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A guide to the Turing morphogenesis papers

Copyright (c) Jonathan Swinton 1998-2004.

This document provides a guide to all the material relevant to morphogenesis written by Turing, together with a small amount written by others, and consists largely of an annotated list of the King's College archive. Please let me know of the inevitable errors.

[June 2004: King's have reorganised their website, and some of the links to there are broken; but the material is still there for the persistent].

Contents

Sources

Published material

The two primary published texts are

Unpublished material

The major archive source is the Turing archive at King's College Cambridge, and there is a small amount of additional material at the Manchester National Archive for the History of Computing.

Annotated list of material in King's

This list complements the catalogue for the entire King's Turing archive, originally compiled by the Contemporary Scientific Archives Centre (CSAC), and now online and maintained by the King's archivist. It attempts to note all material relevant to morphogenesis.

AMT/A/6

From the Nature obituary by Wardlaw:

'He had already published a version of the theory for distributions round a ring, and was at work on the case of a cylinder; using a machine to solve the appropriate differential equations, he was hoping to exhibit the spiral pattern based on the Fibonacci series which are so frequently found in plants.'

AMT/A/7

Letter from Hoskin to Newman (Oct 3rd) re Newman obituary: suggests phrase 'in particular he showed that if a phyllotactic system is Fibonacci in character then it must change, if at all, to another Fibonacci system'. Discusses writing of Richards and Hoskin paper.

AMT/A/8

Letter from Gandy to Newman.

'I am particularly anxious that the work on morphogenesis on a cylinder should not be lost… When I was staying with Alan the weekend before Whitsun he also told me more or less where the computations had got to; but since his methods were so individual, he was unmethodical, I imagine it will be almost impossible for anyone to go on with the programme where he left off'

AMT/A/11

Times obituary 1954

…A mathematical theory of the chemical basis of organic growth which he had lately started to develop has been tragically interrupted, and must remain a fragment…

Manchester Guardian, (MHA Newman)

…In the last two years of his life he began to work out a remarkable chemical theory of the growth of living things. In this work he found the fullest scope for his mathematical powers, his great flare [sic] for machine-computing, and his power of tearing his way into a subject new to him-- in this case a chemistry of living tissues. This work, though it has already excited a good deal of interest, is still in an early, tentative stage so that it is doubtful if it carried though to completion by another hand. It remains a painful reminder of the loss that science has suffered.

AMT/A/21

Letter from Malcom MacPhail to Sara Turing. 18/9/1960.

..'Alan's zeta-function computer was a device for adding a large number of sines and cosines of various periods and amplitudes to get a numerical approximation to the zeta function. The gears, of which there were to be hundreds were to be approximations to the required periods. These were irrational numbers: Alan obtained rational approximations to them (that is ratios of integers such as 22/7 for \pi) by the method of continued fractions…'

Letter from Anthony D J Robertson to Mrs Turing 18/10/1966.

'… At the moment I am working with a mathematician. We are collaborating in designing experiments to test various models of brain function it seems likely that some of your son's unpublished work on morphogenesis is related to our approach. As far as I can tell he developed non-linear differential equations very similar to those with which my colleague is working. Would it be possible to see any of this material? I understand that the projected collection of his work, both published and unpublished, has not yet materialised, so if you could allow us to see the morphogenesis manuscript we would be extremely grateful. …'

AMT/C/7

Typescript of 'A diffusion-reaction theory of morphogenesis in plants'. This typescript contains a description for a biological audience of the pattern-formation mechanism spelled out mathematically in CBM.

The opening paragraph of C7 (quoted in the CSAC catalogue) states that AMT and Wardlaw are joint authors, but Turing probably contributed little to the writing of this paper. Saunders recognised this but included it in the Collected Works anyway.

The Manchester archive contains a earlier draft entitled Pattern in plants and a commentary on Turing's Diffusion-Reaction theory of morphogenesis, with Wardlaw as the only author, along with a note from Wardlaw dated 1/52 asking for comments on a preliminary draft `for massacring'. This Manchester draft does contain a few comments in Turing's hand, which were not incorporated into C7. C7 bears a few comments in Turing's hand which were incorporated into the paper eventually published as the single author paper Wardlaw (1952), with an acknowledgement to Turing at the end. Thus the Manchester draft was written by Wardlaw under his name alone, sent to Turing for comments, then revised by Wardlaw into C7 under both names, and then finally revised by Wardlaw for publication with Turing's name removed.

The Wardlaw paper was received by the journal on 21 May 1952, so C7 can be dated between January and May 1952.

AMT/C/8

Typescript of MTP I made by Hoskin and used by Saunders. This typescript is based largely on the Turing typescript in C25. The C25 typescript contains two section 8s, as does this typescript.

Turing used 2pi/J rather than kappa throughout C25, changed by hand (by Hoskin?) on this typescript.

A number of elements in C24 and C27 drafted by Turing are likely to have been intended for MTPI but were not incorporated here by Hoskin, who took a conservative but sensible approach to the text.  There are some corrections of the typescript by Hoskin and some by a third hand.

AMT/C/9

Typescript of MTP II made by Hoskin and used by Saunders for MTPII Sections 1-3.

There seems to have been an attempt by Hoskin to simplify the notation used in the derivation of the modal equation. C9/10 has a pencil comment on II.2.12 (II.2.11 in Saunders): 'indices wrong…unnecessary with simplified symbols'.

AMT/C/24

A plus (+) means that the sheet follows the one above in sequence, a minus (-) that it follows some other sheet. The labels after the foliation numbers refer to the themes to which I have tentatively assigned them, listed below.

The most coherent material here is that on daisies: see below and the daisy theme for comments on how it relates to the text in Saunders.

AMT/C/25

C25 contains mainly Turing's typescript of MTPI, used in C8, with some extras at the end.

AMT/C/26

C26 contains Turing's TS of MTPII. Used by Saunders for sections 4-6 of his MTPII.

AMT/C/27

The foliation of C27 was fortunately unchanged. 

AMT/C/4

Letter from AMT to Gandy. November 23 1952.
'Had quite a jolly time lecturing on fir cones' [in Cambridge].

AMT/C/13

Letter from AMT to Philip Hall describing morphogenesis work.

AMT/K/1 78

Letter from AMT to JZ Young. Quoted in entirety Hodges p436-437 and EST p144-147.

AMT/K/3

The diagrams folder.

K3/1 Labelled 1. A (2,3) parastichy. Blue and red wash and pencil.

K3/1v Labelled 3. A (3,5) parastichy. Blue and pink wash.

K3/2 Labelled 2. A (3,5) parastichy. Blue and pink wash with pencil.

K3/3 (Labelled 0) The daisy ring diagram. Blue and pink wash with pencil, on double foolscap. Note how the numbering starts at the centre.

K3/4 Labelled 4 and MRA. Real and Fourier space diagram. Double sheet with half sheet attached

K3/5 Labelled 5 and MR. Real and Fourier space diagram. Red and blue ink. Double sheet with half sheet attached.

K3/6 Labelled 6 and MRS. Real and Fourier space diagram. Red green and blue ink. Double sheet with half sheet attached.

K3/7 Contour plot diagram.

K3/8 Contour plot diagram.

Tiny fragment of same plot on reverse.

Coxeter letter

Not part of the King's archive, but recorded here for convenience, is a quotation from a letter Turing wrote on 28th May 1953, cited in HSM Coxeter, The Role of Intermediate Convergents in Tait’s Explanation for Phyllotaxis, Journal of Algebra, 20:167-172, 1972. There is no record of this letter in the King's archive; Professor Coxeter died in early 2003.
 

According to the theory I am working on now there is a continuous advance from one pair of parastichy numbers to another, during the growth of a single plant ... You will be inclined to ask how one can move continuously from one integer to another. The reason is this-on any specimen there are different ways in which the parastichy numbers can be reckoned; some are more natural than others. During the growth of a plant the various parastichy numbers come into prominence at different stages. One can also observe the phenomenon in space (instead of in time) on a sunflower. It is natural to count the outermost florets as say 21+34, but the inner ones might be counted as 8+13. Church is hopelessly confused about it all, and I don’t know any really satisfactory account, though I hope to get myself one in about a year’s time.

List of Manchester material

Held by the John Rylands University Library, Manchester, and briefly catalogued at the Manchester National Archive for the History of Computing. The catalogue is incorrect about folder C2 and C3.

The sheets are not foliated, and I have made up my own partial numbering, based on photocopies sent by Dr Jon Agar. I finally saw this archive in 2003, and these are my rough note supplements to the NAHC catalogue. To my surprise the interesting bits were the reprints Turing had in his possession.

NAHC/TUR/A. Biographical and Personal Papers

A1 Reprints of The Escalator Process for the Solution of Damped Lagrangian Frequency Equations, Phil Mag Ser 7 vol xxxvii 106-120, Feb 1946 and An Escalator Process for the Solution of Linear Simultaneous Equations, Phil Mag Ser 7 vol xxxviii 275, April 1947. MS note 'To A.M. Turing from J. Morris 3/12/48'.

A2 MHA Newman, "Alan Mathison Turing 1912-1954", Biographical Memoirs of Fellows of the Royal Society, 1 (1955).

A3 Sara Turing, Alan M. Turing (Cambridge: Heffer, 1959), pp.157. Inscribed inside: "C.W. Wardlaw (who, alas! wrote his obit. for "Nature")."

A4 BE Carpenter and RW Doran, The Other Turing Machine, Massey University Computer Unit Report No 23, typescript, pp. 41, 1975. A paper on the NPL ACE.

NAHC/TUR/B. Correspondence

B1 Photocopy of letter, 12 April 1950, from Turing to Rainford at Manchester University, requesting a duplicating machine and a desk calculating machine.

NAHC/TUR/C. Drafts, Working Papers, Reports

NAHC/TUR/C1

A.M. Turing, "Mark I Programming Manual" (ca.1950). Typescript, pp.110. 1976 Photocopy.

In contrast to the NAHC catalogue, there are actually four folders that follow, numbered C2, C2/C3, C3 and C3/C2.

NAHC/TUR/C2

Contains reprints by others:

The Mercer paper is about fibril packing.

From the Wrigglesworth paper:

'new plaques appear at more or less a fixed distance from existing plaques... each plaque appears to inhibit the emergence of new plaques within a certain distance';

'this process, like the Lisegang phenomenon, acting in a substrate of uniform potentialities, will result in periodic inductions'

Room card for Regent Palace Hotel, Piccadilly (printed 21/8/53) with name inscribed.

Reprint requests (6).

NAHC/TUR/C2/C3

In order:

Typescript of the Wardlaw paper (cf King's AMT/C/7); computer printout, N1 to N12 and Q1 to Q9 (see table)

NAHC/TUR/C3

Also reprint of Size in relation to internal morphology 1, CW Wardlaw, Trans Roy Soc Ed LIII-III-(25)-503. Note that many of these figures itch to be modelled by a reaction-diffusion model; Size in relation to internal morphology 3, CW Wardlaw, Trans Roy Soc Ed LVI-I(2);

NAHC/TUR/C3/C2

Ofprint: Pontecorro, Genetical analysis of cell organisation, Symp Soc Exp Biol Bristol July 1951.

NAHC/TUR/C4

A.M. Turing, "Proposed Electronic Calculator". Typescript of Turing's proposal for the ACE.

NAHC/TUR/D. Publications

Turing's own offprints of the period, all apparently unmarked.

The Saunders volume

The Morphogenesis volume edited by Saunders contains most of the most coherent manuscripts left by Turing, and is an extremely valuable resource. Saunders' task was largely thankless at the time, but anyone tackling the material now has reason to be grateful both to Saunders and to the herculean efforts of Hoskin who established most of the text. The reader should be aware of some odd editorial judgements by Saunders.

Introduction

Minor points:

MTPI

Saunders used the typescript in C8 for his version of MTPI. He took it largely verbatim, although a section was silently omitted and there are some misleading typos. Saunders omits one of the section 8s, about Fibonacci numbers (oddly, he then provides a description of them in his Introduction). The missing section defines the symbol $\omega$, used in subsequent sections, to be the golden mean $\omega\approx 1.618$.

There are many minor differences. Typos and other features include

MTPII

Saunders says 'there are two versions of Part II in the archive, one a draft of the other. The revised version ends after section 3'; the fuller version is the Turing typescript at AMT/C/26 and the revised version is the Hoskin typescript at AMT/C/9. Taken apparently verbatim into Saunders apart from some typos.

Typos:

Daisy

Taken from part of C24.

AMT/C/24/4 (entitled `Outline of the development of the Daisy') to AMT/C/24/14 are in Turing's typescript but with a page ordering later proposed by Gandy written at the top right (and corresponding to the foliation number). Saunders deleted at least two pages and changed the order of the remainder. Saunders takes the order 1,2,3,4,5,7,8,11,10, and omits 6 and 9. Gandy's order seems more logical. The missing pages AMT/C/24/9 incorporates a linking paragraph between the lattice model and the reaction-diffusion system, and AMT/C24/12 is also directly relevant. AMT/C/24/6 includes a figure (for $G(\eta^2)$) not reproduced in Saunders.

There then follows some further typescript which are likely part of the same paper. (In this folder, there is an unfoliated note by Hoskin whose place is lost asking 'Does this continue on'). AMT/C24/15 is indeed likely to continue AMT/C/24/14 (but is again not in Saunders), although the next page, AMT/C/24/16, is part of EIGEN not DAISY. AMT/C/24/27, AMT/C/24/28 and AMT/C/24/29 read as an alternate (perhaps earlier) draft of AMT/C/24/10.

Typos

Saunders does not reproduce the final handwritten line on AMT/C/24/13: "(This function calculated in `Subgroup[?] smooth' )".

Jonathan Swinton 01/06/04 |  Home | News | Sitemap | Copyright | Comments