-musicologist.

Ernest has written several books and numerous papers on musical structures in world mythology and religious texts.

His three books are available for anyone to download from this link below.


Ernest used to live in Washington DC area and had friendship with Professor Nasr.

Professor Nasr has written many books on Islam, Islamic Science, Philosophy, and Cosmologies which are available to download through the Internet Archive.  See the link below.


I must admit that I have not yet read books by Prof. Nasr but they are on my reading list.

Professor Nasr uses perspective of perennial philosophy (sophia perennis) to guide his views on comparing world religions.  In Hinduism, the perennial philosophy is known as Sanatan Dharma, the eternal immutable law.

Please see the link below to access works published by the Perennial Philosophy school:


I share below a biographical essay available from the website of his foundation.



About Seyyed Hossein Nasr

Seyyed Hossein Nasr, currently University Professor of Islamic Studies at the George Washington University, Washington D.C. is one of the most important and foremost scholars of Islamic, Religious and Comparative Studies in the world today. Author of over fifty books and five hundred articles which have been translated into several major Islamic, European and Asian languages, Professor Nasr is a well known and highly respected intellectual figure both in the West and the Islamic world. An eloquent speaker with a charismatic presence, Nasr is a much sought after speaker at academic conferences and seminars, university and public lectures and also radio and television programs in his area of expertise. Possessor of an impressive academic and intellectual record, his career as a teacher and scholar spans over four decades.

Born in 1933, Professor Nasr began his illustrious teaching career in 1955 when he was still a young and promising, doctoral student at Harvard University. Over the years, he has taught and trained an innumerable number of students who have come from the different parts of the world, and many of whom have become important and prominent scholars in their fields of study.

He has trained different generations of students over the years since 1958 when he was a professor at Tehran University and then, in America since the Iranian revolution in 1979, specifically at Temple University in Philadelphia from 1979 to 1984 and at the George Washington University since 1984 to the present day. The range of subjects and areas of study which Professor Nasr has involved and engaged himself with in his academic career and intellectual life are immense. As demonstrated by his numerous writings, lectures and speeches, Professor Nasr speaks and writes with great authority on a wide variety of subjects, ranging from philosophy to religion to spirituality, to music and art and architecture, to science and literature, to civilizational dialogues and the natural environment.

For Professor Seyyed Hossein Nasr, the quest for knowledge, specifically knowledge which enables man to understand the true nature of things and which furthermore, “liberates and delivers him from the fetters and limitations of earthly existence,” has been and continues to be the central concern and determinant of his intellectual life.

Brief Biography
Seyyed Hossein Nasr was born on April 7, 1933 (19 Farvadin 1312 A.H. solar) in Tehran into a family of distinguished scholars and physicians. His father, Seyyed Valiallah, a man of great learning and piety, was a physician to the Iranian royal family, as was his father before him. The name “Nasr” which means “victory” was conferred on Professor Nasr’s grandfather by the King of Persia. Nasr also comes from a family of Sufis. One of his ancestors was Mulla Seyyed Muhammad Taqi Poshtmashhad, who was a famous saint of Kashan, and his mausoleum which is located next to the tomb of the Safavid king Shah Abbas, is still visited by pilgrims to this day.

As a young boy, Nasr attended one of the schools near his home. His early formal education included the usual Persian curriculum at school with an extra concentration in Islamic and Persian subjects at home, as well as tutorial in French. However for Nasr, it was the long hours of discussion with his father, mostly on philosophical and theological issues, complemented by both reading and reaction to the discourses carried on by those who came to his father’s house, that constituted an essential aspect of his early education and which in many ways set the pattern and tone of his intellectual development. This was the situation for the first twelve years of Nasr’s life.

Nasr’s arrival in America at the young age of twelve marked the beginning of a new period in his life which was totally different and therefore, discontinuous from his early life in Iran. He attended The Peddie School in Highstown, New Jersey and in 1950 graduated as the valedictorian of his class and also winner of the Wyclifte Award which was the school’s highest honor given to the most outstanding all-round student. It was during the four years at Peddie that Nasr acquired his knowledge of the English language, as well as studying the sciences, American history, Western culture and Christianity.

Nasr chose to go to M.I.T. for college. He was offered a scholarship and was the first Iranian student to be admitted as an undergraduate at M.I.T. He began his studies at M.I.T in the Physics Department with some of the most gifted students in the country and outstanding professors of physics. His decision to study physics was motivated by the desire to gain knowledge of the nature of things, at least at the level of physical reality. However, at the end of his freshman year, although he was the top student in his class, he began to feel oppressed by the overbearingly scientific atmosphere with its implicit positivism.

Furthermore, he discovered that many of the metaphysical questions which he had been concerned with were not being asked, much less answered. Thus, he began to have serious doubts as to whether physics would lead him to an understanding of the nature of physical reality. His doubt was confirmed when the leading British philosopher, Bertrand Russell, in a small group discussion with the students following a lecture he had given at M.I.T, stated that physics does not concern itself with the nature of physical reality per se but with mathematical structures related to pointer readings.

The shock of discovering the real nature of the subject he had chosen to study, together with the overbearingly scientific atmosphere at his Department, led Nasr to experience a major intellectual and spiritual crisis during his second year. Although the crisis did not destroy his belief in God, it shook certain fundamental elements in his worldview, such as his understanding of the meaning of life, the significance of knowledge and the means to find the Truth. He was prepared to leave the field of physics and M.I.T. and depart from America in quest of the Truth. However, the strong discipline in him, inculcated by his father, prevented him from abandoning his studies altogether. He remained at M.I.T. and graduated with honors, but his heart was no longer with physics.

Having realized in his second year that a study of the physical sciences would neither lead him to an understanding of the nature of physical reality nor deal with some of the metaphysical questions he was concerned with, Nasr decided to look at other fields of study for his answers. He started to read extensively and to take many courses in the humanities, especially those taught by Professor Giorgio Di Santillana, the famous Italian philosopher and historian of science. Under Professor Di Santillana’s instruction, Nasr began his serious study of not only the ancient Greek wisdom as contained in the philosophies of Pythagoras, Plato, Aristotle and Plotinus but also European, Medieval philosophy, Dante’s highly mystical and symbolic Divine Comedy, Hinduism and a critique of modern Western thought. It was also Di Santillana who first introduced him to the writings of one of the most important traditionalist writers of this century, Rene Guenon. Guenon’s writings played a decisive role in laying the intellectual foundation of Nasr’s traditionalist perspective. Nasr also had the great fortune of having access to the library of the late Ananda K. Coomaraswamy, the outstanding Singhalese metaphysician and historian of art. The library had an incredible collection of works on traditional philosophy and art from all over the world. It was in this library that Nasr first discovered the works of the other traditionalist writers such as Frithjof Schuon, Titus Burckhardt, Marco Pallis and Martin Lings and who were to have tremendous and enduring intellectual and spiritual influence on Nasr.

According to Nasr, it was the discovery of traditional metaphysics and the philosophia perennis through the works of these figures which settled the crisis he had experienced and gained an intellectual certitude which has never left him since. From then on, he was certain that there was such a thing as the Truth and that it could be attained through knowledge by means of the intellect which is guided and illuminated by divine revelation. His childhood love for the attainment of knowledge returned to him but on a higher and deeper plane. The traditional writings of Schuon with their singular emphasis on the need for the practice of a spiritual discipline as well as theoretical knowledge, were especially instrumental in determining the course of Nasr’s intellectual and spiritual life from that time onward.

Upon his graduation from M.I.T., Nasr enrolled himself in a graduate program in geology and geophysics at Harvard University. After obtaining his Master’s degree in geology and geophysics in 1956, he went on to pursue his Ph.D. degree in the history of science and learning at Harvard. Nasr wanted to study other types of sciences of nature apart from the modern Western and also to understand why modern science had developed as it had. He planned to write his dissertation under the supervision of George Sarton, a great authority on Islamic science. However, Sarton passed away before he could begin his dissertation work and since there was not another specialist in Islamic science at Harvard then, he wrote his dissertation under the direction of three professors. They were I. Bernard Cohen, Hamilton Gibb and Harry Wolfson.

It was also at Harvard that Nasr resumed his study of classical Arabic which he had left since coming to America. He struggled with philosophical Arabic while getting some assistance from Wolfson and Gibb. However, the mastery of philosophical Arabic was only attained after he studied Islamic philosophy from the traditional masters of Iran after his return to his homeland in 1958.

During his Harvard years, Nasr also traveled to Europe, especially to France, Switzerland, Britain, Italy and Spain, widening his intellectual horizon and establishing important and fruitful contacts. It was during these travels to Europe that Nasr met with the foremost traditionalist writers and exponents of the philosophia perennis, Frithjof Schuon and Titus Burckhardt, who made a tremendous impact and decisive contribution to his intellectual and spiritual life. He also traveled to Morocco in North Africa, which had great spiritual significance for Nasr who embraced Sufism in the form taught and practiced by the great Sufi saint of the Maghrib, Shaykh Ahmad al-Alawi. Thus, the years at Harvard witnessed the crystallization of the major intellectual and spiritual elements of Nasr’s mature worldview, elements which have since dominated and determined the course and pattern of his scholarship and academic career.

At twenty-five, Nasr graduated with a Ph.D. degree from Harvard and on the way to completing his first book, Science and Civilization in Islam. His doctoral dissertation entitled “Conceptions of Nature in Islamic Thought” was published in 1964 by Harvard University Press as An Introduction to Islamic Cosmological Doctrines. Although he was offered a position as assistant professor at M.I.T., Nasr decided to return permanently to Iran.

Back in Iran, Nasr was offered a position as an Associate Professor of philosophy and the history of science at the Faculty of Letters in Tehran University. A few months after his return, Nasr married a young woman from a respected family whose members were close friends of his family. Five years later at the age of thirty, Nasr became the youngest person to become a full professor at the University. He used his position and influence to bring major changes to strengthen and expand the philosophy program at Tehran University which like many of its other programs, was very much dominated by and limited to French intellectual influence. Nasr initiated the important move of teaching Islamic philosophy on the basis of its own history and from its own perspective and to encourage his Iranian students to study other philosophies and intellectual traditions from the point of view of their own tradition. He maintains that one cannot hope to understand and appreciate one’s own intellectual tradition from the viewpoint of another, just as one cannot see oneself through the eyes of another person. He also created greater awareness and interest in the study of Oriental philosophies among the students and faculty members. Since Tehran University was the only university in Iran to offer a doctorate in philosophy, these changes introduced by Nasr had far reaching influence. Many universities in Iran integrated these changes into their philosophical studies and until today Nasr’s perspective that Iranian students should study other philosophical traditions from the view of their own tradition instead of studying their tradition from the perspective of Western thought and philosophy remains widely influential. The students he has trained and who have become scholars and university professors of philosophy have enabled this perspective to have enduring influence in Iran.

Apart from the philosophy program, Nasr was also involved in the university’s doctoral program in Persian language and literature for those whose mother tongue was not Persian. He strengthened the philosophical component of this program and had many outstanding students from outside of Iran to receive training, not only in Persian language, but also the rich treasury of philosophical and Sufi literature written in Persian. Many of the students trained in this program have since become important scholars in this field such as the American scholar, William Chittick and the Japanese woman scholar, Sachiko Murata.

Furthermore, from 1968 to 1972, Nasr was made Dean of the Faculty and for a while, Academic Vice-Chancellor of Tehran University. Through these positions, he introduced many important changes which all aimed at strengthening the university programs in the humanities generally and in philosophy, specifically. In 1972, he was appointed President of Aryamehr University by the Shah of Iran. Aryamehr University was then the leading scientific and technical university in Iran and the Shah, as the patron, wanted Professor Nasr to develop the university on the model of M.I.T. but with firm roots in Iranian culture. Consequently, a strong humanities program in Islamic thought and culture, with a particular emphasis upon an Islamic philosophy of science, was established at Aryamehr University by Nasr. Nasr’s pioneering effort has led Aryamehr to create one of the first graduate programs in the Islamic world in the philosophy of science based upon the Islamic philosophy of science, some ten years ago. In 1973, the Queen of Iran appointed Professor Nasr to establish a center for the study and propagation of philosophy under her patronage. Hence, the Imperial Iranian Academy of Philosophy was established and very soon became one of the most important and vital centers of philosophical activities in the Islamic world, housing the best library of philosophy in Iran and attracting some of the most distinguished scholars in the field, both from the East and the West, such as Henry Corbin and Toshihiko Izutsu. The Academy also organized important seminars and lecture series given by philosophers, offered fellowships for short and long term research work in Islamic philosophy, and comparative philosophy and undertook a major publication program of works in this field in Persian, Arabic, English and French.

Another very important dimension to Nasr’s intellectual activities after his return to Iran in 1958, was his program in re-educating himself in Islamic philosophy by learning it at the feet of the masters through the traditional method of oral transmission. He studied hikmah for twenty years under some of the greatest teachers in Iran at the time, reading traditional texts of Islamic philosophy and gnosis, three days a week at the Sepahsalar madrasah in Tehran and also in private homes in Tehran, Qom and Qazwin. Among his venerable teachers were Sayyid Muhammad Kazim Assar, an alim who was an authority on Islamic law, as well as philosophy, and a very close friend of Professor Nasr’s father; the great luminary and master of gnosis, Allamah Sayyid Muhammad Husayn Tabatabai and Sayyid Abul-Hasan Qazwini, a great authority on Islamic law and the intellectual sciences who knew mathematics, astronomy and philosophy extremely well. Nasr read and studied several of the major texts of Islamic philosophy under these masters such as the al-Asfar al-arbaah of Mulla Sadra and the Sharh-i manumah of Sabziwari and benefited greatly from the invaluable insights and commentaries provided by them orally. In this way, Nasr had the best educational training both from the modern West and the traditional East, a rare combination which put him in a very special position to speak and write with authority on the numerous issues involved in the encounter between East and West, and tradition and modernity, as demonstrated very clearly by his writings and lectures.

During the years Professor Nasr was in Iran, he wrote extensively in Persian and English and occasionally in French and Arabic. His doctoral dissertation was rewritten by him in Persian and it won the royal book award. Nasr also brought out the critical editions of several important philosophical texts such as the complete Persian works of Suhrawardi and of Mulla Sadra and the Arabic texts of lbn Sina and al-Biruni. Nasr’s great interest in the philosophy of one of the greatest later Islamic philosophers, Mulla Sadra resulted in the publication of the Mulla Sadra written by the traditional masters of Islamic philosophy. Nasr was also the first person to introduce the figure of Mulla Sadra to the English speaking world.

With the assistance of William Chittick, Nasr prepared An Annotated Bibliography of Islamic Science in three volumes, with Persian and English annotations. He also wrote Three Muslim Sages and completed and published Science and Civilization in Islam which he had written while still a student at Harvard. Both of these books were translated into several languages very quickly and were reprinted in Iran many times and have been used for the past three decades as textbooks for courses in Islamic philosophy and science in Iranian universities. Three Muslim Sages, which presents the whole of the Islamic intellectual tradition from within, grew out of three lectures which Nasr gave in 1962 as the first visiting professor at the Center for the Study of World Religions at Harvard University. Ideals and Realities of Islam, which is one of Nasr’s most widely read book on the Islamic religion and which opens up the world of Islam, revealing some of its most universal and profound dimensions, was based on the text of the first six of fifteen lectures which he delivered at the American University in Beirut as the first Aga Khan Professor of Islamic studies in 1964-65.

In 1966 Nasr was invited to deliver the Rockefeller Lectures at the University of Chicago and to speak on some aspects of the relation between religion, philosophy and the environmental crisis. Consequently, Man and Nature: The Spiritual Crisis of Modern Man, which deals with the philosophical and spiritual roots of the question and the first work to predict the coming of the environmental crisis was written for the occasion. Nasr also brought out Islam and the Plight of Modern Man, Sufi Essays and The Transcendent Theosophy of Sadr al-Din Shirazi. Both Islam and the Plight of Modern Man and Sufi Essays have proved to be very popular and have been translated into many European and Islamic languages and reprinted several times since their first appearance.

In 1964-65, Nasr spent an academic year at the American University of Beirut as the first Aga Khan professor of Islamic Studies. Besides Ideals and Realities of Islam, Nasr also brought out Islamic Studies, which is a collection of articles discussing several fundamental aspects of the Islamic tradition. This work was later expanded and published under the title, Islamic Life and Thought. During this period in Lebanon, Nasr also met with and had intellectual discourses with several important Catholic and Shi`ite thinkers and scholars. He also had the opportunity to meet with the woman Sufi saint Sayyidah Fatimah Yashrutiyah, daughter of the founder of the Yashrutiyah order, a branch of the Shadhiliyah Sufi order.

Although Nasr lived in Iran, he maintained strong contacts with America and many of the major universities in the country. He taught at Harvard in 1962 and 65 and conducted short seminars at Princeton University and the University of Utah. He also had close associations with several important American scholars such as Huston Smith, professor of philosophy and comparative religion, Jacob Needleman, editor of the well-known work, Sword of Gnosis which includes Nasr’s essays, and a number of Catholic and Protestant philosophers and theologians. Nasr also helped with the planning and expansion of Islamic and Iranian studies in several universities such as Princeton, the University of Utah and the University of Southern California. In 1977, he delivered the Kevorkian Lectures on Islamic art at New York University on the meaning and philosophy of Islamic art.

In 1979 at the time of the Islamic Revolution in Iran, Nasr moved with his family to the United States where he would rebuild his life again and secure a university position to support himself and his family. By 1980, Nasr began to write again. He started to work intensively on the research and text of the prestigious Gifford Lectures at the University of Edinburgh to which he received an invitation shortly before the Iranian Revolution took place. Nasr had the honor of being the first non-Westerner to be invited to deliver the most famous lecture series in the fields of natural theology and philosophy of religion in the West. Thus, Knowledge and the Sacred, one of Nasr’s most important philosophical works, one which had a great impact on scholars and students of religious studies, came to be prepared amidst the strain of trying times and the strenuous commute between Boston and Philadelphia. However, Nasr discloses that the actual writing of the text of Knowledge and the Sacred came as a gift from heaven. He was able to write the texts of the lectures with great facility and speed and within a period of less than three months, they were completed. Nasr says that it was as though, he was writing from a text he had previously memorized.

In 1982, Nasr was invited to collaborate on a major project to bring out the Encyclopedia of World Spirituality together with Ewert Cousins, chief editor and professor of Medieval philosophy at Fordham University, and many other leading philosophers and scholars of religion. Nasr accepted to edit the two volumes on Islamic Spirituality, which came out in 1989 and 1991. Both volumes have since become invaluable reference material in English for those interested in this subject. In 1983, Nasr delivered the Wiegand Lecture on the philosophy of religion at the University of Toronto in Canada. He also helped in the establishment of the section on Hermeticism and perennial philosophy at the American Academy of Religion.

Nasr was soon recognized in American academic circles as a traditionalist and a major expositor and advocate of the perennialist perspective. Much of his intellectual activities and writing since being in exile in America, are related to this function and also in the fields of comparative religion, philosophy and religious dialogue. He has participated in many debates and discussions with eminent Christian and Jewish theologians and philosophers such as Hans Kung, John Hick and Rabbi Izmar Schorch. In 1986, Nasr edited The Essential Writings of Frithjof Schuon and in 1990, he was selected as a patron of the Center for the Study of Islam and Christian-Muslim relations of the Sally Oaks College in Birmingham. In addition, he has played an active role in the creation and activities of the Center for Muslim-Christian Understanding at Georgetown University in Washington D.C. He has also attended many conferences on this subject including the famous 1993 Parliament of World Religions.

He continues to travel to Europe often, giving lectures and being involved with intellectual activities. He gives lectures at Oxford, University of London and a few other British universities and is a member of the Temenos Academy. In 1994, he was invited to deliver the Cadbury Lectures at the University of Birmingham and a major work entitled Religion and the Order of Nature was produced by Nasr for this occasion.

Nasr also continues to travel to Spain, especially southern Spain which still has an Islamic presence and which reminds him very much of his home country, Iran. It was also during some of his journeys to Spain, that Nasr was inspired to compose several poems related to Spanish themes. Nasr has brought out recently a collection of forty English poems on spiritual themes, which were written within the past fifteen years, under the title Poems of the Way.

Although Professor Nasr continues to have a very busy teaching and lecturing schedule, he still manages to allocate much of his time and energy to writing. 1987 saw the publication of two of his books: Islamic Art and Spirituality and Traditional Islam in the Modern World. Islamic Art and Spirituality which deals with the metaphysical and symbolic significance of Islamic art, poetry and music is Nasr’s first book on this subject. Traditional Islam in the Modern World discusses several important dimensions of the Islamic tradition and its relation to the West. Nasr also wrote a book specifically for young Muslims entitled, A Young Muslim’s Guide to the Modern World which addresses some of the major problems and challenges which the modern world presents to them.

Recently, Nasr together with the British scholar of Islamic and Jewish philosophy, Oliver Leaman, edited a two volume work, History of Islamic Philosophy which consists of articles written by important scholars in this field, discussing the different aspects and schools of Islamic philosophy and its development in the different parts of the Islamic world. Nasr’s continued interest in science is made evident by his latest book on this subject, The Need for a Sacred Science. Also, together with one of his former students, Mehdi Amin Razavi, Nasr is now bringing out a major four volume work, An Anthology of Philosophy in Persia which will be published by Oxford University Press. Razavi also edited earlier, The Islamic Intellectual Tradition in Persia, which is a collection of Nasr’s articles on Islamic philosophy in Persia written during the last forty years.

Another important aspect to Nasr’s intellectual activities in Washington D.C. is his active involvement in the activities of the Foundation for Traditional Studies. The Foundation which is devoted to the dissemination of traditional thought was established in 1984 under the direction of a board presided by Nasr. The Foundation has published several books including the festschrift of Frithjof Schuon entitled, Religion of the Heart, edited by Nasr and William Stoddart and In Quest of the Sacred: The Modern World in the Light of Tradition which Nasr co-edited with the executive director of the Foundation, Katherine O’Brien. In Quest of the Sacred is a collection of essays presented by some of the major traditionalist writers in an important conference held in Peru, organized by the Foundation and the Peruvian Instituto de Estudios Tradicionales. The Foundation also publishes the journal, “Sophia,” which carries essays on traditional thought written by the leading authorities in this field. Together with the Foundation, Nasr is also involved in the production of a major documentary television series on “Islam and the West,” which deals with some of the more important and profound aspects of the encounter between the Islamic and Western civilizations.

At sixty-six, Seyyed Hossein Nasr leads an extremely active intellectual life with a very busy schedule of teaching at the university and lecturing at many institutions in America and around the world, writing scholarly works, being involved in several intellectual projects simultaneously and meeting individuals who are interested in traditional thought. At the same time, he leads a very intense spiritual life spent in prayer, meditation and contemplation and also providing spiritual counsel for those who seek his advice and guidance. Exiled from his homeland, Seyyed Hossein Nasr has found his home in the inviolable and sacred Center which is neither in the East nor the West.


Please see my related posts:

Myth of Invariance: Sound, Music, and Recurrent Events and Structures

Integral Philosophy of the Rg Veda: Four Dimensional Man



Sources of Research

Frithjof Schoun





Sophia Perennis




Ananda K Coomaraswamy





Rene Guenon




Titus Burckhardt




Recursion, Incursion, and Hyper-incursion

Recursion, Incursion, and Hyper-incursion


How do Past and Future inform the present?

What happens in the Present is not only determined by the Past but also by the Future.  Karma and Destiny both play a role as to what is going on in your life Now.

Key Terms

  • Recursion
  • Incursion
  • Hyper Incursion
  • Discrete Processes
  • Cellular Automata
  • Fractal Machine
  • Hypersets
  • Interpenetration
  • Turing Machine
  • Symmetry
  • Non Well Founded Set Theory
  • Sets as Graphs
  • Leela
  • Predetermined Future
  • Bhagya
  • Fate
  • Destiny
  • Karma
  • Anticipation
  • Four Causes of Aristotle
  • Material Cause
  • Formal Cause
  • Efficient Cause
  • Final Cause
  • Left Computer
  • Right Computer
  • Parallel Computing
  • Fifth and the Fourth in Music Theory
  • Bicameral Brain
  • Hemispheric Division of Brain
  • One, Two, Three.  Where is the Fourth?


The recursion consists of the computation of the future value of the variable vector X(t+l) at time t+l from the values of these variables at present and/or past times, t, t-l, t-2 ….by a recursive function :

X (t+ 1) =f(X(t), X(t-1) …p..)

where p is a command parameter vector. So, the past always determines the future, the present being the separation line between the past and the future.

Starting from cellular automata, the concept of Fractal Machines was proposed in which composition rules were propagated along paths in the machine frame. The computation is based on what I called “INclusive reCURSION”, i.e. INCURSION (Dubois, 1992a- b). An incursive relation is defined by:

X(t+l) =f(…, X (t+l), X(t), X(t-1) ..p..).

which consists in the computation of the values of the vector X(t+l) at time t+l from the values X(t-i) at time t-i, i=1, 2 …. , the value X(t) at time t and the value X(t+j) at time t+j, j=l, 2, …. in function of a command vector p. This incursive relation is not trivial because future values of the variable vector at time steps t+l, t+2 …. must be known to compute them at the time step t+ 1.

In a similar way to that in which we define hyper recursion when each recursive step generates multiple solutions, I define HYPERINCURSION. Recursive computational transformations of such incursive relations are given in Dubois and Resconi (1992, 1993a-b).

I have decided to do this for three reasons. First, in relativity theory space and time are considered as a four-vector where time plays a role similar to space. If time t is replaced by space s in the above definition of incursion, we obtain

X(s+ l) =f( …, X(s+ 1), X(s), X (s-l) …p.).

and nobody is astonished: a Laplacean operator looks like this. Second, in control theory, the engineers control engineering systems by defining goals in the future to compute their present state, similarly to our haman anticipative behaviour (Dubois, 1996a-b). Third, I wanted to try to do a generalisation of the recursive and sequential Turing Machine in looking at space-time cellular automata where the order in which the computations are made is taken into account with an inclusive recursion.

We have already proposed some methods to realise the design of any discrete systems with an extension of the recursion by the concept of incursion and hyperincursion based on the Fractal Machine, a new type of Cellular Automata, where time plays a central role. In this framework, the design of the model of any discrete system is based on incursion relations where past, present and future states variables are mixed in such a way that they define an indivisible wholeness invariant. Most incursive relations can be transformed in different sets of recursive algorithms for computation. In the same way, the hyperincursion is an extension of the hyper recursion in which several different solutions can be generated at each time step. By the hyperincursion, the Fractal Machine could compute beyond the theoretical limits of the Turing Machine (Dubois and Resconi, 1993a-b). Holistic properties of the hyperincursion are related to the Golden Ratio with the Fibonacci Series and the Fractal Golden Matrix (Dubois and Resconi, 1992). An incursive method was developed for the inverse problem, the Newton- Raphson method and an application in robotics (Dubois and Resconi, 1995). Control by incursion was applied to feedback systems (Dubois and Resconi, 1994). Chaotic recursions can be synchronised by incursion (1993b). An incursive control of linear, non- linear and chaotic systems was proposed (Dubois, 1995a, Dubois and Resconi, 1994, 1995). The hyperincursive discrete Lotka-Voiterra equations have orbital stability and show the emergence of chaos (Dubois, 1992). By linearisation of this non-linear system, hyperincursive discrete harmonic oscillator equations give stable oscillations and discrete solutions (Dubois, 1995). A general theory of stability by incursion of discrete equations systems was developed with applications to the control of the numerical instabilities of the difference equations of the Lotka-Volterra differential equations as well as the control of the fractal chaos in the Pearl-Verhulst equation (Dubois and Resconi, 1995). The incursion harmonic oscillator shows eigenvalues and wave packet like in quantum mechanics. Backward and forward velocities are defined in this incursion harmonic oscillator. A connection is made between incursion and relativity as well as the electromagnetic field. The foundation of a hyperincursive discrete mechanics was proposed in relation to the quantum mechanics (Dubois and Resconi, 1993b, 1995).

This paper will present new developments and will show that the incursion and hyper-incursion could be a new tool of research and development for describing systems where the present state of such systems is also a function of their future states. The anticipatory property of incursion is an incremental final cause which could be related to the Aristotelian Final Cause.

Aristotle identified four explicit categories of causation: 1. Material cause; 2. Formal cause; 3. Efficient cause; 4. Final cause. Classically, it is considered that modem physics and mechanics only deal with efficient cause and biology with material cause. Robert Rosen (1986) gives another interpretation and asks why a certain Newtonian mechanical system is in the state (phase) Ix(t) (position), v(t) (velocity)]:

1. Aristotle’s “material cause” corresponds to the initial conditions of the system [x(0), v(0)] at time t=0.

2. The current cause at the present time is the set of constraints which convey to the system an “identity”, allowing it to go by recursion from the given initial phase to the latter phase, which corresponds to what Aristotle called formal cause.

3. What we call inputs or boundary conditions are the impressed forces by the environment, called efficient cause by Aristotle.

As pointed out by Robert Rosen, the first three of Aristotle’s causal categories are tacit in the Newtonian formalism: “the introduction of a notion of final cause into the Newtonian picture would amount to allowing a future state or future environment to affect change of state in the present, and this would be incompatible with the whole Newtonian picture. This is one of the main reasons that the concept of Aristotelian finality is considered incompatible with modern science.

In modern physics, Aristotelian ideas of causality are confused with determinism, which is quite different…. That is, determinism is merely a mathematical statement of functional dependence or linkage. As Russell points out, such mathematical relations, in themselves, carry no hint as to which of their variables are dependent and which are independent.”

The final cause could impress the present state of evolving systems, which seems a key phenomenon in biological systems so that the classical mathematical models are unable to explain many of these biological systems. An interesting analysis of the Final Causation was made by Emst von Glasersfeld (1990). The self-referential fractal machine shows that the hyperincursive field dealing with the final cause could be also very important in physical and computational systems. The concepts of incursion and hyper-incursion deal with an extension of the recursive processes for which future states can determine present states of evolving systems. Incursion is defined as invariant functional relations from which several recursive models with interacting variables can be constructed in terms of diverse physical structures (Dubois & Resconi, 1992, 1993b). Anticipation, viewed as an Aristotelian final cause, is of great importance to explain the dynamics of systems and the semantic information (Dubois, 1996a-b). Information is related to the meaning of data. It is important to note that what is usually called Information Theory is only a communication theory dealing with the communication of coded data in channels between a sender and a receptor without any reference to the semantic aspect of the messages. The meaning of the message can only be understood by the receiver if he has the same cultural reference as the sender of the message and even in this case, nobody can be sure that the receiver understands the message exactly as the sender. Because the message is only a sequential explanation of a non-communicable meaning of an idea in the mind of the sender which can be communicated to the receiver so that a certain meaning emerges in his mind. The meaning is relative or subjective in the sense that it depends on the experiential life or imagination of each of us. It is well- known that the semantic information of signs (like the coding of the signals for traffic) are the same for everybody (like having to stop at the red light at a cross roads) due to a collective agreement of their meaning in relation to actions. But the semantic information of an idea, for example, is more difficult to codify. This is perhaps the origin of creativity for which a meaning of something new emerges from a trial to find a meaning for something which has no a priori meaning or a void meaning.

Mind dynamics seems to be a parallel process and the way we express ideas by language is sequential. Is the sequential information the same as the parallel information? Let us explain this by considering the atoms or molecules in a liquid. We can calculate the average velocity of the particles from in two ways. The first way is to consider one particular particle and to measure its velocity during a certain time. One obtains its mean velocity which corresponds to the mean velocity of any particle of the liquid. The sec- ond way is to consider a certain number of particles at a given time and to measure the velocity of each of them. This mean velocity is equal to the first mean velocity. So there are two ways to obtain the same information. One by looking at one particular element along the time dimension and the other by looking at many elements at the same time. For me, explanation corresponds to the sequential measure and understanding to the parallel measure. Notice that ergodicity is only available with simple physical systems, so in general we can say that there are distortions between the sequential and the parallel view of any phenomenon. Perhaps the brain processes are based on ergodicity: the left hemisphere works in a sequential mode while the right hemisphere works in a parallel mode. The left brain explains while the right brain understands. The two brains arecomplementary and necessary.

Today computer science deals with the “left computer”. Fortunately, the informaticians have invented parallel computers which are based on complex multiplication of Turing Machines. It is now the time to reconsider the problem of looking at the “right computer”. Perhaps it will be an extension of the Fractal Machine (Dubois & Resconi, 1993a).

I think that the sequential way deals with the causality principle while the parallel way deals with a finality principle. There is a paradox: causality is related to the successive events in time while finality is related to a collection of events at a simultaneous time, i.e. out of time.Causality is related to recursive computations which give rise to the local generation of patterns in a synchronic way. Finality is related to incursive or hyperincursive symmetry invariance which gives rise to an indivisible wholeness, a holistic property in a diachronic way. Recursion (and Hyper recursion) is defined in the Sets Theory and Incursion (and Hyperincursion) could be defined in the new framework of the Hypersets Theory (Aczel, 1987; Barwise, Moss, 1991).

If the causality principle is rather well acknowledged, a finality principle is still controversial. It would be interesting to re-define these principles. Causality is defined for sequential events. If x(t) represents a variable at time t, a causal rule x(t+l) = f(x(t)) gives the successive states of the variable x at the successive time steps t, t+l, t+2, … from the recursive functionf(x(t)), starting with an initial state x(0) at time t=0. Defined like this, the system has no degrees of freedom: it is completely determined by the function and the initial condition. No new things can happen for such a system: the whole future is completely determined by its past. It is not an evolutionary system but a developmental system. If the system tends to a stable point, x(t+l) = x(t) and it remains in this state for ever. The variable x can represent a vector of states as a generalisation.

In the same way, I think that determinism is confused with predictability, in modern physics. The recent fractal and deterministic chaos theory (Mandeibrot, 1982; Peitgen, Jurgens, Saupe, 1992) is a step beyond classical concepts in physics. If the function is non-linear, chaotic behaviour can appear, what is called (deterministic) chaos. In this case, determinism does not give an accurate prediction of the future of the system from its initial conditions, what is called sensitivity to initial conditions. A chaotic system loses the memory of its past by finite computation. But it is important to point out that an average value, or bounds within which the variable can take its values, can be known;

it is only the precise values at the successive steps which are not predictable. The local information is unpredictable while the global symmetry is predictable. Chaos can presents a fractai geometry which shows a self-similarity of patterns at any scale.

A well-known fractal is the Sierpinski napkin. The self-similarity of pattems at any scale can be viewed as a symmetry invariance at any scale. An interesting property of such fractals is the fact that the final global pattern symmetry can be completely independent of the local pattern symmetry given as the initial condition of the process from which the fractal is built. The symmetry of the fractal structure, a final cause, can be independent of the initial conditions, a material cause. The formal cause is the local symmetry of the generator of the fractal, independently of its material elements and the efficient cause can be related to the recursive process to generate the fractal. In this particular fractal geometry, the final cause is identical to the final cause. The efficient cause is the making of the fractal and the material cause is just a substrate from which the fractal emerges but this substrate doesn’t play a role in the making.

Finally, the concepts of incursion and hyperincursion can be related to the theory of hypersets which are defined as sets containing themselves. This theory of hypersets is an alternative theory to the classical set theory which presents some problems as the in- completeness of G6del: a formal system cannot explain all about itself and some propositions cannot be demonstrated as true or false (undecidability). Fundamental entities of systems which are considered as ontological could be explain in a non-ontological way by self-referential systems.

Please see my related posts

On Anticipation: Going Beyond Forecasts and Scenarios

Autocatalysis, Autopoiesis and Relational Biology

Key sources of Research


Computing Anticipatory Systems with Incursion and Hyperincursion

Daniel M. DUBOIS


Click to access cd554835f0ae367c3d3e3fa40f3e5e5f5f11.pdf




Anticipation in Social Systems:

the Incursion and Communication of Meaning

Loet Leydesdorff 

Daniel M. Dubois

Click to access casys03.pdf





Daniel M. Dubois


Click to access dubois.pdf




Non-wellfounded Set Theory



  • Jon Barwise &
  • Larry Moss


Non-well-founded set theory


Rituals | Recursion | Mantras | Meaning : Language and Recursion

Rituals | Recursion | Mantras | Meaning : Language and Recursion


Key Terms

  • Rituals
  • Recursion
  • Mantras
  • Japa
  • Puja
  • Prayer
  • Psychological Development
  • Meaning
  • Meditation
  • Yoga
  • Vedas
  • Tantra
  • Hindu Vedic Rituals
  • Hindu Tantric Rituals
  • Language and Recursion
  • Speech
  • Communication
  • Organizing principle
  • Cognitive science
  • Linguistics
  • Panini
  • Pingala
  • Sanskrit
  • Act of walking and Talking
  • Take a walk
  • Vedic Meters
  • Gayatri Mantra
  • Grammer
  • Panini’s Asthadhyayi
  • Univarsal Grammer
  • Noam Chomsky
  • Frits Staal
  • Subhash Kak
  • Word and World
  • Structure of a Mantra
  • Structure of a Ritual
  • Prose and Poetry
  • Gadya and Padya
  • Fractals
  • Holarchy
  • Holons

Why Language and Thought Resemble Russian Dolls

Michael Corballis is a professor emeritus at the University of Auckland, who has written extensively on the evolution of language and the origins of thought.


Michael Corballis is a professor emeritus at the University of Auckland, who has written extensively on the evolution of language and the origins of thought. In his 2011 book The Recursive Mind, he wrote about how the structure of human language allows for recursion—in which ideas are nested within each other: “He thinks that I think that he thinks.” Recursion allows the construction of sentences of of theoretically unlimited complexity

The main argument of The Recursive Mind is that recursion applies to thought processes and actions that are not limited to language itself, but characterize other aspects of human thought, such as our ability to imagine ourselves in the past or future. I queried Corballis on some of these ideas. An edited transcript follows:

You wrote a book in the last few years called The Recursive Mind. What is recursion and why is it important?

Recursion can refer to a procedure that references itself or similar procedures, or to a structure that contains an embedded structure of the same type. It is important because it allows one to build entities of theoretically unlimited complexity by progressively embedding structures within structures.

Some scholars think that language may be built on the use of recursive building blocks. Isn’t that a fundamental tenet of the modern linguistics pioneered by Noam Chomsky.

Yes, Chomsky’s view of language is that it is recursive, and this gives language its potential for infinite complexity—or what he has also called “discrete infinity.” In recent formulations, this is achieved through an “unbounded Merge,” in which elements are merged, and the merged elements can then themselves be merged—a process that can be repeated without bounds. To Chomsky, though, language is essentially internal thought, a mental monologue, known as I-language, and not a means of communication. The structure of language is therefore a by-product of internal thought. This implies a common structure, called “universal grammar,” that underlies all languages. But there is growing doubt as to whether such a structure exists.

Wasn’t one of the flaws of Chomsky’s work that he thought that the recursion as found in internal self-talk was not linked in any way to natural selection and evolution

Yes, I think so. In Chomsky‘s view, the recursive principle emerged in a single step—a “great leap forward”—at some particular point in human evolution, probably within the time span of our own species. This is contrary to Darwinian evolution, which postulates that change occurs in small, incremental steps, implying that something as complex as language could not have occurred in a single step. Chomsky also argues that language could not have evolved through natural selection because the elements of language are fundamentally symbolic, with no direct reference to the external world, and could not have been “selected” by natural events.

I think these are flawed arguments. The idea of a single step is based in part on evidence of a sudden appearance of symbolic thought in the archeological record, but one can just as easily make a case for a gradual rise. Apes and dogs can fairly easily learn to respond meaningfully to spoken words, suggesting that they, like humans, have innate, mental concepts, to which words are easily attached. Even in humans, the apparently abstract nature of language might well have arisen from more iconic or even pantomimimed representations used as forms of communication that have become “conventionalized” through the generations, sustained through culture rather than biological endowment

I would argue instead that recursive thought may have origins in our ape-like forebears over 6 million years ago, but gained increasing complexity during the Pleistocene, dating from over 2 million years ago, largely as an adaptation toward increasing social complexity. Language depended on this broader recursive capacity, but also owed its earlier origins to manual gesture, perhaps developing a more complex structure through pantomime, with the iconic structure eventually replaced by vocal gestures (speech) or the more arbitrary signs of signed languages.

Hasn’t more recent work extended the idea of recursion? Isn’t it thought that recursive behaviors may actually precede language—and may have led to the emergence of language?

To some extent Chomsky’s view can be seen as consistent with this, since recursion applies to what he calls I-language, which is the language of thought rather than of communication. Communicative language can then regarded as a means of externalizing thought so that we can share our thoughts with others. It may have emerged after the “great leap forward” that gave us I-language, although Chomsky does not, as far as I know, suggest that thought and communicative language arose in sequence.

My own view is that recursive thoughts preceded language in broader ways. So-called theory of mind (“I know what you’re thinking”) or mental time travel (mentally reliving what I did yesterday, or foreseeing what I will do tomorrow) involve the embedding of thoughts within thoughts, and otherwise seem quite independent of what we understand as language. Navigation may be another example, as we embed maps within maps (my office in my house in my city in my country in the world). As I understand it, I-language is based on the structuring of internal symbols, whereas the extended examples I have given are more spatial or iconic than abstract, and therefore have a more direct relation with the external world. It is entirely plausible that they could have emerged through natural selection, which rescues language evolution from the “miracle” of a sudden great leap forward.

Since writing the book, I have moved further from the Chomskyan notion that language is uniquely human to finding the basis of mental time travel even in the ability of rats to “replay” and perhaps “preplay” trajectories in spatial environments. The aim of researchers should be to develop an account of the evolution of thought and language through natural selection, and not through some miraculous event within the past 100,000 years. .

Why are the origins of human language, considered to be one of the hardest problems in science?

The reason is that grammatical language, with its recursive structure, is considered unlike any other form of animal communication, and is restricted to our own species. Part of the difficulty is that we have no historical evidence to go on, since we are the only remaining species among the 20 or so hominins that split from the line leading to the great apes, and even the great apes closest to us do not appear to have grammatical language.

My sense is that we now have enough information from fossil evidence, primate communication, ancient DNA, and the structure of human cognition to begin to construct plausible Darwinian scenarios of how language might have evolved over the past 6 million or so years, and perhaps even earlier, without having to postulate a one-off miracle within the past 100,000.

From Art and Cosmology in India


General equivalences

The view that the arts belong to the domain of the sacred and that there is a connection between them is given most clearly in a famous passage in the Vishnudharmottara Purana in which the sage Markandeya instructs the king Vajra in the art of sculpture, teaching that to learn it one must first learn painting, dance, and music:

Vajra: How should I make the forms of gods so that the image may always manifest the deity?

Markandeya: He who does not know the canon of painting (citrasutram) can never know the canon of image-making (pratima lakshanam).

Vajra: Explain to me the canon of painting as one who knows the canon of painting knows the canon of image-making.

Markandeya: It is very difficult to know the canon of painting without the canon of dance (nritta shastra), for in both the world is to be represented.

Vajra: Explain to me the canon of dance and then you will speak about the canon of painting, for one who knows the practice of the canon of dance knows painting.

Markandeya: Dance is difficult to understand by one who is not acquainted with instrumental music (atodya).

Vajra: Speak about instrumental music and then you will speak about the canon of dance, because when the instrumental music is properly understood, one understands dance.

Markandeya: Without vocal music (gita) it is not possible to know instrumental music.

Vajra: Explain to me the canon of vocal music, because he, who knows the canon of vocal music, is the best of men who knows everything.

Markandeya: Vocal music is to be understood as subject to recitation that may be done in two ways, prose (gadya) and verse (padya). Verse is in many meters.

Please see my related posts

Sounds True: Speech, Language, and Communication

Reflexivity, Recursion, and Self Reference

Society as Communication: Social Systems Theory of Niklas Luhmann

Consciousness of Cosmos: A Fractal, Recursive, Holographic Universe

On Holons and Holarchy

Indra’s Net: On Interconnectedness

The Great Chain of Being


Key Sources of Research

God in the Fractals: Recursiveness as a Key to Religious Behavior

in Method & Theory in the Study of Religion



Eternal Recursion, the Emergence of Metaconsciousness, and the Imperative for Closure

Consciousness of Cosmos: A Fractal, Recursive, Holographic Universe

Consciousness of Cosmos: A Fractal, Recursive, Holographic Universe

From Consciousness in the Universe is Scale Invariant and Implies an Event Horizon of the Human Brain

 Our brain is not a “stand alone” information processing organ: it acts as a central part of our integral nervous system with recurrent information exchange with the entire organism and the cosmos. In this study, the brain is conceived to be embedded in a holographic structured field that interacts with resonant sensitive structures in the various cell types in our body. In order to explain earlier reported ultra-rapid brain responses and effective operation of the meta-stable neural system, a field-receptive mental workspace is proposed to be communicating with the brain. Our integral nervous system is seen as a dedicated neural transmission and multi-cavity network that, in a non-dual manner, interacts with the proposed supervening meta-cognitive domain. Among others, it is integrating discrete patterns of eigen-frequencies of photonic/solitonic waves, thereby continuously updating a time-symmetric global memory space of the individual. Its toroidal organization allows the coupling of gravitational, dark energy, zero-point energy field (ZPE) as well as earth magnetic fields energies and transmits wave information into brain tissue, that thereby is instrumental in high speed conscious and sub-conscious information processing. We propose that the supposed field-receptive workspace, in a mutual interaction with the whole nervous system, generates self-consciousness and is conceived as operating from a 4th spatial dimension (hyper-sphere). Its functional structure is adequately defined by the geometry of the torus, that is envisioned as a basic unit (operator) of space-time. The latter is instrumental in collecting the pattern of discrete soliton frequencies that provided an algorithm for coherent life processes, as earlier identified by us. It is postulated that consciousness in the entire universe arises through, scale invariant, nested toroidal coupling of various energy fields, that may include quantum error correction. In the brain of the human species, this takes the form of the proposed holographic workspace, that collects active information in a ”brain event horizon”, representing an internal and fully integral model of the self. This brain-supervening workspace is equipped to convert integrated coherent wave energies into attractor type/standing waves that guide the related cortical template to a higher coordination of reflection and action as well as network synchronicity, as required for conscious states. In relation to its scale-invariant global character, we find support for a universal information matrix, that was extensively described earlier, as a supposed implicate order as well as in a spectrum of space-time theories in current physics. The presence of a field-receptive resonant workspace, associated with, but not reducible to, our brain, may provide an interpretation framework for widely reported, but poorly understood transpersonal conscious states and algorithmic origin of life. It also points out the deep connection of mankind with the cosmos and our major responsibility for the future of our planet.

Key Sources of Research:

The Folding of Life Proteins: On the role of long- and short range electromagnetic pilot mechanisms

Dirk K.F. Meijer* and Hans (JH) Geesink

Click to access The-Folding-of-a-Life-Proteins-On-the-role-of-long-and-short-range-electromagnetic-pilot-mechanisms.pdf



Nature Unites First, Second and Third Harmonics to Organize Coherent Electromagnetic Frequency Patterns that are Crucial for Health and Disease

A Soliton Algorithm with Discrete Frequencies for Ordering and Therapeutic Restoration of Life Processes

Hans (J H) Geesink and Dirk K F Meijer




Bio-Soliton Model that predicts distinct non-thermal Electromagnetic Radiation Frequency Bands, that either Stabilize or Destabilize Life Conditions

J.H. Geesink and D.K.F. Meijer


Click to access 2016_Bio-Soliton_Model_that_predicts_distinct_non-thermal_Electromagnetic_Radiation_Frequency_Bands.pdf



Phonon Guided Biology: Architecture of Life and Conscious Perception Are Mediated by Toroidal Coupling of Phonon, Photon and Electron Information Fluxes at Discrete Eigenfrequencies

Dirk K. F. Meijer and Hans J. H. Geesink

NeuroQuantology | December 2016 | Volume 14 | Issue 4 | Page 718-755



Quantum Wave Information of Life Revealed: An Algorithm for Electromagnetic Frequencies that Create Stability of Biological Order, with Implications for Brain Function and Consciousness


Hans J. H. Geesink and Dirk K. F. Meijer





The Universe as a Cyclic Organized Information System: John Wheeler’s World Revisited


Dirk K. F. Meijer






The Extended Brain: Cyclic Information Flow in a Quantum Physical Realm

Dirk K. F. Meijer




Immortality: myth or becoming reality? On the conservation of information

Dirk K.F. Meijer


Click to access 2013-eng-3-04.pdf


Information: what do you mean? On the formative element of our universe

Dirk K. F. Meijer


Click to access 2013-eng-3-01.pdf



Quantum Physics in Consciousness Studies

Dirk K. F. Meijer and Simon Raggett

Click to access 56e0121478004572db4d42edc9ec6555fa33.pdf



Quantum modeling of the mental state: the concept of a cyclic mental work space

Dirk K. F. Meijer and Jakob Korf


Click to access 2013-eng-1-1.pdf



Consciousness in the Universe is Scale Invariant and Implies an Event Horizon of the Human Brain

Dirk K.F. Meijer, Hans J H Geesink
2017 Vol 15 no 3

Integral Philosophy of the Rg Veda: Four Dimensional Man

Integral Philosophy of the Rg Veda: Four Dimensional Man


Meditations Through the Rg veda:  Four Dimensional man was published in 1976.  In 1999, Antonio de Nicholas published a review of his work.  See below.


From Forward to the book.


From Infinity Foundations website

Meditations Through The Rg Veda: A Retrospective
(Philosophy East and West. Vol.49. Number 2. April 1999)
by Antonio T. de Nicolas, PhD

Paradigm, Theory, Ritual

It is now twenty five years since Meditations Through the Rg Veda: Four-Dimensional Man was first published in the United States. My earlier work on the Rg Veda was published in 1971 in Bangalore, India. Though the structures of the book were born during my twelve years of consecutive living in India, these structures did not become a paradigm until later. The structures I refer to are the word things and the order of their arrangement I was embodying as I lived there, a context at a time. It was the way the sun rose or the dawn arrived, the slow-motion for the sun to set and the sudden night; the lines of movement, of people, animals, wind or rain; the sudden appearance of forms, by the river, a well, in the sky; the dissolution of familiar and unfamiliar names, in the rhythm of language, Gujarati, Sanskrit, even English or Spanish names; but above all, the new habit of listening with my eyes to the movements in the sky, the forest, the streets, the homes; for the world, and my body, were a musical string plucked at every turn, in every silence , in every sight, sound, smell, touch and movement. Hidden geometries became human flesh, unnoticed. It was a silent world longing to become language, but can a multiplicity of embodied languages be expressed as one? After a while it was life in the twilight; which was the shadow, which the real object? One has to gain distance, and none farthest than an American Ph.D. Nonetheless, dispite the distance, and dispite the academic language, a new paradigm was born, in the Bronx, of all places. The structures I embodied gave way to an experienced, embodied geometry, sustaining all the structures, texts and statements I silently learned in India. Of course, when I set down this paradigm in writing, be it the Rg Veda or the Gita, the actual written text was already a theory, no longer a paradigm, though perhaps the most accurate translation of the paradigm. Those who disagreed kept silent and those who agreed, the majority, repeated my theory as participants in a ritual. In short, the acts by which the paradigm was born in me, or is born in any one giving birth to an original text like the Rg Veda, is not the written text. The act of creation is silent. The text of the Rg Veda, however, as written down is only a theory of itself, an invitation to a ritual. It is not even one text, or one language, but several and can only be expressed in plural linguistic wholes. Paradigms may be tested; they leave invariant epistemologies, but they can never be taught; they are sheer creation. Theories, as short hand of possible paradigms, on the other hand, we learn in the classroom. They are the easy ones to repeat. Those who follow the path of creation, of embodied-vision, follow the path of the gods. The others follow the path of the fathers, the path of pro-creation, the path of ritual, as the Rg Veda indicates. One leads to immortality, the other to rebirth. On which of these two paths stands the author of the text, rsi, commentator, priest or scholar? Besides, the Rg Veda is the sruti (revelation) tradition of India. As such, it is earlier than any other claim of revelation from any of the canonical texts, from the East, Middle East or West. The paradigm of the interpreter, if it coincides with that of the Rg Veda, should also give birth to those gods that gave humans sensation, inspiration and immortality, not just life to a priesthood that changes ritual as the mood strikes, bent on the act of pro-creation for, after all, the immortality of the ritual is more important than the immortality of the soul. Nor is it legitimate while interpreting to disband these earlier gods in the name of a later one, nor the heart-ethics of these original people for the head-ethics of those who came later, and if done it should be made evident. And this is how the “written” Rg Veda began. The priests wrote it down thousands of years later (depending on which initial date you choose). Ideographic language gave way to alphabetic writing, criteria of sound to those of sight, the path of the gods to that of the fathers, the structures of immortality to those of reincarnation, paradigm to theory repeated in ritual. Which Rg Vedic text are we talking about? What is recoverable from such a text? In the end, all we are left with are the technologies by which we recreate either text. Which path do they open for us? Now, once this is said, however, the modern interpreter cannot be blamed for not being a rsi. Let the reader be free to decide between the two paths, and let the interpreter be aware of both.

The Myth of Invariance

The first scholar to find my 1971 edition of the Rg Vedic world “captivating” was Ernest McClain. His interest was my claim that every statement in the oral/aural Rg Veda was tied to a language grounded on musical criteria. Music was once, at the origin of human language, the epistemology of oral cultures. This was all Ernest McClain needed to make a life and a project of his training as a musicologist. We started collaborating, getting together for brunch at the Waldorf Astoria in New York, in 1974. His first book appeared in 1976, The Myth of Invariance: The Origin of the Gods, Mathematics and Music from the Rg Veda to Plato (Nicolas-Hays, N.Y.) In 1978 he brought out with the same publisher, The Pythagorean Plato: Prelude to the Song Itself, and in 1981 Meditations Through the Quran: Tonal Images in an Oral Culture. When we last spoke he had already found confirmation of his work and mine, not only in Greece, but also in Chinese and Biblical texts. In his words: The Rg Veda is the original epistemology upon which humans built knowledge and also immortality. And thus by the hand of music the Rg Veda re-entered human consciousness.

The Artful Universe

At first glance this book is a most welcome addition to Vedic studies. It covers a territory in Indian Studies few dare to tread, and in doing so the author brings to the discussion almost everyone, ancient or modern, who has written anything on the Vedas. The writing style is beautiful and the translations from the Sanskrit have a modern ring that makes the original less intimidating. There is a definitive purpose by the author in the writing and interpreting of these texts! On the one hand, and this is the thesis of the book, the Vedas are the product of the imagination, and on the other this imagination expresses itself as ritual, as the religious imagination of the Vedic religion. Professor William K. Mahoney takes six chapters to develop this thesis. The first two are a preparation to understanding the religious imagination, the third and fourth chapters cover the Rg Veda and the last two the Upanishads. The book, however, does not end here. The Notes that follow these six chapters are yet another book within the book which allow the reader to follow the inner footsteps of Prof. Mahoney in the composition of his book. It is easy here to admire his delicate scholarship and his flare for the happy phrase in translating or interpreting the work of others. While my intention in writing this essay is a celebration of the human effort carried out in getting to the origins of our species, I wish also to sharpen the debate in the hope that “embodied structures” take over where simple or simplistic statements become the origin of the dialogue.

The modern scholar dealing with the Vedic period has several options: Translations of individual hymns under arbitrary categories, as it has been done and can be found in the Bibliography of The Artful Universe; or corrections, very important, as to the date of the Vedas, as In Search of the Cradle of Civilization, as G. Feuerstein, S. Kak and D. Frawley have successfully done, or he/she may try to uncover for us the paradigm and mental faculty through which the Vedic seers “composed” the original hymns. This is what Prof. Mahoney promises us:

“To Vedic visionary poets, the world is – or could be – an integrated whole, a unified structure and process of being in which there are no unbridgeable distances separating the divine, natural, and human worlds” (p.2).

And this world is held together by ” mental abilities or processes associated with what I will call the imagination” (p.5), ” the divine imagination… and the human imagination – especially the poetic, sacerdotal, and contemplative imagination… (and) whether divine or human, it is precisely the imagination that fashions and recognizes the universe as meaningful, abiding, and valuable, that is to say, real” (p. 7).

Here are my first questions. When we take, say the Rg Veda, for examination or commentary, which “text” are we recreating? The oral text the rsis chanted, the written text the priests codified in ten mandalas and became a ritual, or a new mongrel text that repeats a lot of names and quotes but can be used, at most, as the weekend comfort of New Age Evangelicals? And if so, where are the priests in the Upanishads when the Ksatriya instruct them? But above all, if the imagination is the faculty used by the seers in the composition (creation) of the oral, original hymns, which is the faculty that the priests use when they write down the text and when they repeat the same written text in ritual after ritual? But above all, if the imagination is a faculty, how does it work, which are its movements besides naming it, which are its structures, and are these structures the same or different from our own, and if the same why, and if different, how can we understand the Vedic imagination? How many priests does the author know with imagination? Isn’t their job to repeat a ritual imagined by others, deadening thus not only their senses but their faculties too? An imaginative priest is known as a heretic!

These remarks are not to be answered by Prof. Mahoney. He has written his beautiful text. But is this text the Rg Veda, or is it the case that any attempt at writing down one Rg Veda will give us of necessity several texts? It is obvious that this study fluctuates between the “creation” text of the original rsis and the “pro-creation” text of the later, codifying priests. Where once we had sheer power of creation, through an active imagination, giving birth to gods, powers and continuities, very soon we descend to the repetitive ritual of procreation through human semen, and the danger is that this becomes the ritual we celebrate today:

O holy drop!
You are the master of ecstasies!
You are the immortal god’s favorite drink!
Show us the way to success,
as a friend to a friend. (p. 85)

But it is the Rg Veda itself which admonishes us a few hymns later than the one quoted above by Prof. Mahoney (R.V.9.112) to be weary of one single text, be it rituals or anything else:

l. Our thoughts wander in all directions
And various are the ways of men:
The cartwright looks for accidents,
The physician for the sick,
And the brahman for a rich patron.
For the sake of Indra,
Flow, Indra, flow.

4. The horse draws a swift carriage,
The generous host an easy laugh and play.
The penis seeks a hairy slot
And the frog (brahman) hankers for a flood.
For the sake of Indra,
Flow, Indra, flow.

(My translation in Meditations through the Rg Veda).

How does this effort in all of us at producing “one single” text fail when dealing with Indian classical texts, particularly the Rg Veda?

As regards the Notes of this book I have only admiration. It is almost heroic the effort of Prof. Mahoney to footnote his conclusions. It is as if footnoting he were building a path for others to follow. The way he does it, however, may raise serious questions. Is not this the “path of the fathers” leading to the re-incarnation of all ritual, including the ritual of scholarship? Take, for example, part of the footnote he dedicates to my book Meditations through the Rg Veda:

” …The “four dimensions” of the Vedic intentional life outlined by de Nicolas are similar in some ways to the poetic and ritual aspects of the Vedic World I discuss in Chapters Three and Four, below. We overlap most in regards to what de Nicolas calls the “language of embodied vision.” My approach is different from his, however, in that, whereas he concentrates on the linguistic nature of visionary knowledge, I focus my intention on the visionary background of linguistic expression.” (Emphasis mine) (p.238). Does Prof. Mahoney understand that no matter how he “overlaps” me, (ritualizes my writing?) my work antedates his by twenty five years, and supplies him not only with the pertinent Rg Vedic hymns he quotes but also with the secondary sources he needs to gather the community of scholars that will testify to his thesis? Furthermore, was not my book the one to establish not only the “imagination” as a rational intelligence of oral cultures, but also the “moves” it must make to be an imagination in movement, able to keep a diverse society in continuity within the discontinuity of sensation? If this is my thesis where is his? In the ritual of repetition of the original text? I would most probably let this point go were it not for the fact that this “tracing” over other people’s work seems to count these days as scholarship. It seems to be a mind-set of the times. But is this the “text” that gave birth to the Rg Veda? Scholarship is not a ritual, and more so, a thesis is not a ritual. Where is the imagination to get out of other people’s rituals, to rise to ” the path of the gods”? Let’s go on with our conversation. Prof. Mahoney will rejoin us later in the dialogue.

The New Theogony And The Heresy Of Oedipus

” Let us with tuneful skill
Proclaim the origin of the gods,
So that in future generations these origins
May be seen, when these songs are sung.” (R.V. l0.72.1)

Dr. Colavito, in The New Theogony, perhaps the best book on myth written in English, universalizes the “languages” of the Rg Veda, Asat, Sat, Yajna and Rta, to cover the study of all myth.

“What we call “myth”,” she writes “is a fourfold cluster of actions and mental properties that individually and together account for the necessary and sufficient conditions of the mythopoetic worldview, of the nature and workings of the cosmos, and of the individuals and groups of individuals within this cosmos.” For the sake of clarity she summarizes these languages thus: ” These four fundamental acts defining myth are: maia, mythos, mimesis and logos. Each act is a single focus or mental habit; together the four account for the totality of human and divine acts, or mental habits, that have guided the human species to the present shores. Though strictly speaking myth is merely one of the (four) acts in myth making, even this act is incomprehensible unless the other three mental operations are included in the narratives of myth…” She then goes on outline the four “languages”:

“Maia (Gr. midwife) is the term used to signify the bringing forth of action from inaction, cosmos out of chaos, the initial spark that kindles the mind to transform from nothig to something. It is the midwife between the divine realm of immortality and potentiality and the human realm of temporality and human existence. The aspect of maia in the human sphere is represented by the human faculty of imagining. It is the expression of the creative experience; it cannot be described,, it has no form, its proper abode is the midregion between the human and the immortal. Once an individual begins to interpret or reflect upon the experience, maia disappears and the experience receives an existence of its own, outside the real of potentiality, and it is given a form, name, boundary. In short, the reflective act heralds in the aspect of mythos. And with mythos the world moves from chaos to cosmos.

Mythos (from the Gr. delivered by word of mouth) primarily describes the initial reflection of the creative experience. It is the oral transmission of the experience… The first “scream of individuation,” to quote Nietzsche… Mythos, also, represents the original fall from grace, the first act that breaks from the unity of the beginning, from the glory of immortality; for the telling of the experience now has another element, an experiencer, a self, through whom the experience flowed. Thus… the telling of the experience is not the experience… and only those who have had the same experience may truly understand the full import of the teller’s tale…so that communities of experiencers can share common revelations.

Mimesis (Gr. to make a copy) is the aspect that describes the mythopoetic action of re-membering or re-creating… In this manner the story is told with an intent, a moral… What becomes important now is the story not so much as it relates to the original creative experience of individuals, but as it relates to the desire to make a point… The mimetic phase is … the first frozen form: the pictographic mode… geared toward establishing the social mores of the collective group.

Finally, logos, (Gr. the word by which the inward thought is expressed), taking as its origins these mores, completely eradicates the level of personal experience and uses the rules derived from the mimetic to create theories about human action. These mores are founded on human experience, but only on hypothetically universal experience – in other words, experience filtered through the sieve of a collective interpretation. As such, then , no origin in logos has the certainty of an origin in maia… Logos ceases to be a pictographic representation; it transforms into a symbolic or alphabetic system that has only its own correlatives within its own framework, with no derivative capacity from the experiential realm of the individual… Logos has always been the shadow of maia in the mythopoetic world.

This fourfold division is neither a convenient devise for classification, nor an arbitrary tool for interpretation; it is the fabric itself of myth… an abstraction, that, though distinguishable, is inseparable from myth. From a biological perspective this fourfold division is the neurophysiological equipment of the species, its mental habits accumulated through the repetition of the past: imagining, fantasizing, narrating, following the discursive path of logic… )(Thus) while maia stands for an original experience… mythos, mimesis and logos stand for different ways (languages) of making this experience public, either through narrative (mythos), visual forms (mimesis), or… theory…alphabetic substitutions, or conceptual analyses (logos)… Finally, this fourfold system of acts corresponds to the scientific operations functioning within the oral/aural worldview, which has as its verification the ancient science of acoustics.” (pp.6-8)

Using the model of the one dismembering itself or the model of the zero as an addition of objects, Dr. Colavito makes evident the model through which dialogue and understand of myth is possible, and this not in just a few cases, Classical India, Rg Veda, Upanishads, but also the Greek gods and goddesses ending with the education of Pythagoras as imparted on his students, and the acoustic verification in Plato. A breathless trip that ends in the frustrating realization that while simple acts may lead to overwhelming “oceanic” experiences, the unity of maia, once broken, can never be recovered in one single language, but we must learn to move with of plurality of at least four irreconcilable and irreducible languages. Or is this a frustration or a temptation, the temptation to be the shadow of a god, if not god him/her self?

These are very strong claims. If true they may lead to the mobility of the Rta to perform the good act (sukrta), the original act of creation. Can they also lead to reconstructing the original Rg Veda? Where do we find the verification? Dr. Colavito took it upon herself to get to the bottom of the issue. Equipped with two Ph.Ds – one in Comparative Literature and the other in Psychology, her next book, The Heresy of Oedipus and the Mind/Mind Split, introduces us to her “Biocultural Paradigm.” She starts with the Nature/nurture controversy raging in the biological sciences to conclude that neither one nor the other works in isolation, but that nurture opens Nature, and Nature is not activated without nurture. In other words, the neural passages of the brains are open or forever shut if there is not a mutual fecundation. This interaction is limited, and almost chronologically developed in every child from conception to the age of l2; after that, what nurture has not activated in the right hemisphere of the neocortex is forever destroyed, though the left hemisphere, the seat of logic and discourse and the place of the “interpreter module,” keep developing abstract substitutions based on information received by these other brains or by its own conceptual loops, forever. What in her first book was called maia, in the second is the reptilian brain; mythos becomes the limbic brain; mimesis she divides into two: the visual, right hemisphere and the conceptual left hemisphere; and logos is the interpreter module located in the left hemisphere of the neocortex. In this she follows MacLean and Gazzaniga and the latest discoveries in neurobiology. But for the purpose of our discussion, in what way is this relevant to the Vedas and Prof. Mahoney’s or McClain’s books?

Revelation, individual experience, is an affair of the right side of the brains. The left hemisphere can only interpret, translate what the right hemisphere presents as sensation. Thus, while we have five different brains, (not one as Descartes thought and we presume,) only the three of the right hemisphere deal with original experience. And this in different ways. While maia ( the Asat) is the origin, maia is also wired with a geometry capable of letting forms appear, while mythos, the place of gods and heroes, is already a world of forms. However, and this is the point of our discussion, when these two original and originating brains are translated by the right hemisphere of the neocortex they are translated as “visual images;” they are seen as images even if originally they were waves and movement and tactility. In other words, by the time the ritual priests take on the “visual images” to the sacrifice and the ritual, these visual images, originally, were neither images nor visual. Thus by constituting these images as the original text, the followers are removed from the origin, from the source of sensation and are led into a repetition of acts that may crystallize either in a crisis of faith or in a crisis of dogma. The believers may either end up losing faith,( also sensation) or becoming dogmatic preachers in a game of endless logomachy. And the same with any other “text” bound by single language-games, like Western Theology. Thus, according to the Rg Veda it is precisely because of this tendency that the culture calls for cyclical returns to the Asat: to lose all forms, verbal, audial, or visual and break the dragon Vrta open, again. And that excerise, in the Rg Veda, is the true meaning of sacrifice (yajna). The sacrifice is necessary because these languages are invariant biological epistemologies, irreducible to one another.

Dr. Colavito follows up her neurobiocultural bases with studies on myth, Rg Vedic and principally the Oedipus cycle and the whole history of the House of Cadmus, after the mind/mind split took place in the species with the repetition of the technologies developed to introduce alphabetic writing in our mental habits. The paradigm is so explosive that Time magazine (Feb. 1997) could not avoid making a full use of it to describe the early development of the different brains in children, the contrary pole of Dr. Colavito’s thesis as she verified it through earlier cultures, in the infancy of the species. Of special interest in our discussion is her Appendix 2.3 making visible the hidden geometries of the Asat and the two ways of reading those texts: as from the “path of the fathers” or as from the “path of the gods.” How can we overcome the temptation of one single language, and how do we learn to be open to a plurality of four?

The Human Potential

“We can’t put it together; it is together.”

“What we need to understand may only be expressible in a language that we do not know.”

The Encyclopedia of World Problems and Human Potential, is a mammoth ongoing enterprise to cover all human problems ( l2,000 profiles with 120,000 hyperlinks), strategies and solutions (29,500 profiles with 91,000 hyperlinks), human development (4,400 profiles with l5,000 hyperlinks) and human values (1,900 profiles with 23.000 hyperlinks). The Encyclopedia hypertexts are currently edited at the Union of International Associations (UIA) by Nadia McLaren. It is now in its fourth hardback edition, first CD-ROM edition, and is available in demo version on the Web (http://www.uiaorg/homeency.htm/), although all texts have been accessible since l998. Profiles on the Web can be translated through Alta Vista into a variety of major languages.

It is in this global environment that the paradigm of “languages” in the Rg Veda has found a home. The Director of the Union of International Associations, Dr. Anthony J.N. Judge, in article after article, profile after profile, conference after conference has articulated, and compiled in the Encyclopedia, the modern consequences of academic attempts at synthesis when these attempts are expressed in one common language, namely the one engendering the problems in the first place. Dr. Judge’s point of departure is the need to start from the experiential human origins as described in the Rg Veda and then articulate the ensuing insights in the plurality of languages available for their manifestation in the Rg Vedic model. Thus, the model or paradigm, is part of the “answer” proposed by the Editors of the Encyclopedia. Contrary to the position academics take of locating themselves within the “web” of a discipline, research, culture, department or, at times, a simple desk, Dr. Judge travels with ease the “lines of the webs” linking the totality of squares, within which the rest of us seem to be trapped, to a knowledge that seems to come only to those who are able to travel in his manner. He is at home in the East and in the West, in music and in science, in Christianity, Islam, Judaism, Hinduism, Buddhism or the Tao, and he seems to know a “knowledge” that comes only to those who travel the “lines” of the “web,” never squeezed by the particular generalizations of the cubicles within each “web’s square.” His summary of the “languages” of the Rg Veda for contemporary guidance to those looking to solve the problems, individual, communal or global, or contemporary life is appealing to him because it takes into account: ” The interrelated formal languages based on tone; (they lead ) toward reintegrating the individual in action; (make ) this integration embodied: re-imagining man; (take care) of the pluralism through an integration of community dialogue; (guarantee) this integrative renewal through sacrifice (of perspectives); (account) for an integrative vision that is encountered in the movement.” ( ” Liberation of Integration, Universality and Concord through pattern, oscillation, harmony and embodiment.” Originally delivered for the 5th Network Meeting of the United Nations University, project of Goals, Processes and Indicators of Development (Montreal) as a contribution to the discussion on integration of the findings of the Project.) And he includes in his remarks the fact that:

” Integration modelled on sound may be inherently more comprehensible to more people than integration modelled on sight.” (Ibid.) In view of Dr. Colavito’s previous discussed work, this conclusion is not so far fetched since the structures of the Rg Veda are the original embodied structures of the humanity that gave us birth, and as such they are embodied structures, bio-culturally invariant, not only in each of us but also in the earlier cultures that preceeded us.

Conclusion: Regathering The Fragments

The Artful Universe provided the occasion for a round discussion of the earliest structures of human “languages” we carry in our genes.

Any one particular language joining the discussion does not only show us the empirical grounding of their speech, experience, academic construction, but also the imperialistic tendency of such mono-linguistic speech universalizing itself. Contemporary discoveries in neurobiology and the paradigm based on them of Dr. Colavito make it clear that life, that is, human life, is life in community. This community is formed through interaction or dismemberment of a sensorium that is plural by its very bio-culture base and becomes integrated through dialogue. All dialogue, all language carries with it the possibility of sharing in the embodied vision of a paradigm that has been with us from the beginning, since through it we had to break through the “experience of separating earth and sky.” In this manner there is no need, as Prof. Mahoney does in one of his initial footnotes with a humility rarely present in Sanskrit scholars, to apologize for not being ethnically Indian while interpreting the Vedas. Interpretation, like everything else, is biocultural not ethnical. We are dealing with neural equipment, genes, receptors and transmitters, not the color of one’s skin, or the geography of one’s birth. And finally, if there is any hope in preserving the integrity of the University or returning it to its original call, especially in the humanities, this hope resides in the work of scholars like Profs. Mahony, Colavito and Judge who through their work in the classical myths were able to avoid the “empiricist languages” of the present Academic fashion and return to us the memories of our distant progenitors with the structures that made them live in innovation and continuity in the company of the gods. If we form the communities to carry these traditions forward, we might be able to share in the glory and celebration of life that once was ours. I am glad and grateful that Meditations Through the Rg Veda was an inspiration to them. But even more so the reiteration that our human makeup is larger, deeper and more full of sensation in the plurality we are than in the oppression of one single language-community-creed.

This is what William Irwin Thompson called, commenting on my work: “the planetization of the esoteric.”


The Artful Universe: An Introduction to the Vedic Religious Imagination.
by William K. Mahoney , Ph.D.
SUNY Press, Albany N.Y. l998

The New Theogony: Mythology for the Real World.
by Maria M. Colavito, PhD
SUNY Press, Albany, N.Y. l992

The Heresy of Oedipus and The Mind/Mind Split: A Study of the Biocultural Origins of Civilization.
by Maria M. Colavito, PhD
The Edwin Mellen Press, Lewiston,N.Y. l995

Encyclopedia of World Problems and Human Potential.
Edited by The Union of International Associations
4th. Edition, K.G. Saur Verlag, Munchen, New Providence,
London, Paris l994-95

The Myth of Invariance: The Origin of the Gods, Mathematics and Music from the Rg Veda to Plato.
by Ernest McClain, Ph.D.
Nicolas-Hays Ltd. N.Y. 1976

The Pythagorean Plato: Prelude to the Song Itself.
by Ernest McClain
Nicolas-Hays Ltd. N.Y. 1978

Meditations Through the Quran: Tonal Images in an Oral Culture.
by Ernest McClain
Nicolas-Hays Ltd. N.Y. 1981

Coming Into Being: Artifacs and Texts in the Evolution of Consciousness
by William Irwin Thompson
St. Martin’s Press, New York , 1996. (p.187)

Antonio T. de Nicolas was educated in Spain, India and the United States, and received his Ph.D. in philosophy at Fordham University in New York. He is Professor Emeritus of philosophy at the State University of New York at Stony Brook.

Dr. de Nicolas is the author of some twenty- seven books, including Avatara: The Humanization of Philosophy through the Bhagavad Gita,a classic in the field of Indic studies; and Habits of Mind, a criticism of higher education, whose framework has recently been adopted as the educational system for the new Russia. He is also known for his acclaimed translations of the poetry of the Nobel Prize-winning author,Juan Ramon Jimenez, and of the mystical writings of St. Ignatius de Loyola and St. John of the Cross.

A philosopher by profession, Dr. de Nicolas confesses that his most abiding philosophical concern is the act of imagining, which he has pursued in his studies of the Spanish mystics, Eastern classical texts, and most recently, in his own poetry.

His books of poetry: Remembering the God to Come, The Sea Tug Elegies, Of Angels and Women, Mostly, and Moksha Smith: Agni’s Warrior-Sage. An Epic of the Immortal Fire, have received wide acclaim. Critical reviewers of these works have offered the following insights:

from, Choice: “…these poems could not have been produced by a mainstream American. They are illuminated from within by a gift, a skill, a mission…unlike the critico-prosaic American norm…”

from The Baltimore Sun: “Steeped as they are in mythology and philosophy these are not easy poems. Nor is de Nicolas an easy poet. He confronts us with the necessity to remake our lives…his poems …show us that we are not bound by rules. Nor are we bound by mysteries. We are bound by love. And therefore, we are boundless”

from William Packard, editor of the New York Quarterly: ” This is the kind of poetry that Plato was describing in his dialogues, and the kind of poetry that Nietzsche was calling for in Zarathustra.”

Professor de Nicolas is presently a Director of the Biocultural Research Institute, located in Florida.


Please see my related posts:


Meta Integral Theories: Integral Theory, Critical Realism, and Complex Thought

Truth, Beauty, and Goodness: Integral Theory of Ken Wilber

Myth of Invariance: Sound, Music, and Recurrent Events and Structures

Sounds True: Speech, Language, and Communication

Mind, Consciousness and Quantum Entanglement

Semiotics, Bio-Semiotics and Cyber Semiotics

Systems and Organizational Cybernetics

Society as Communication: Social Systems Theory of Niklas Luhmann


Key sources of Research:



Meditations Through The Rg Veda: A Retrospective

(Philosophy East and West. Vol.49. Number 2. April 1999)

by Antonio T. de Nicolas, PhD









Antonio de  Nicolás



Myth of Invariance: Sound, Music, and Recurrent Events and Structures

Myth of Invariance: Sound, Music, and Recurrent Events and Structures


From  Philosophy and Science of Music in Ancient Greece:
The Predecessors of Pythagoras and their Contribution

 One of the ironies of twentieth-century thought is that the final dethronement of Pythagoras as a ‘father’ of western science and philosophy and the ‘inventor’ of music and mathematics should be accompanied by a world-wide revival of Pythagorean research and speculation. During the seventeenth century, the ‘harmony of the spheres’, which had remained an article of faith until the age of Shakespeare and even Louis XIV [Isherwood 1973, Ch. 1], was suddenly overwhelmed by the mighty discoveries of Kepler and Newton; but this traumatic ‘Untuning of the Sky’ [Hollander 1970] did not entirely obliterate the Pythagorean tradition (to which both Kepler and Newton were sympathetic).

Since the pioneering studies of Thomas Taylor (1758-1835), Antoine Fabre d’Olivet (1767-1825) and Albert von Thimus (1806-1878), there has been a steady renewal of interest in the old science of harmonics, culminating in the work of Hans Kayser (1891-1964) and his two most influential successors, Rudolf Haase and Ernest G. McClain (both of whom are living in retirement). Neo-pythagoreanism is now a conspicuous feature of post-modern philosophy and science: the revival of musica speculativa, part of a larger resurgence of neo-classicism, is well represented in the writings of Joscelyn Godwin [Godwin 1987, 1993, etc.]. To his extensive bibliographies could be added not only impressive results of recent mainstream research into Pythagoras and the Pythagoreans, e.g., Huffman [1993], but also the publications of several ‘alternative’ thinkers, including the French-American composer, music theorist, and astrologer, Dane Rudhyar, the French ‘neo-astrologer’ Michel Gauquelin, the English numerologist John Michell, and the English geneticist Rupert Sheldrake. Sheldrake’s notion of ‘morphic resonance’ — forms resonating in Nature’s memory — is a very Pythagorean-Platonic alternative to mechanistic causality. His wife, Jill Purce, is a music therapist [Purce 1974]; so both sides of the Pythagorean tradition — the ‘hard’ and the ‘soft’ sciences — are here reunited in the work of one family.

Though hardly any of these writers would describe themselves as Pythagoreans, their ideas have important connections with the old tradition and all are symptomatic of a new era in the history of thought when mechanistic and reductionist paradigms are giving way to a holistic and organic world-view. This emergent rationality is fundamentally ecological and its impact is being felt from metaphysics to everyday manners. The new paradigms of the Age of Ecology are already transforming the professions, sciences, arts, academic disciplines, and human enterprises generally — from the minute study of bird-song and insect music to the utopian vision of planet Earth designed and managed as a single, organic Gesamtkunstwerk [Pont 1997].

Central to this new understanding of the world is the concept of the ‘Biosphere’, which is the very antithesis of Newton’s mechanistic universe [Teilhard de Chardin 1955]. Thus the Pythagorean vision of the living cosmos — or Plato’s ‘World Soul’ — has reappeared in new vitalist theories, including the Gaia hypothesis of James E. Lovelock [1979]. The modern world-view and its vast astronomical time-frame have changed our conception of humanity itself, if only in recognising our evolutionary affiliations with, and biological dependence on, other species in the terrestrial ecosystem. And it has also transformed the idea of the ‘humanities’: never again can they be taught as just a narrow study of the ‘classical’ texts or litterae humaniores of Greece, Rome, and the Renaissance. No longer can the ancient Greeks be contemplated, in museum-like isolation, as perfect models of everything European.

With the growth of modern archaeology, prehistory, anthropology, linguistics, and other comparative studies, the marmoreal idols of Eurocentric scholarship are now revealed in something like their original gaudy splendour — a Joseph’s coat of distinctly oriental hues. Most of the discoveries traditionally ascribed to Pythagoras were Asiatic in origin; and, in a recent survey, Music and Musicians in Ancient Greece [Anderson 1994], the Pythagoreans have been reduced to four passing references and Pythagoras himself is omitted altogether!

The innovations still plausibly credited to the historical Pythagoras include the coining of the terms ‘philosophy’ and ‘theory’ which, in his case, must have referred to the dogmatic teachings and pre-scientific wisdom of a guru rather than genuinely theoretical inquiries like those of Heraclitus and the Eleatics. Pythagoras was also credited with inventing the term Kosmos, but the idea of the beautiful world-order (above and below) must surely have been Egyptian in origin [Cf. Plato, Laws II, 656a-657b]. Our admiration of the Greeks is now tempered by a better understanding of their true historical circumstance and actual indebtedness to other civilisations [Cf. Bernal 1987].

Just as Whitehead saw western philosophy as ‘a series of footnotes to Plato’, so modern scholarship has established that most of the doctrines traditionally ascribed to Pythagoras were really the contributions of the older high civilisations, particularly of Mespotamia and Egypt.[1] The rise and dissemination of these perennially influential doctrines remains one of the most formidable problems for the historian of ideas.

Many of these ideas had already been explored in my General Studies courses at the University of New South Wales, particularly in ‘The Philosophy of Music’ (Australia’s first academic course on the subject, 1974-1988) and, more recently, in shorter courses on ‘Ancient Rationality’ and ‘Modern Rationality’ (1988-1995). It was with their arguments and conclusions in mind that I undertook during 1997 my last course at the University, entitled ‘The Predecessors of Pythagoras’. This aimed to examine the origins and analogies of Pythagorean traditions in Mesopotamia, Egypt, China, and India. The lectures contained little that was new and the literature survey was, unavoidably, far from exhaustive; but, even so, the course had the unintended effect of changing the lecturer’s point of view — and, indeed, his whole approach to Greek philosophy and science of music.

Instead of burdening the class with the meagre texts of the early Pythagorean school and the interminable difficulties of their interpretation, lectures took a broad view of ancient history and prehistory, in an attempt to answer two very large and necessarily speculative questions: first, what might have been the origins of the famous ‘analogy of the macrocosm and the microcosm’? And, second, how and when was this world-view ‘mathematised’? — that is, when was it precisely articulated with a system of musical numbers or harmonic ratios that eventually constituted the ‘harmony of the spheres’? Most of the fifteen students had some background in history and philosophy of science but no prior musical knowledge was required for the course and readings had to be confined to material available in English. The only set text was The Pythagorean Sourcebook and Library [Guthrie 1987].


Key People:

  • Hans Kayser
  • Ernest G McClain
  • Jay Kappraff
  • Barabara Hero
  • John Goldman
  • Guy Beck
  • Joscelyn Godwin
  • Joachim-Ernst Berendt
  • Antonio De Nicolas
  • Richard Heath
  • Subhash Kak
  • Rachel Wells Hall
  • Amba Kulkarni
  • Leon Crickmore
  • Pingala
  • Rudolf Haase


Key Concepts:

  • Scale Invariance
  • Time Invariance
  • Cycles
  • Fractals
  • Power Laws
  • Periodicity
  • Multi-scales
  • Octave Invariance
  • Octave Equivalence
  • Self Similarity
  • Pattern Formation
  • Harmonics
  • Repeating Patterns
  • Repeating Events
  • Attractors
  • Basin of Attraction
  • Prosody
  • Vedic Sanskrit Meters
  • Sounds, Numbers, Akshara
  • Ratios and Proportions
  • Harmonic Mean
  • Geometric Mean
  • Arthematic Mean
  • Symmetry



Key Sources of Research:


Ancient Harmonic Law

Jay Kappraff


Click to access bridges2007-311.pdf


Ancient Harmonic Law (version 2)

Jay Kappraff


Click to access report0809-19.pdf


Philosophy and Science of Music in Ancient Greece:
The Predecessors of Pythagoras and their Contribution

Graham Pont



The Lost Harmonic Law of the Bible

Jay Kappraff

Click to access bridges2006-481.pdf



Sound: A Basis for Universal Structure in Ancient and Modern Cosmology

Jung Hee Choi


Click to access Sound_JHC_v2013.pdf



Beck, Guy L.

Sonic theology: Hinduism and sacred sound.

Motilal Banarsidass Publ., 1995.


Beck, Guy L.

Sacred sound: Experiencing music in world religions.

Wilfrid Laurier Univ. Press, 2006.


Beck, Guy L.

Sonic Liturgy: Ritual and Music in Hindu Tradition.

Univ of South Carolina Press, 2012.


“Vedic perspective of Sound—Science and Spirituality.”

Prasad, M. G.

Click to access mgpSoundArticles.pdf


Conch-shells, bells, and gongs in Hindu temples

Marehalli Prasad

Click to access 54b3ebc70cf2318f0f969b1f.pdf


Acoustical studies on conch shells.

M.G. Prasad and B. Rajavel


On the role of acoustics in the Vedic Hindu tradition and philosophy.

M. G. Prasad and B. Rajavel


Conch‐shell and bamboo flute: Spiritual and musical expressions of acoustics in Vedic Hinduism

M. G. Prasad


Philosophical and cultural perspectives on acoustics in Vedic Hinduism

M. G. Prasad


Vedic chanting and vowel intrinsic pitch: Evidence from an ancient source

T. V. Ananthapadmanabha1, Kim Silverman1 and M. G. Prasad


“Perspectives on sound in Sanskrit literature on natural philosophy.”

Prasad, M. G.

The Journal of the Acoustical Society of America 85.S1 (1989): S147-S147.


Akasha (Space) and Shabda (Sound): Vedic and Acoustical perspectives

Prasad, M. G.

Click to access workshop2009_speakermgp5.pdf

The Golden Mean and the Physics of Aesthetics


Early Indian Music

Subhash Kak

Click to access manila.pdf


Hero, B. F.

“Design in nature from Pythagoras to Helmholtz to the Cantor musical array.”

WIT Transactions on Ecology and the Environment 57 (2002).

Click to access DN02021FU.pdf


Hero, Barbara, and Robert Miller Foulkrod.

“The Lambdoma matrix and harmonic intervals.”

IEEE engineering in medicine and biology magazine 18.2 (1999): 61-73.

Click to access Lambdoma%20Matrix%20Harmonic%20Intervals.pdf


Goldman, Jonathan.

Healing sounds: The power of harmonics.

Inner Traditions/Bear & Co, 2002.


Heath, Richard.

Matrix of Creation: Sacred Geometry in the Realm of the Planets.

Inner Traditions/Bear & Co, 2004.


Heath, Richard.

Sacred Number and the Origins of Civilization: The Unfolding of History Through the Mystery of Number.

Inner Traditions/Bear & Co, 2006.


Heath, Richard.

Precessional Time and the Evolution of Consciousness: How Stories Create the World.

Inner Traditions/Bear & Co, 2011.


A re-valuation of the ancient science of harmonics


Psychology of Music 31.4 (2003): 391-403.


Crickmore, Leon.

“New Light on the Babylonian Tonal System.”

CONEA 2008: Proceedings of the International Conference of Near Eastern Archaeomusicology, held at the British Museum, December 4-6. Vol. 24. 2008.


Crickmore, Leon.

“A possible Mesopotamian origin for Plato’s World Soul.”

Hermathena 186 (2009): 5-23.


Crickmore, Leon.

“The musicality of Plato.”

Hermathena 180 (2006): 19-43.


Crickmore, Leon.

“A New Hypothesis for the Construction and Tuning of Babylonian Musical Scales.”

Journal of Ancient Civilizations (2007): 006.


Recursion and Combinatorial Mathematics in Chandashastra

Amba Kulkarni

Click to access 0703658.pdf



Math for Poets and Drummers

Rachel Wells Hall


The Sound of Numbers

A Tour of Mathematical Music Theory

Rachel Wells Hall



Bhattacharya, Aryya.

“Vāc–Its Ontological Status and Importance in Prayers and Rituals of Śakti Oriented Tantric Tradition.”

The Polish Journal of the Arts and Culture. New Series 2015.1 (2015) (2015): 7-22.


Godwin, Joscelyn.

The Harmony of the Spheres: The Pythagorean Tradition in Music.

Inner Traditions/Bear & Co, 1992.


Breath, Voice, and Embodiment of the Metaphysical in Hindu Tradition

Gabe C. Alfieri

Click to access Breath_Voice_Embodiment.pdf


Franklin, Ellen, and Donna Carey.

“From Galaxies to Cells: Bridging Science, Sound Vibration and Consiousness Through the Music of the Spheres.”

Subtle Energies & Energy Medicine Journal Archives 16.3 (2005)


McClain, Ernest G.

“The bronze chime bells of the Marquis of Zeng: Babylonian biophysics in Ancient China.”

Journal of Social and Biological Structures 8.2 (1985): 147-173


McClain, Ernest G.

“The Myth of Invariance: The Origin of the Gods, Mathematics and Music from the Rg Veda to Plato.”



McClain, Ernest G., and Ming Shui Hung.

“Chinese cyclic tunings in late antiquity.”

Ethnomusicology 23.2 (1979): 205-224


McClain, Ernest G.

“Structure in the ancient wisdom literature: The holy mountain.”

Journal of social and biological structures 5.3 (1982): 233-248



“Pythagorean Paper Folding: A Study in Tuning and Temperament.”

The Mathematics Teacher 63.3 (1970): 233-237


McClain, Ernest G.

“Musical Theory and Ancient Cosmology.”

The World and I (1994): 371-391.


McClAIN, Ernest G.

“The pythagorean Plato: prelude to the song itself.

York Beach, Mine: Nicolas-Hays.” (1978).


Pont, Graham.

“Plato’s philosophy of dance.”

Dance, Spectacle, and the Body Politick, 1250–1750 (2008): 267-281.


McClain, Ernest G.

“The ancient Chinese “calendrical” pitchpipes.”

The Journal of the Acoustical Society of America 80.S1 (1986): S101-S101.


The World Is Sound: Nada Brahma
Music and the Landscape of Consciousness

By Joachim-Ernst Berendt



De Nicolás, Antonio T.

Meditations through the Rig Veda: Four-dimensional man.

iUniverse, 2003.


De Nicolás, Antonio T.


iUniverse, 2003.

Sounds True: Speech, Language, and Communication

Sounds True:  Speech, Language, and Communication


At the end of the cosmic dance Lord Shiva the Lord of dance, sounded his damaru fourteen times.

For the sake of the upliftment of sages like sanaka.

According to legend, these sutras were written by Panini upon mystically hearing the beats of Siva-Nataraja’s damaru (hourglass-shaped drum). It has been said that the mantra is God in the form of sound. Therefore, words are an extension of that power. The tantra stresses the importance of sound as a divine substance and vehicle for salvation. And hence, entire cosmos is in the form of these sutras.

The Fourteen Verses Of Maheswara Sutra.

“The Maheshwara Sutra is the most ancient known Sanskrit alphabet sequence. This alphabet sequence is at the same time a powerful Mantra and the vibrations of its sound has healing powers.

1. अ इ उ ण् |
2. ऋ ऌ क् |
3. ए ओ ङ् |
4. ऐ औ च् |
5. ह य व र ट् |
6. ल ण् |
7. ञ म ङ ण न म् |
8. झ भ ञ् |
9. घ ढ ध ष् |
10. ज ब ग ड द श् |
11. ख फ छ ठ थ च ट त व् |
12. क प य् |
13.श ष स र् |
14. ह ल् |

The fourteen sutras contain all the letters of the Sanskrit varnamala- the svaras (vowels) a, i, u, R^i, lR^i, e, ai, o, au and all the vyanjanas (consonants). As per the Rig Veda Lord Shiva brought this Sanskrit alphabet sequence, and the Sanskrit language to earth. The sounds of the alphabet originated from Lord Shiva’s ‘damru’, probably some kind of a sound device.


Among those present at Nataraja’s dance was Panini. For him these 14 sounds meant the fourteen cardinal sutras of Grammer and on them he based his “Ashtadhyayi”. Given are the 16 vowels and 33 consonants that are evolved from these 14 Shiva Sutras.

16 vowels (a – ch)

a, Aa, i, Ii, u, Uu, ri, rii, lri, lrii, e, ai, o au, am, ah

33 consonants (ha l)

ka, kha, ga, gha, gna
ca, cha, ja, jha, jna
ta, ttha, da ddha, nna
ta, tha, da, dha, na
pa, pha, ba, bha, ma
ya, ra, la, va
sa, sha, sa


Key People:

  • Yaska
  • Panini
  • Pingala
  • Patanjali
  • Katyayana
  • Bhartrhari
  • Leonard Bloomfield
  • Ferdinand de Saussure
  • Harold G Coward
  • Harvey Alper
  • Jan Gonda
  • George Cardona
  • Frits Staal
  • Johannes Bronkhorst
  • Ashok Aklujkar
  • Paul Kiparsky
  • Subhash Kak


Key Contributions:

  • Yaska wrote Nirukta
  • Panini wrote Ashta-adhyayi
  • Pingala wrote Chhandashastra
  • Katyayana wrote Varttikas
  • Patanjali wrote Vyakaran-Mahabhashya
  • Bhartrhari wrote Vakyapadiyas



Key Sources of Research:


Karakatattva of Sesacakrapani an edition and study

Leela, K N



Indian Linguistics

Click to access Indian_Linguistics-1.pdf


Old ideas of language


Click to access oldideasoflanguage.pdf


Alper, Harvey P., ed.

Understanding mantras.

Motilal Banarsidass Publ., 1991.


The word is the world: Nondualism in Indian philosophy of language

By Ashok Aklujkar



LANGUAGE AND REALITY On an episode in Indian thought

Johannes Bronkhorst

Click to access Language%20and%20Reality_On%20an%20Episode%20in%20Indian%20Thought_Bronkhorst.pdf


Coward, Harold G.

The sphota theory of language: a philosophical analysis.


Motilal Banarsidass Publ., 1980.


Coward, Harold.

“The meaning and power of mantras in Bhartrhari’s Vãkyapadiya.”

Studies in Religion Toronto 11.4 (1982): 365-375.


Coward, Harold.

“Derrida and Bhartrhari’s Vākyapadīya on the Origin of Language.”

Philosophy East and West 40.1 (1990): 3-16.


Coward, Harold G.

The philosophy of the grammarians.

Vol. 5. Motilal Banarsidass Publ., 1990.






Pånini’s grammar: from meaning to utterance




Johannes Bronkhorst

Click to access 02e7e53a1c07adb71d000000.pdf



Staal, Frits.

Universals: Studies in Indian logic and linguistics.

University of Chicago Press, 1988.


Staal, Frits.

“Oriental ideas on the origin of language.”

Journal of the American Oriental Society (1979): 1-14.


Staal, Frits.

Discovering the Vedas: Origins, Mantras, Rituals, Insights.

Penguin Books India, 2008.


Staal, Frits.

“The science of language.”

The Blackwell companion to Hinduism (2003): 348-359.


Cardona, George.

“Some principles of Pānini’s grammar.”

Journal of Indian Philosophy 1.1 (1970): 40-74.


On the structure of Pånini’s system

George Cardona


Click to access Cardona.pdf



Marco Ferrante


On movements of language−−within self, of self, about self, and between selves: Commentary on language and self

Lakshmi Bandlamudi


Staal, J. F.

“Sanskrit philosophy of language.”

History of linguistic thought and contemporary linguistics (1976): 102-136.


Sabda-Pramana: The Written and Spoken Word as Means for Right Knowledge.
An Issue of Nyaya Epistemology

Augustine Thottakara


Click to access P%20Pl%206%20Sabda-Pramana.Library%20Seminar.pdf


Language and Grammar

Click to access 334_03_05_09_KTPI_XII-Language.pdf


Economy and the Construction of the Sivasutras

Paul Kiparsky



Paul Kiparsky




Singh, Jaideva.

Spanda Karikas.

Motilal Banarsidass Publ., 1980.


Dyczkowski, Mark SG.

“The Stanzas on Vibration.”

State of NY Press, Albany (1992).



“The Relationships Between Music and Language According to Hindu Theory.”

The World of Music 17.1 (1975): 14-23.


A Mathematical Analysis of Panini’s Sivasutras



Click to access petersen_jolli_proof.pdf