Ratio Club: A Brief History of British Cyberneticians

Ratio Club: A Brief History of British Cyberneticians

 

The Ratio Club: a melting pot for British cybernetics

 

When Alan Turing was working at Manchester University, he was invited to join the Ratio Club, a dining club comprised of a mixture of biologists and engineers with an interest in cybernetics. There he was able to air and discuss new research and draw inspiration from an eclectic, yet extremely gifted group of individuals. Wired.co.uk examines the club and its influence on British science.

The Ratio Club was a group of young academics who came together to discuss cybernetics. It was founded in September 1949 by neurologist John Bates at the National Hospital for Nervous Diseases.

The club gathered in a basement room below the nurses’ accommodation and over beer and food participants would listen to a speaker, then have a discussion. The other members were a combination of neurobiologists, engineers, mathematicians and physicists and included Alan Turing. Professors were banned; anyone promoted to this position was supposed to resign their membership.

The aim was to keep the atmosphere informal and the discussion flowing.

Cybernetics is the science concerned with the study of systems and how they interact with each other — be they natural systems or machines. The club met regularly between 1949 and 1955, with a final reunion meeting in 1958. Founder Bates believes that cybernetic ideas could be important tools for developing new insights into the nervous system.

The Ratio Club was noteworthy because many of its 21 members went on to become extremely prominent scientists. These included: – Horace Barlow, the great-grandson of Charles Darwin who has become an enormously influential neuroscientists specialising in the field of vision. – Thomas Gold, who became one of the most prominent astrophysicists of the 20th century and gave the first explanation of pulsars among many other contributions. At the time of the Ratio Club he was working in the Zoology department of Cambridge University studying the inner ear. – John Pringle was a leading invertebrate neurobiologist and was the first scientists to get recordings from single neurons in insects.

He went on to become professor of zoology at Oxford University. – Albert Uttley, who researched radar and automatic tracking during WWII. He later became the head of the Autonomics Division at the National Physical Laboratory where he researched machined intelligence and brain modelling. Later he became Professor of Psychology at Sussex University. – William Grey Walter was a world leader in EEG research, discovering the theta and delta brain waves and developing the first EEG brain topography machine. When he wasn’t researching EEG, he was developing the world’s first autonomous mobile robots, called Elmer and Elsie, which he used to study ideas about brain function.

Many of the club members were interested in developing brain-like devices as a means to either understand biological brains or develop machine intelligence. As a result the conversation tended to focus on the mechanisation of the mind.

Most Ratio talks provided an opportunity for members to discuss their current research. Turing led three different talks. His second talk on Educating a Digital Computer, which took place on 7 December 1950, introduced the Turing Test and focused on how intelligent machines might be developed. Turing suggested using adaptive machines that could learn over their lifetime.

Turing’s third talk at the club, in February 1952, described his then unpublished work on reaction-diffusion models of morphogenesis. This launched him into new directions of theoretical biology and was incredibly influential in the field of computer modelling.

It was through Turing’s communications with fellow Ratio Club members that he expressed his interest in using a computer such as the ACE to study the brain. In a letter to psychiatrist William Ross Ashby, he said: “In working on the ACE I am more interested in the possibility of producing models of the action of the brain than in the practical applications of computing.”

The club had run its course by the summer of 1955, a year after Turing had died from cyanide poisoning after having been chemically-castrated. By that point many of the members’ research had been recognised internationally and cybernetics had become a respected discipline.

You can find out more about the organisation in The Ratio Club: A Hub of British Cybernetics, by Phil Husbands and Owen Holland.

 

Members of Ratio Club

  • W. Ross Ashby
  • John Bates
  • George Dawson
  • Thomas Gold
  • I.J. Jack Good
  • W. E. Hick
  • Victor Little
  • Donald Mackay
  • Turner McLardy
  • Pat Merton
  • John Pringle
  • William Rushton
  • Harold Shipton
  • D.A.Sholl
  • Eliot Slator
  • Alan Turing
  • Albert Uttley
  • W. Grey Walter
  • John Westcott
  • Philip M. Woodward
  • Horace Barlow

 

From The Ratio Club: A Hub of British Cybernetics

The definitive list of twenty-one members, with very brief details of expertise and achievements, is given below. Of course these summaries are far too short to do justice to the careers of these scientists. They are merely intended to illustrate the range of expertise in the club and to give a flavour of the calibre of members.

W. Ross Ashby (1903-1972), trained in medicine and psychiatry, is regarded as one of the most influential pioneers of cybernetics and systems science. Author of the classic books Design for a Brain (Ashby 1952a) and An Introduction to Cybernetics(Ashby 1958), some of his key ideas have recently experienced something of a renaissance in various areas of science including Artificial Life and modern AI. At the inception of the club he was director of research at Barnwood House Hospital, Gloucester. He subsequently became a professor in the Department of Biophysics and Electrical Engineering, University of Illinois.

Horace Barlow FRS (1921- ), a great-grandson of Charles Darwin, is an enormously influential neuroscientist, particularly in the field of vision, and was one of the pioneers of using information theoretic ideas to understand neural mechanisms (Barlow 1953, 1959, 1961), a direct consequence of his involvement in the Ratio Club. When the club started he was a PhD student in Lord Adrian’s lab at the department of physiology, Cambridge University. He later became Royal Society Research Professor of Physiology at Cambridge University.

John Bates (1918-1993) had a distinguished career in the neurological research unit at The National Hospital for Nervous Diseases, London. He studied human EEG in research into voluntary movement and became the chief electroencephalographer at the hospital. The Club was his idea and he ran it with quiet efficiency and unstinting enthusiasm.

George Dawson (1911-1983) was a clinical neurologist at the National Hospital, Queen square. At the time of the Ratio Club he was a world leader in using EEG recordings in a clinical setting. He was a specialist in ways of averaging over many readings which allowed him to gather much cleaner signals than was possible by more conventional methods (Dawson 1954). He became Professor of Physiology at UCL.

Thomas Gold FRS (1920-2004) was one of the great astrophysicists of the 20thcentury, being a co-author, with Bondi and Hoyle, of the steady state theory of the universe and having given the first explanation of pulsars, among countless other contributions. However, he had no time for disciplinary boundaries and at the time of the Ratio Club he was working in Cambridge University Zoology Department on a radical positive feedback theory of the working of the inner ear (Gold 1948) – a theory that was, typically for him, decades ahead of its time. He went on to become Professor of Astronomy at Harvard University and then at Cornell University.

I.J. (Jack) Good (1916- 2009) was recruited into the UK top secret code cracking operation at Bletchley Park during the second world war, where he worked as the main statistician under Alan Turing and Max Newman. Later he became a very prominent mathematician, making important contributions in Bayesian methods and early AI. During the Ratio years he worked for British Intelligence. Subsequently he became Professor of Statistics at Virginia Polytechnic Institute.

W.E. Hick (1912-1974) was a pioneer of information theoretic thinking in psychology. He is the source of the still widely quoted Hick’s law which states that the time taken to make a decision is proportion to the log of the number of alternatives (Hick 1952). During the Ratio years he worked in the Psychology laboratory at Cambridge University. He went on to become a distinguished psychologist.

Victor Little (1920-1976) was a physicist at Bedford College, London, who worked in acoustics and optics before moving on to laser development.

Donald Mackay (1922-1987), trained as a physicist, was a very highly regarded pioneer of early machine intelligence and of neuropsychology. He was also the leading scientific apologist for Christianity of his day. At the birth of the club he was working on a PhD in the Physics department of King’s College, London. He later became a professor at Keele University where he founded the Department of Communication and Neuroscience.

Turner McLardy(1913-1988) became an international figure in the field of clinical psychiatry. He emigrated to the USA in the late 1950s to develop therapeutic techniques centred around planned environments and communities. Later he became a pioneer of understanding the role of zinc in alcoholism and schizophrenia. At the inception of the club he worked at Maudsley Hospital, London.

Pat Merton FRS (1921-2000) was a neurophysiologist who did pioneering work on control theoretic understandings of the action of muscles (Merton 1953). Later he carried out a great deal of important early research in magnetic stimulation of the cortex for which he is justly celebrated (Merton and Morton 1980). During the Ratio years he worked in the neurological research unit at the National Hospital. He later became Professor of Human Physiology at Cambridge University.

John Pringle FRS (1912-1982) was one of the leading invertebrate neurobiologists of his day. He was the first scientist to get recordings from single neurons in insects, something that had previously been thought to be practically impossible (Pringle 1938). He did much important work in proprioception in insects, insect flight and invertebrate muscle systems. At the birth of the club he worked in the Zoological laboratory, Cambridge University. He subsequently became Professor of Zoology at Oxford University.

William Rushton FRS (1901-1980) is regarded as one of the great figures in 20thcentury vision science. He made enormous contributions to understanding the mechanisms of colour vision, including being the first to demonstrate the deficiencies that lead to colour blindness (Rushton 1955). Earlier he did pioneering work on the quantitative analysis of factors involved in the electrical excitation of nerve cells, helping to lay the foundations for the framework that dominates theoretical neuroscience today (e.g. Rushton 1935). He worked at Cambridge University throughout his career where he became Professor of Visual Physiology.

Harold Shipton (1920- 2007) worked with Grey Walter on the development of EEG technology at the Burden Neurological Institute, Bristol. He was the electronics wizard who was able to turn many of Walter’s less than precise designs into working realities. Later he became a professor at The University of Washington at St. Louis, where he worked on biomedical applications. At the time of the Ratio meetings, his father-in-law, Clement Attlee, was prime minister of Great Britain.

D.A. Sholl (1903-1960) did classic research on describing and classifying neuron morphologies and growth patterns, introducing the use of rigorous statistical approaches (Sholl 1956). Most of the classification techniques in use today are based on his work. He also published highly influential papers on the structure and function of the visual cortex. He worked in the Anatomy department of University College, London where he became Reader in Anatomy before dying young.

Eliot Slater (1904-1983) was one of the most eminent British psychiatrists of the twentieth century. He helped to pioneer the use of properly grounded statistical methods in clinical psychiatry. Slater’s work with Rudin on the genetic origins of schizophrenia, carried out in Munich in the 1930s, still underpins all respectable Anglo-American work in psychiatric genetics, a field to which Slater made many important contributions (Slater et al. 1971). He worked at the National Hospital for Nervous diseases, London.

Alan Turing FRS (1912-1954) is universally regarded as one of the fathers of both computer science and artificial intelligence. Many regard him as one of the key figures in twentieth century science and technology. He also anticipated some of the central ideas and methodologies of Artificial Life and Nouvelle AI by half a century – for instance, he proposed artificial evolutionary approaches to AI in the late 1940s (Turing 1950) and published work on reaction-diffusion models of the chemical origins of biological form in 1952 (Turing 1952). At the inception of the club he was working at Manchester University, where he was part of a team that had recently developed the world’s first stored-program digital computer.

Albert Uttley (1906-1985) did important research in radar, automatic tracking and early computing during WWII. Later he became head of the pioneering Autonomics Division at the National Physical Laboratory in London where he did research on machine intelligence and brain modeling. However, he also became well know as a neuropsychologist, having made several important contributions to the field (Uttley 1979). At the birth of the club he worked at TRE, Malvern, the main British military telecommunications research establishment. Later he became Professor of Psychology at Sussex University.

W. Grey Walter (1910-1977) was a world leader in EEG research. He discovered theta and delta brain waves and, with Shipton, developed the first EEG brain topography machine (Walter and Shipton 1951). At the time of the Ratio Club he was at the Burden Neurological Institute, Bristol, where, alongside his EEG research, he developed the first ever autonomous mobile robots, referred to as tortoises, which were controlled by analogue electronic nervous systems (Walter 1950a). This was the first explicit use of mobile robots as a tool to study ideas about brain function, a style of research that has become very popular in recent times.

John Westcott FRS (1920- ) made many very distinguished contributions to control engineering, including some of the earliest work on control under noisy conditions. He also worked on applications of control theory to economics which resulted in his team developing various models used by the UK Treasury. At the inception of the club he was doing a PhD in the department of Electrical Engineering, Imperial College, London, having just returned from a year in Wiener’s lab at MIT. He later became Professor of Control Systems at Imperial College.

Philip M. Woodward (1919- ) is a mathematician who made important contributions to information theory, particularly with reference to radar, and to early computing. His gift for clear concise explanations can be seen in his elegant and influential 1953 book on information theory (Woodward 1953). He worked at TRE, Malvern throughout his entire distinguished career (one of the buildings of the present day successor to TRE is named after him). In retirement Woodward has come to be regarded as one of the world’s greatest designers and builders of mechanical clocks (Woodward 1995).

Bates’ own copy of his typed club membership list of 1st January 1952 has many hand-written corrections and annotations (Bates 1952a). Among these, immediately under the main list of members, are the following letters, arranged in a neat column: ‘Mc’, ‘P’, ‘S’ and then a symbol that may be a ‘U’ or possibly a ‘W’. If we assume it is a ‘W’, then a possible, admittedly highly speculative, interpretation of these letters is: McCulloch, Pitts, Shannon, Wiener. The first three of these great American cyberneticists attended club meetings – McCulloch appears to have taken part whenever travel to Britain allowed. Wiener was invited and intended to come on at least one occasion but travel difficulties and health problems appear to have got in the way. The ‘W’, if that’s what it is, could also refer to Weaver, co-author with Shannon of seminal information theory papers and someone who was also well known to the club. Of course the letters may not refer to American cyberneticists at all – they may be something more prosaic such as the initials of members who owed subs – but it is just possible that Bates regarded them as honorary members.

It is clear from the membership listed above that the centre of gravity of the club was in the brain sciences. Indeed the initial impetus for starting the club came from a neurologist (Bates) who believed that emerging cybernetic ideas and ways of thinking could be very important tools in developing new insights into the operation of the nervous system. Many members had a strong interest in developing ‘brain-like’ devices, either as a way of formalizing and exploring theories about biological brains, or as a pioneering effort in creating machine intelligence, or both. Hence meeting tended to centre around issues relating to natural and artificial intelligence and the processes underlying the generation of adaptive behaviour – in short, the mechanisation of mind. Topics from engineering and mathematics were usually framed in terms of their potential to shed light on these issues. This scope is somewhat different to that which had emerged in America, where a group of mathematicians and engineers (Wiener, von Neumann, Bigelow, Shannon, Pitts) and brain scientists (Lorente de No, Rosenblueth, McCulloch) had formed an earlier group similar in spirit to the Ratio Club, although smaller and with a centre of gravity further towards the mathematical end of the spectrum. Their influence soon spread, via Frank, Mead, Bateson and others, into the social sciences, thereby creating a much wider enterprise (Heims 1991). This difference in scope helps to account for the distinct flavour of the British scene in the late 1940s and for its subsequent influences.

 

Key Sources of Research:

 

 

Ratio Club

https://en.wikipedia.org/wiki/Ratio_Club

 

 

 

The Ratio Club: a melting pot for British cybernetics

http://www.wired.co.uk/article/ratio-club-turing

 

 

The Mechanical Mind in History

Phil Husbands, Owen Holland, and Michael Wheeler

http://mitpress.universitypressscholarship.com/view/10.7551/mitpress/9780262083775.001.0001/upso-9780262083775-chapter-6

 

 

The Ratio Club: A Hub of British Cybernetics

 

Phil Husbands and Owen Holland

 

Click to access Ratio2.pdf

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.732.3291&rep=rep1&type=pdf

http://www.emeraldinsight.com/doi/full/10.1108/03684921111117951

 

 

 

Alan Turing and the Ratio Club

http://blog.wellcomelibrary.org/2012/02/alan-turing-and-the-ratio-club/

 

 

 

Warren McCulloch and the British cyberneticians

 

Click to access McCullochBritCyberneticsV3-final.pdf

Shapes and Patterns in Nature

Shapes and Patterns in Nature

 

There are so many colors, shapes, and patterns in nature.

  • Seashells
  • Animal Skins (Zebra, Leopard)
  • Butterflies
  • Shape of Plants
  • Flowers (Sun Flower)
  • Fruits (Pineapple)

How do we explain these from perspective of science?  There are several branches of science which have explored these questions for decades.  There are Reaction Diffusion Models and Cellular Automata models explaining development of patterns on seashells, plants and animal skins.  There is L-system developed by Aristid Lindenmayer to explain development of plants.  It is a fascinating subject.

 

From Exploring Complex Forms in Nature Through Mathematical Modeling: A Case on Turritella Terebra

There are several studies have been carried out in a number of scientific disciplines, such as mathematics, biology, paleontology and computer engineering to understand and decipher the relations of the seashells complex forms. Starting with Descartes, Figure 4 shows a time line in which many investigators having focused on the curves of these shells and their mathematical properties. They all outlined a number of mathematical relations that control the overall geometry of seashells.

After examining the existing seashell models in literature it is seen that they all followed Raup’s model which roughly abstracts the seashell form using three parameters; whorl (rate of expansion of the generating curve), distance (relative distance between the generating curve and axis of coiling), and translation (the change of the cone’s movement along an axis with respect to the whorl), an ellipse as the whorl cross-section as well. However, it is clear from the observations of actual shells (Figure 5) that the cross-section is more complex than the input that the three parameters allow. In the pursuit of realistic visualizations, Kawaguchi enhanced the appearance of shell models using filled polygons which represented the surface of shells more convincingly than line drawings. Similar techniques were used subsequently by Oppenheimer (1986). A different approach was adopted by Pickover (1989) who approximated shell surfaces by using interpenetrating spheres. Illert (1989) introduced Frenet Frames (Bronsvoort, 1985) to precisely orient the opening of a shell. His model also captured a form of surface sculpture. Cortie (1989) studied the pattern forms on the surface of the shell model (Meinhardt, 2003). Finally, the model of seashell geometry by Fowler et al. (2003) was similar to that introduced by Raup, and was the first to implement free-form cross sections using a Bézier curve (Farin, 2002 Rogers, 2001) as the input. It can be claimed that, studies above all focused on modeling the appearance of the shell surface.

All these approaches can be considered as a milestone for their era, as each model reflects the observation and tools of measurement, modeling and technologies of their time. In all these approaches seashells were modeled as a single surface, as a twodimensional object, and embedded in three-dimensional space. Today, such modeling research should be carried out employing observation tools, knowledge, information, and computational technologies to the maximum extent. For this reason, we developed a mathematical model that can be transformed into a computational model for further studies (such as overall behavior of shells, form-structure relations, form finding explorations etc.) to explore potentials of such optimized forms.

 

From Exploring Complex Forms in Nature Through Mathematical Modeling: A Case on Turritella Terebra

 

patterns

 

From Computational models of plant development and form

A broad program of using mathematical reasoning in the study of the development and form of living organisms was initiated almost 100 yr ago by D’Arcy Thompson (1942) in his landmark book On Growth and Form (see Keller, 2002, for a historical analysis). One of his most influential contributions was the ‘theory of transformations’, which showed how forms of different species could be geometrically related to each other. The theory of transformations was extended to relate younger and older forms of a developing organism (Richards & Kavanagh, 1945), but did not incorporate the formation and differentiation of new organs. This limitation was addressed a quarter of a century later by Lindenmayer (1968, 1971), who introduced an original mathematical formalism, subsequently called L-systems, to describe the development of linear and branching structures at the cellular level. By the mid 1970s, computational models based on Lsystems and other formalisms had been applied to study several aspects of plant development, including the development of leaves and inflorescences, and the formation of phyllotactic patterns (Lindenmayer, 1978). The questions being asked included the impact of distinct modes of information transfer (lineage vs interaction) on plant development, and the relationship between local development and global form. Similar interests underlied the independent pioneering work of Honda and co-workers on the modeling of trees (Honda, 1971; Borchert & Honda, 1984).

Another class of models was pioneered by Turing (1952), who showed mathematically that, in a system of two or more diffusing reagents, a pattern of high and low concentrations may spontaneously emerge from an initially uniform distribution. This was a surprising result, as it appeared to contradict the second law of thermodynamics: the general tendency of systems to proceed from more organized states toward disorder (the apparent paradox is resolved by jointly considering the reaction–diffusion system and its surroundings). Related models were introduced, under the name of activator–inhibitor and activator-substrate (depletion) systems, by Gierer & Meinhardt (1972), and extensively investigated by Meinhardt (1982). Reaction–diffusion systems showed how, in principle, molecular-level interactions may lead to morphogenesis and differentiation. In plants, reaction– diffusion-type models have been used to explain the patterning of trichomes in leaves and hair cells in roots (Digiuni et al., 2008; Savage et al., 2008; Jo¨nsson & Krupinski, 2010; Benı´tez et al., 2011). Nevertheless, the extent to which reaction–diffusion models apply to the plant kingdom appears to be limited (Kepinski & Leyser, 2005; Berleth et al., 2007). A significant role is played instead by mechanisms involving active transport of the plant hormone auxin (Section V). In some cases, such as the generation of phyllotactic patterns, this reliance on active transport is difficult to explain in evolutionary terms, as reaction–diffusion systems can generate the same patterns. Spatio-temporal coordination of other developmental processes, however, such as bud activation, requires long-distance signaling. Active transport may thus have evolved to overcome the limitations of diffusion, which is very slow over long distances (Crick, 1971).

In the last decade, computational modeling has become a mainstream technique in developmental plant biology, as reflected in numerous reviews (e.g. Prusinkiewicz, 2004b; Prusinkiewicz & Rolland-Lagan, 2006; Grieneisen & Scheres, 2009; Chickarmane et al., 2010; Jo¨nsson&Krupinski, 2010; Jo¨nsson et al., 2012). On the one hand, the sequencing of the human genome put in focus the chasm between knowing the genome of an organism and understanding how this organismdevelops and functions.Computational models bridge this chasm. On the other hand, successes of early conceptual models that relate patterns of gene expression to the form of animals (Lawrence, 1992) and plants (Coen & Meyerowitz, 1991) have prompted a quest for a comprehensive, mechanistic understanding of development (Coen, 1999). Current experimental techniques for tracking growth and observing marked proteins in living tissues (Reddy et al., 2004; Fernandez et al., 2010) are yielding a wealth of data that correlate molecular-level processes with plant development and form. Computational models play an increasingly important role in interpreting these data.

The use of models has been accelerated by the advancements in computer hardware, software, and modeling methodologies. General-purpose mathematical software (e.g. Mathematica and MATLAB), modeling programs built on the basis of this software (e.g. GFtbox, Kennaway et al., 2011) and specialized packages for modeling plants (e.g. the Virtual Laboratory and L-studio (Prusinkiewicz, 2004a), OpenAlea (Pradal et al., 2008) and VirtualLeaf (Merks et al., 2011)) facilitate model construction, compared with general-purpose programming languages. Furthermore, current computers are sufficiently fast to simulate and visualize many models at interactive or close-to-interactive rates, which is convenient for model exploration.

 

From The reaction-diffusion system: a mechanism for autonomous pattern formation in the animal skin

In his paper entitled ‘The chemical basis of morphogenesis’ Turing presented a ground-breaking idea that a combination of reaction and diffusion can generate spatial patterns (Turing 1952). In the paper, he studied the behaviour of a complex system in which two substances interact with each other and diffuse at different diffusion rates, which is known as the reaction–diffusion (RD) system. Turing proved mathematically that such system is able to form some characteristic spatio-temporal patterns in the field. One of the most significant deviations is s formation of a stable periodic pattern. He stated that the spatial pattern generated by the system might provide positional information for a developing embryo.

In spite of the importance of the idea in the developmental biology, his model was not accepted by most experimental biologists mainly because there were no experimental technologies available to test it. Therefore, most of those who took over and developed the Turing’s idea were applied mathematicians and physicists. They proposed various types of model that developed Turing’s original equation to fit real, naturally occurring phenomena (Meinhardt 1982; Murray & Myerscough 1991; Murray 1993; Nagorcka & Mooney 1992). Although the equations for each model differ, they all share the basic requirement of the original model; that is, ‘waves’ are made from the interactions of two putative chemical substances which we refer to here as the ‘activator’ and the ‘inhibitor’ (Meinhardt 1982).

 

Key Terms

  • Development Biology
  • Mathematical Biology
  • Biomathematics
  • Morphogenesis
  • Phyllotaxis
  • Evolutionary Biology
  • Nonlinear dynamical systems
  • Cellular Automata
  • Fractals
  • Iterated Systems
  • L-Systems
  • Pattern Formation
  • IFS (Iterated Functions Set)
  • Theoretical Biology
  • diffusion–reaction (DR) model
  • Systems Biology
  • Code Biology
  • Computational Biology
  • Algorithmic Biology
  • Complex Systems
  • Turing Patterns

 

 

Key People:

  • D’Arcy Wentworth Thompson
  • Aristid Lindenmayer
  • Alan Turing
  • Hans Meinhardt
  • Philip Ball
  • Przemyslaw Prusinkiewicz
  • Murray JD
  • Stephen Wolfram

 

 

Key Sources of Research:

 

On Growth and Form

Thompson D’Arcy W.

(1952)

 

 

The Algorithmic Beauty of Plants

Prusinkiewicz, Przemyslaw, Lindenmayer, Aristid

 

 

The Algorithmic Beauty of Seashells

Meinhardt H, Prusinkiewicz P, Fowler D

(2003)

(Springer, New York), 3rd Ed.

 

 

The Algorithmic Beauty of Seaweeds, Sponges and Corals

Kaandorp, Jaap A., Kübler, Janet E.

 

 

Mathematical Biology

Murray JD

(2003)

 

 

Models of biological pattern formation

Meinhardt H

(1982)

 

 

The chemical basis of morphogenesis.

Turing A

(1952)

Click to access Turing.pdf

 

 

Pattern formation by coupled oscillations: The pigmentation patterns on the shells of molluscs

Hans Meinhardt, Martin Klingler

 

 

The Self-Made Tapestry Pattern formation in nature

Philip Ball

1999

 

 

Models of biological pattern formation in space and time

Hans Meinhardt

2014

Click to access Meinhardt.pdf

 

 

Models of biological pattern formation

Hans Meinhardt,

Click to access Hans_Meinhardt.pdf

 

 

 

Cellular Automata, PDEs, and Pattern Formation

 

Click to access 1003.1983.pdf

 

 

The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation

By Gary William Flake

 

 

The Curves of Life

Cook, T

1979

Dover Publications, Inc. New York.

 

 

Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation

Shigeru Kondo1* and Takashi Miura

2010

Click to access reaction-diffusion_model_as_a_framework_for_understanding_biological_pattern_formation.pdf

Click to access kondomiura10science.pdf

 

 

The Hegemony of Molecular Biology

PHILIP KITCHER

 

Click to access kitcher99-hegemony.pdf

 

 

Modeling seashells

 

Deborah R. Fowlery􏰣, Hans Meinhardtz and Przemyslaw Prusinkiewicz

Click to access shells.sig92.pdf

 

 

The neural origins of shell structure and pattern in aquatic mollusks

Alistair Boettigera, Bard Ermentroutb, and George Oster

2009

Click to access 6837.full.pdf

 

 

Mechanical basis of morphogenesis and convergent evolution of spiny seashells

Régis Chirata, Derek E. Moultonb,1, and Alain Goriely

2013

 

Click to access 6015.full.pdf

 

 

The Geometry and Pigmentation of Seashells

S Coombes

2009

Click to access Seashells09.pdf

 

 

PATTERNS IN NATURE

Richie Khandelwal

Sahil Sahni

 

Click to access P7.pdf

 

 

Forging patterns and making waves from biology to geology: a commentary on Turing (1952) ‘The chemical basis of morphogenesis’

Philip Ball

 

Click to access f989a13264a455ec2898ed361b1c435b5f0c.pdf

 

 

Mollusc Shell Pigmentation: Cellular Automaton Simulations and Evidence for Undecidability

INGO KUSCH AND MARIO MARKUS

1995

 

Click to access KuschMarkus1996.pdf

 

 

Pattern Formation in Reaction-Diffusion Systems

Masayasu Mimura

 

Click to access 7adbe7e696d4ba9ad3a89fed4ba15549a091.pdf

 

 

The Natural 3D Spiral

Gur Harary and Ayellet Tal

 

Click to access 11-HararyTal.pdf

 

 

A Model for Pattern Formation on the Shells of Molluscs

HANS MEINHARDT AND MARTIN KLINGLER

1987

Click to access Meinhardt_1987.pdf

 

 

 

The reaction-diffusion system: a mechanism for autonomous pattern formation in the animal skin

 

Shigeru Kondo

Click to access The%20reaction-diffusion%20system_%20a%20mechanism%20for%20autonomous.pdf

 

 

Mechanical growth and morphogenesis of seashells

Derek E. Moulton, Alain Goriely and R ́egis Chirat

 

Click to access finalOR01.pdf

 

 

Scaling of morphogenetic patterns in continuous and discrete models

 

Click to access RasolonjanaharyMan_Sep2013_17293.pdf

 

 

On the Dynamics of a Forced Reaction-Diffusion Model for Biological Pattern Formation

A A Tsonis, JB Elsner, P A Tsonis

Click to access TsonisElsnerTsonis1989.pdf

 

 

 

A Model for Pattern Formation on the Shells of Molluscs

H M

Click to access 5_doc.pdf

 

 

Impact of Turing’s Work

Maini

Click to access 172.pdf

 

 

The possible role of reaction–diffusion in leaf shape

Nigel R. Franks1* and Nicholas F. Britton

 

Click to access 10972123.pdf

 

 

Pattern regulation in the stripe of zebrafish suggests an underlying dynamic and autonomous mechanism

Motoomi Yamaguchi*†, Eiichi Yoshimoto‡, and Shigeru Kondo

 

Click to access yamaguchi2007.pdf

 

 

Turing Patterns

P Ball

Click to access Turing_long.pdf

 

 

 

MODELS FOR PIGMENT PATTERN FORMATION IN THE SKIN OF FISHES

K.J. PAINTER

Click to access kjp006.pdf

 

 

 

Web Resource for Algorithmic Botony 

http://algorithmicbotany.org/papers/

 

 

FRACTAL GEOMETRY AND SUPERFORMULA TO MODEL NATURAL SHAPES

Nicoletta Sala

2013

Click to access ijrras_16_4_09.pdf

 

 

The Geometry of Seashells

Dr S Coombes

 

Click to access SeaShells.pdf

 

 

SEASHELLS: THE PLAINNESS AND BEAUTY OF THEIR MATHEMATICAL DESCRIPTION

JORGE PICADO

 

Click to access article.pdf

 

 

Models for the morphogenesis of the molluscan shell

 

Click to access molluscanshell.pdf

 

 

Modeling Seashell Morphology

 

Click to access AE-MKMpre.pdf

 

 

Exploring Complex Forms in Nature Through Mathematical Modeling: A Case on Turritella Terebra

 

Click to access ecaade2009_164.content.pdf

 

 

The Neural Origins of Sea Shell Patterns

Click to access Shells.pdf

 

 

Biological Pattern Formation : from Basic Mechanisms to Complex Structures

A. J. Kochy and H. Meinhardt

 

 

Form-Optimizing in Biological Structures The Morphology of Seashells

EDGAR STACH University of Tennessee

 

 

A Theory of Biological Pattern Formation

A. Gierer and H. Meinhardt

1972

 

Click to access gierer_meinhardt.pdf

 

 

Cellular Automata as Models of Complexity

Stephen Wolfram,

Nature 311 (5985): 419–424, 1984

Click to access 006_Wolfram1984.pdf

 

 

Website on Oliva Porphyria

http://oliva.porphyria.free.fr/menu%20GB.html

 

 

Evolution of patterns on Conus shells

Zhenqiang Gonga, Nichilos J. Matzkeb, Bard Ermentroutc, Dawn Songa, Jann E. Vendettib, Montgomery Slatkinb, and George Oster

 

Click to access 2012%20Evolution%20of%20patterns%20on%20Conus%20shells%20_E234.full.pdf

 

 

Theoretical aspects of pattern formation and neuronal development

http://www.eb.tuebingen.mpg.de/de/forschung/emeriti/hans-meinhardt/home.html

 

 

20+ Photos Of Geometrical Plants For Symmetry Lovers

http://www.boredpanda.com/geometry-symmetry-plants-nature/

 

 

Computational models of plant development and form

Przemyslaw Prusinkiewicz and Adam Runions

 

Click to access tansley.np2012.pdf

 

 

Periodic pattern formation in reaction–diffusion systems: An introduction for numerical simulation

Takashi Miura* and Philip K. Maini

 

Click to access 173.pdf

 

 

Dynamics of Complex Systems

Yaneer Bar-yam

http://necsi.edu/publications/dcs/index.html#fulltext