Alan Turing and the sunflowers

Alan Turing was fascinated by mathematical patterns found in nature. In particular, he noticed that the Fibonacci sequence often occurred in sunflower seed heads. However, his theory that sunflower heads featured Fibonacci number sequences was left unfinished when he died in 1954, but some years ago a citizen science project led by the Museum of Science and Industry in Manchester and the Manchester Science Festival has found examples of Fibonacci sequences and other mathematical sequences in more than 500 sunflowers.
Inspired by this, I suggest a prompt to NightCafe, a text-to-image generator to celbrate Turing and his unstoppable mind:

The entropy and the halting probability problem

The third law of thermodynamics states:

It is impossible for any procedure to lead to the isotherm \(T = 0\) in a finite number of steps.

The theorem, discovered by Walther Nernst, is equal to say:

It is impossible for any process, no matter how idealized, to reduce the entropy of a system to its zero point value in a finite number of operations.

In classical thermodynamics we can define entropy, or the variation of entropy \(\Delta S\), with the following equation: \[\Delta S = \frac{\Delta Q}{T}\] where \(\Delta Q\) is the heat's variation and \(T\) is the temperature.

The quantum Zeno paradox

via Filozofy

The standard axioms of quantum mechanics imply that in the limit of continuous observation a quantum system cannot evolve.
(Andrew Hodges in Alan Turing: the logical and physical basis of computing - pdf)

Initially known as Turing’s paradox, in honor of the mathematician who formulated it in the 1950s, was subsequently identified as quantum Zeno effect, resulting an advanced version of the famous Zeno’s arrow paradox, whose phylosophical result is the negation of motion. A first formulation and derivation of the effect is found in Does the lifetime of an unstable system depend on the measuring apparatus?⁽¹⁾, while George Sudarshan and Baidyanath Misra were the first to identify it as quantum Zeno paradox. The two theoretical physicists established that an unstable particle will not decay as long as it is kept under continuous observation⁽²⁾. However they try to save goat and cabbage:

There is a fundamental principle in quantum theory that denies the possibility of continuous observation.⁽²⁾

On the other hand, Ghirardi, Omero, Weber and Rimini show that:

if the uncertainty relations are properly taken into account the arguments leading to the paradox are not valid.⁽³⁾

Alan Turing's declassified papers

@ulaulaman via @MathisintheAir about #AlanTuring on #arXiv

Recently Ian Taylor has uploaded on arXiv a couple of declassified papers by Alan Turing about statistics, probability and cryptography:

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1. The Statistics of Repetitions

In order to be able to obtain reliable estimates of the value of given repeats we need to have information about repetition in plain language. Suppose for example that we have placed two messages together and that we find repetitions consisting of a tetragramme, two bigrammes, and fifteen single letters, and that the total overlap was 105, i.e. that the maximum possible number of repetitions which could be obtained by altering letters of the messages is 105; suppose also that the lengths of the messages are 200 and 250; in such a case what is the probability of the fit being right, no other information about the day's traffic being taken into consideration, but information about the character of the enciphered text being available in considerable quantity?

2. The Applications of Probability to Cryptography

The theory of probability may be used in cryptography with most effect when the type of cipher used is already fully understood, and it only remains to find the actual keys. It is of rather less value when one is trying to diagnose the type of cipher, but if definite rival theories about the type of cipher are suggested it may be used to decide between them.

Turing's morphogenesis and the fingers' formation

by @ulaulaman http://t.co/9Q5rVkVzEc about #Turing #morphogenesis

On today Science's issue it is published a paper about the application of Turing's morphogenesis to the formation of fingers. In this period I'm not able to download the papers, so I simple publish the editor's summaries. First of all I present you the incipit of the paper by Aimée Zuniga, Rolf Zeller⁽²⁾

Alan Turing is best known as the father of theoretical computer sciences and for his role in cracking the Enigma encryption codes during World War II. He was also interested in mathematical biology and published a theoretical rationale for the self-regulation and patterning of tissues in embryos. The so-called reaction-diffusion model allows mathematical simulation of diverse types of embryonic patterns with astonishing accuracy. During the past two decades, the existence of Turing-type mechanisms has been experimentally explored and is now well established in developmental systems such as skin pigmentation patterning in fishes, and hair and feather follicle patterning in mouse and chicken embryos. However, the extent to which Turing-type mechanisms control patterning of vertebrate organs is less clear. Often, the relevant signaling interactions are not fully understood and/or Turing-like features have not been thoroughly verified by experimentation and/or genetic analysis. Raspopovic et al.⁽¹⁾ now make a good case for Turing-like features in the periodic pattern of digits by identifying the molecular architecture of what appears to be a Turing network functioning in positioning the digit primordia within mouse limb buds.

And now the summary of the results:

Most researchers today believe that each finger forms because of its unique position within the early limb bud. However, 30 years ago, developmental biologists proposed that the arrangement of fingers followed the Turing pattern, a self-organizing process during early embryo development. Raspopovic et al.⁽¹⁾ provide evidence to support a Turing mechanism (see the Perspective by Zuniga and Zeller). They reveal that Bmp and Wnt signaling pathways, together with the gene Sox9, form a Turing network. The authors used this network to generate a computer model capable of accurately reproducing the patterns that cells follow as the embryo grows fingers.

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(1) Raspopovic, J., Marcon, L., Russo, L., & Sharpe, J. (2014). Digit patterning is controlled by a Bmp-Sox9-Wnt Turing network modulated by morphogen gradients Science, 345 (6196), 566-570 DOI: 10.1126/science.1252960
(2) Zuniga, A., & Zeller, R. (2014). In Turing's hands--the making of digits Science, 345 (6196), 516-517 DOI: 10.1126/science.1257501

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Read also on Doc Madhattan:
Doc Madhattan: Matching pennies in Turing's brithday
Turing patterns in coats and sounds
Genetics, evolution and Turing's patterns
Calculating machines
Turing, Fibonacci and the sunflowers
Turing and the ecological basis of morphogenesis

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via phys.org

Turing and the ecological basis of morphogenesis

about #AlanTuring #morphogenesis #ecology

It is recently published a paper (in open access) dedicated to the morphogenesis. The work (and the model) is inspired by Alan Turing:

Our results demonstrate that a simple model implementing counteracting processes acting on different length-scales can indeed recreate branching patterns similar to those of swarming colonies. The kernel-based phenomenological model presented here draws from ecological theory, which has long recognized the relevance of distance-dependent processes as drivers for spatial patterning (Levin 1992). Many concepts from patterning in ecology are intimately related with the chemical basis of morphogenesis first proposed by Turing (1952), who first explained that counteracting positive and negative chemical processes acting on different length-scales can lead to symmetry-breaking that triggers biological patterning (Morelli et al 2012). The model presented here is inspired by Turing's findings but uses the spatial kernel approach of recent population ecology models (e.g. Rietkerk et al 2004, Lindstrom et al 2011) rather than reaction–diffusion processes.

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Pan Deng, Laura de Vargas Roditi, Dave van Ditmarsch, Joao B Xavier (2014). The ecological basis of morphogenesis: branching patterns in swarming colonies of bacteria New Journal of Physics, 16 (1), 1-16 DOI: 10.1088/1367-2630/16/1/015006

Turing, Fibonacci and the sunflowers

posted by @ulaulaman about #FibonacciDay #AlanTuring #TuringSunflower

Today is the american Fibonacci's day, so it could be a good day to write something about one of the last work by Alan Turing.

One of a number of problems [Alan Turing] was trying to solve was the appearence of Fibonacci numbers in the structure of plants.⁽¹⁾

The problem was knwon as the Fibonacci phyllotaxis, and we can state it in this way:

the spiral shapes on the heads of sunflowers seemed to follow the Fibonacci sequence, prompting [Turing's] proposal that by studying sunflowers we might better understand how plants grow

Turing wrote his interest in a letter to the zoologist JZ Young:

About the point (iii) Turing wrote in another letter:

Our new machine is to start arriving on Monday. I am hoping to do something about 'chemical embyology'. In particular I think I can account for the appearence of Fibonacci numbers in connection with fir-cones.⁽¹⁾

The last year Jonathan Swinton, during the Manchester Science Festival in October, announced the results of the great experiment about the Turing's sunflower:

Calculating machines

posted by @ulaulaman about the italian exhibition dedicated to #AlanTuring

The Museum of Science and Technology Leonardo da Vinci in Milano was founded 60 years ago on the 15th February and yesterday this birthday was celebrated with a free admission from 14:30 until 23:30. Inside the museum you can participate in various interactive laboratories usually offered to visitors and visit exhibitions presented in the rooms of the Museum. While some of the exhibits are permanent, such as the one dedicated to Leonardo da Vinci, others are temporary, such as the one dedicated to Alan Turing, which centenary was celebrated during the last year.
The exhibit occupies three small rooms, nothing incredibly wide or long to see, but certainly very interesting. The exhibition, in fact, tells about the evolution of calculating machines, starting from pascaline by Pascal until you reach the final room, explicitly dedicated to the great British mathematician and logician, where thanks to some infographics, the life of Turing is telled to the visitors. Meanwhile on the wall a series of videos posted to YouTube and dedicated to the great scientist are projected.
To complete the room there is also an Enigma machine:

The Enigma machine was patented in 1918 by the German Arthur Scherbius which takes and expand the operating principle of the Italian Leon Battista Alberti's cipher disk.
Created to facilitate the encryption of communications in the financial and commercial, it was officially presented to the Berne International Postal Congress in 1923. Continuously improved, it is adopted first by the German navy and later by the army to make secret military communications. Easy to use, it enjoys a solid reputation as indecipherable that it ensure the wide diffusion.
Enigma is based on the whole system of secrecy of Nazi Germany. The attempt to break the cipher is one of the most famous stories of counterintelligence of the twentieth century. The first attempts are due to the Polish Marian Rejewski but with the capture of the specimen aboard the German submarine U-boat 110, which is able to provide important information to the team of scientists at Bletchley Park, working night and day to this purpose. Alan Turing is among these. Thanks to his skills as a mathematician and cryptographer, Turing can make an electromechanical computer, The Bombs, ables to perform the task with a remarkably efficient. Subsequently he developed a more powerful computer, Colossus, that at the end of the war will be destroyed on the orders of Winston Churchill, and its existence is only made ​​public uprisings years after the end of the war.

In the gallery below there are the photos I took during the visit to the museum:

Genetics, evolution and Turing's patterns

posted by @ulaulaman about #Turing #genetics #biology #evolution #morphogenesis reaction-diffusion system

I've just written a post about the theory of patterns in nature started by Alan Turing, and I describe the reaction-diffusion system: in the system there are an activator and an inhibitor molecule of morphogenesis. The dynamics between activator and inhibitor generates tha patterns and we can describe it with the following mathematica relation: \[u_t = d \Delta u + f (\gamma, u)\] where $u$ is the position of the gene, $u_t$ the diffusion speed, $d$, $\gamma$ real constants.
It's really interesting observe that recently the Turing model about patterns was applied also to the study of the evolution of genes, in particular to study the generation of digit patterning.
The story start from the Hox genes:

Hox genes are a group of related genes that control the body plan of the embryo along the anterior-posterior (head-tail) axis. After the embryonic segments have formed, the Hox proteins determine the type of segment structures (e.g. legs, antennae, and wings in fruit flies or the different vertebrate ribs in humans) that will form on a given segment. Hox proteins thus confer segmental identity, but do not form the actual segments themselves.

So the team try to mute some of the Hox genes in order to see if the number of digits decrease, but they surprisingly observed that they can add more and more digits in their mutant mouses (they arrived at 14 digits!). And they can explain this behavour with the reaction-diffusion model: in the following picture you can see the experimental results (the first three rows) and the computer simulation that used Turing's model:

Turing patterns in coats and sounds

posted by @ulaulaman #AlanTuring #mathematics #biology #genetics #ecology

One hundred years ago was born Alan Turing. He was known essentially for his role during the Second World War: he encrypted the Enigma machine. But He is also a brillant mathematichian and today I would try to describe one of his better model, that today biologists are applying to their research field.

A vibrating object tends to vibrate at certain preferred frequencies, called natural frequencies. These frequencies depend on properties such as the density and tension of the vibrating object. Mathematicians and physicists can determine the natural frequencies of an object when they know the values for these other properties. This article describes new work being done to solve the reverse problem - calculation of properties such as density when the natural frequencies of the object are known.

Elizabeth Veomett about Good Vibrations by Joyce R. McLaughlin. American Scientist, July - August 1998

Cymatics was the study of the waves' patterns. The first interested in this subject was Galileo Galilei:

As I was scraping a brass plate with a sharp iron chisel in order to remove some spots from it and was running the chisel rather rapidly over it, I once or twice, during many strokes, heard the plate emit a rather strong and clear whistling sound: on looking at the plate more carefully, I noticed a long row of fine streaks parallel and equidistant from one another. Scraping with the chisel over and over again, I noticed that it was only when the plate emitted this hissing noise that any marks were left upon it; when the scraping was not accompanied by this sibilant note there was not the least trace of such marks.⁽¹⁾

Some years after Galilei (1680), Robert Hooke

was able to see the nodal patterns associated with the modes of vibration of glass plates.

In 1787 Ernst Chladni repeated Hooke's experiments and published his results in Entdeckungen über die Theorie des Klanges (Discoveries in the Theory of Sound). Finally in 1967 Hans Jenny published Kymatik (Cymatics), a book based on Chladni's work, and cymatics became an interesting science, in particular for artists! For example, Jeff Volk, poet, writes an interesting article about Jenny and the pattern of sound: From Vibration to Manifestation (pdf). In particular he presents an image from Alexander Lauterwasser's Water Sound Images

Pay attention: following Lauterwasser and Volk we could explain the pattern of leopard's coat, but the first explanation come from one of the Alan Turing's paper The Chemical Basis of Morphogenesis⁽²⁾. In this paper Turing is interested in the formation and development of path in biology (the so called phenomenon of morphogenesis).

Any pattern or shape observed in nature, even though governed by genetics, is most likely produced by an unknown mechanism. Thus, determining these mechanisms that generate pattern and shape in organisms is an important goal of theoretical biologists.⁽³⁾

The most used model for this type of systems is the reaction-diffusion system, described by the following formula: \[u_t = d \Delta u + f (\gamma, u)\] where $u$is the position of the gene, $u_t$ the diffusion speed, $d$, $\gamma$ real constants. We can write two similar formulas for every morphogene in the system.

Reaction-diffusion models are particularly compelling with regard to their ability to capture complex evolving patterns.⁽³⁾

Similar equations are really complex to analyze, due to their local and general phenomena. In an intuitively way, we can see the pattern formation like a challenge between reaction mode and diffusion mode. In his paper, Turing

suggested that a system of reacting and diffusing chemicals (morphogens) can interact to produce stable patterns in concentration (Turing patterns).⁽³⁾

Or in a more simple way: we can imagine the presence of an activator molecule of the morphogenesis. This molecule will be produced more and more thanks an autocatalysis process, but the activator will produce also an inhibitor, that will limit the production of the activator. The dynamics between activator and inhibitor will generate the pattern observed in nature (for example tigers' stripes). Tipically the two diffusion velocities are different, so we can explain the great variety of patterns.

Matching pennies in Turing's brithday

Today is Alan Turing's birthday. Turing was the mathematician who breaked Enigma code during the Second World War and he posed the basis for modern computers. In a rare coincidence I play today matching pennies, a game of simulation of a launch of a penny against computer. Indeed player must choice head or tail and confrount his choice with computer. In the first page there is the following suggestion:

The best strategy to win the game is to be as random ad possible.

When you conclude all three levels, you can read the project details. We can read:

It is common knowledge that human subjects have difficulty in generating random sequences. This hypothesis has been studied in detail for some time by psychologists and mathematicians. This project takes that theory further by studying it from a game theoretical point of view and supposes that humans are less able to randomise under increasing pressure than under normal circumstances.
Matching pennies is an ideal game for this study because there is no pure strategy in the game that gives one player an advantage over the other.
(...)

I play the game. In the first level I'm losing when I'm trying to apply the suggested strategy, but I win (also the two next levels) when I apply the following simple strategy: change my choice when computer wins!
I don't know if my strategy is casual or it is the best to simulate a casual strategy, but this is the results. And you?

Thanks to Marco Gaudenzi for sharing the game.