This is a partial reference guide for researchers interested in Alan Turing’s work on morphogenesis.
Back in 1998 I made for myself a list of the Turing morphogenesis papers, both published and unpublished. Quite soon afterwards, I put the list online, mainly because I could. The pages I hand-built in HTML then went down permanently around 2005 when I got tired of cleaning up after all the PHP hacks of the time. In those days it was having a blog, not being on Twitter, that made you an expert, and when I put the list online I suddenly found that people were referring to me as not just ‘a’ but ‘the’ specialist on the papers;. To be clear, I am an expert on them, and I will be upset if you don’t link to my page and invite me to give the conference plenary. But I’m definitely not the only expert and there have been a variety of voices commenting on the material since. This page does not survey all of those secondary works but does include my own.
In some places it is rather out of date. In particular I understand more of the theory of cylindrical lattices than I did when I wrote it . This page will be silently updated when it can be.
You are welcome to reproduce, under a Creative Commons CC-BY 3.0 licence without fee, any portions of this page for which I own the copyright. Portions of this page, especially quotations and images, may be copyright by others. The copyright in Turing’s work specifically is administered by King’s College, who you will now need to contact for such permission. Historically King’s have been admirably generous in allowing use of Turing material, which is all the more reason why you should get in touch with them if you plan to reproduce it.
A guide to the Turing morphogenesis papers
Published primary sources
The primary texts are
The Chemical Basis of Morphogenesis, AM Turing, Philosophical Transactions of the Royal Society B (London), 237, 37--72, 1952. (Here abbreviated CBM.) Also reprinted in Saunders.
Morphogenesis: Collected Works of AM Turing, Volume 3, ed PT Saunders, North-Holland, 1992. (Here abbreviated Saunders.) This contains in turn:
An introduction by Saunders
A reprint of CBM
A diffusion-reaction theory of morphogenesis in plants
Taken from MS AMT/C/7, which as discussed below, had been the basis of the article published as C. Wardlaw, ‘A Commentary on Turing’s Diffusion-Reaction Theory of Morphogenesis’, New Phytologist, 1953, and though explicitly influenced by Turing was written by Wardlaw.
The morphogen theory of phyllotaxis I (here abbreviated MTPI).
The morphogen theory of phyllotaxis II (here abbreviated MTPII).
The Saunders version of MTPI and MTPII were broadly taken from AMT/C/8 and AMT/C/9 respectively, which are typescripts prepared by Nick Hoskin, at some time between the 1950s and eventual publication of the long-planned Collected Works, probably earlier rather than later. Hoskin in turn worked primarily from Turing’s drafts in AMT/C25 and AMT/C26 with a number of minor editorial changes: for example changing 2 pi/J to kappa throughout; in my view he did an excellent job. While Sections 1–3 of MTPII were taken from the Hoskin draft, Sections 4–7 of the Saunders text were taken direct from AMT/C26.
‘The morphogen theory of phyllotaxis III’ (here abbreviated Richards).
Saunders implies by this title that Turing wrote this text and intended it as part of the same publication as MTPI and MTPII. This is unlikely because 1) it is not by Turing and 2) it is not about phyllotaxis. Instead it is an account of the work on Radiolaria set by Turing for his 1953 MSc student, Bernard Richards, which made up his MSc thesis still held by the University of Manchester.
Outline of the development of the daisy (here abbreviated DAISY).
Alan Turing: his life and work, ed S. Barry Cooper, Elsevier, 2013. (Here Cooper). I contributed a book chapter to this (see below) but I also made some editorial changes from the Saunders version of MTPI, MTPII, and DAISY. References of the form AMT/x are all to the King’s archive.
The Morphogen Theory of Phyllotaxis, (‘MTPI’, ‘MTPII’), the Radiolaria work (‘Richards’) and Outline of the Development of the Daisy (‘ODD’), are reprinted with some changes, most notably for ODD, from Saunders.
Minor points which I silently updated from the preface of Saunders into Cooper
pXIV: The (*) footnote at Saunders p38 is by Wardlaw and not Turing
pXVIII: Richards (1948) not Richards (1984)
pXXI (bottom of page): Integer, not real
pXXI: Turing did not suppose that most readers would be familiar with Fibonacci sequences and angles: he included a discussion in his draft.
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. I restored one of the section 8s, about Fibonacci numbers. The missing section defines the symbol $\omega$, used in subsequent sections, to be the golden mean $\omega\approx 1.618$. Other MTP1 changes (page refs to Saunders)
p52 garden pink (Dianthus) not maiden pink. Figure 1 appears to have been created by Saunders himself. Figure 2 was intended to be a diagram like Figure 1.
p66: third equation is a continued fraction (as it is in the MS C25/26 and in C8).
p73: the inverse lattice has a matrix given by the transpose of the inverse matrix of the lattice (as it is in C25/49 and in C8) ie the inverse of
a b
c d
is
A C
B Dp73: $A_u$ not $Au$.
p74: first equation should have last matrix element $-m \tau_1$ not $-m \tau_2$. (From p65 top equation).
p74: correct version of I.12.4 is given at C25/51 as $f(x,y)=\sum_{m=-\infty}^\infty \int_{u=-\infty}^\infty F_m(u) e^{ \frac {i m u}{\rho}+{i \nu y} }\, dy$.
MTPII typos:
p97: II.2.12.a,b not II.12.a,b
p98: right hand side of II.2.14 should have a plus sign before the $G$.
p100: l-5 missing citation to Bartlett
p107: $V=\bar{U^2}$ not $\bar U_2$.
The version of ODD in Cooper includes material not in Saunders. ODD is drawn from AMT/C24, primarily sheets 1–15. These sheets were numbered by ‘ROG’ (Robin Gandy), but this ordering was not adopted in full by Saunders who omitted a number of sheets and figures. I reordered the second half based largely on Gandy’s ordering which is more coherent and includes an additional material from C24/6 and C24/9, and recognised that C24/12 continues C24/11. C24/3 includes a figure (for $G(\eta^2)$) which I also placed in the text. C24/13 is not part of this draft article, while C24/21, C24/22 and C24/23 read as an alternate (perhaps earlier) draft of C24/7. C24/12 is indeed likely to continue C24/11 (but is again not in Saunders). A handful of typos were corrected from Saunders
p120: $(\rho,\theta z)$ changed to $(\rho\theta,z)$.
p121: denominator in final equation changed to $1-\nabla^2/R^2$.
Saunders does not reproduce the final handwritten line on C24/10: "(This function calculated in `Subgroup[?] smooth' )".
Richards has a short additional comment by Bernard Richards relative to the Saunders version.
My fee for both article and editorial work was a single copy of the book. Only after publication did I discover that Elsevier wouldn’t even spring for a physical copy and just sent me a PDF… But at least copyright in my contribution to Cooper was retained by me despite what Elsevier say in the book itself.
Unpublished primary sources
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. For a while there was a considerable amount of non-biological Turing material online at alanturing.net, somewhat surprisingly stamped with the copyright mark of the site maintainer. Although the site is extremely useful on other topics, from memory it never had any primary Turing material on morphogenesis.
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 College Modern Archives Centre. The catalogue itself is now at the Cambridge University archives tool, ArchiveSearch which may be more convenient for searching, and is in some cases more detailed.
This list only attempts to note all material relevant to morphogenesis.
A6
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.'
A7
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.
A8
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 on with the programme where he left off'
A11
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.
A21
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. …'
C7
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 for some reason 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.
So almost certainly the Manchester draft was written by Wardlaw under his name alone, was sent to Turing for comments, then revised by Wardlaw into C7 under both names, and then there was some agreement that it should be finally revised by Wardlaw for publication under his name alone. The Wardlaw paper was received by the journal on 21 May 1952, so C7 can be dated between January and May 1952.
After Turing’s death, Wardlaw continued to advertise and propound Turing’s theories to botanists; a substantial proportion of his 1968 Essays on Form draws on Wardlaw’s understanding of Turing’s work.
C8
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.
C9
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'.
C24 to C27
C24 to C27 are folders full of loose sheets which are unlikely ever to be completely made sense of, and most of which are unlikely to be very important if they are. But some provided me (& others) with vital clues as to Turing’s thinking, especially in combination with the Manchester papers.
THE NUMBERING OF THESE SHEETS WAS CHANGED AFTER I MADE THESE NOTES. (I know). THE NUMBERING BELOW IS THE OBSOLETE VERSION.
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 like DAISY refer to the themes to which I assigned them, back in 1998: I would probably do this differently now and may silently change these labels in the future. 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.
C24/1 DAISY `Outline of the development of the daisy'
C24/2 + DAISY
C24/3 + DAISY
C24/4 + DAISY
C24/5 + DAISY
C24/6 + DAISY
C24/7 + DAISY `Considerations governing the choice of parameter'
C24/8 + DAISY
C24/9 + DAISY
C24/10 + DAISY
C24/11 + DAISY
C24/12 + DAISY 'Early stages in pattern formation'. Incorporates lattice plot.
C24/13 EIGEN 'The above theory may be clarified'
C24/14 + EIGEN
C24/15 MTPII '13 Stationary waves in continuous tissue and
abstract space' ...
Surely continues MTP II (Where are 7-12?)
C24/16 + MTPII
C24/17 + MTPII (perhaps followed by C24/26v?)
C24/18 MTPII Ms: 'Forced waves' adds $\sigma s_{mn} \psi_n$ to master
r-d eqn
C24/19 MTPII Ms: 'An assumption that will often be helpful
is that some of the diagonal elements of C_t are very much smaller'
C24/19v DISCARD Discarded plot of $\half U_0+120 U_0^2 -3600 U_0^3$
C24/20 DISCARD 'discarded FIRSTART values" {where is it in my photocopy?)
C24/20v 2 line ms
and if an .. of the .. are much smaller
than the others the .. .. by the method of ....
C24/21 DAISY 'The equation chosen for computation'
C24/22 + DAISY
C24/23 + DAISY
C24/24 LATDYN 'The determinant is' (deals with $F$, $G$, $X$, $Y, $\xi$, $\eta$)
compare with C27/50
C24/24v 'The point where the Jacobian vanishes are those where there are
double roots of $\xi$, $\eta$ for given $X$, $Y$, $H$.
C24/25 Graph paper plot?
C24/26 DISCARD Discarded draft typescript (of \S 12).
C24/26v MTPII Refers to section 12.7 as general discussion of
form of solution
C24/27 MTPII MS discussion of optimum wavelength (cf MTP II.2)
C24/28 MTPII a draft similar to Saunders MTP II.2
C24/29 - MTPII follows C24/27 (not 28)
C24/30 MTPII MS is an (earlier?) draft of Saunders p101
C24/31 + MTPII
C24/32 MTPII MS
C24/33 MTPII '4. The equations applied to a plane'
(Earlier) draft of Saunders p99
C24/34 DISCARD discarded typescript (Penguin Science News)
C24/35 + DISCARD
C24/36 + DISCARD
C24/37 + DISCARD
C24/38 + DISCARD
C24/39 + DISCARD
C24/40 + DISCARD
C24/34v MTPII 'Effect of quadratic terms' a different draft of
processes in MTP II Section 2
C24/35v + MTPII
C24/36v + MTPII
C24/37v + MTPII and gets to the master equation.
C24/38v + MTPII ( perhaps is a redraft of C24/37v)
C24/39v + MTPII
C24/40v discussions of dU/dt=phi(U)+GU^2 -HUV dV/dt=psi_1(V)+AV^2 +CV^2
C24/41 DYN3
C24/42 DYN3
C24/42v family and friends
C24/43 'Stability and approximations'; contrary to catalogue, doesn't look
like Turing's hand and the coding form is similar
to that used by Hoskin at C.26/1
C24/44 `Hex stability with different values of $\gamma$'.
Stability of wavepattern with three given wavenumbers
C24/45 +
C24/46 'Damping due to the J term'
C24/47
C24/48 DYN3
C24/47v DISCARD Draft of 'solvable and unsolvable problems',
C24/48v + DISCARD Penguin Science News 31 (1954 pp7-23)
C24/49 LATDYN 'Lattice solutions and their stability' -
an unpublished section of MTP II?
C24/50 + LATDYN
C24/51 + LATDYN
C24/52 - LATDYN A nomogram (Fig X); follows C24/54
C24/53 - LATDYN
C24/54 - LATDYN mentions fig X
C24/55 - LATDYN follows C24/53
C24/56 + LATDYN
C24/57 may follow C24/56v
C24/58 + LATDYN specialises to hexagonal lattice
C24/59 + LATDYN
C24/60 + LATDYN
C24/56v LATDYN Tabulation of $U$,$U^2$, $UV$ etc for general inverse lattice
C24/60 MS working
C24/61 Contour plot on lattice
C24/62 LATDYN MS working
C24/62 LATDYN
C24/63 LATDYN
C24/64 LATDYN
C24/65 + LATDYN
C24/66 LATDYN
C24/67 LATDYN
C24/68
C24/68v LATDYN
C24/69
C24/69v LATDYN
C24/70 + LATDYN follows C24/69vC25
C25 contains mainly Turing's typescript of MTPI, used in C8, with some extras at the end.
C25/3
C25/10 interpolations in another hand
C25/13v 55,89,…(F0=0,F1=1,Fn+1=Fn+Fn-1) cf Saunders p57 - check with Hoskin
C25/14 interpolations in another hand
C25/16 Table 1 to follow defintion of 'convenient rule'
C25/17 Table 1
C25/18 has log (tau_2/tau_1) etc rather than xi - check with Hoskin version. Doesn't say limiting angle is xi=0.
C25/23 interpolations in another hand. Numbers `Naturally occurring
phyllotactic patterns' as $7.
C25/26 FIBONACCI A (different) section $8,
`Some properties of Fibonacci numbers'
C25/27 + FIBONACCI Includes Table 3 and a definition of $\omega$
C25/28 + FIBONACCI
C25/29 + FIBONACCI
C25/30 $9 lattice parameters has been renumbered from $8. 'As regards the order of the vectors it would be natural to require that the shorter vector be mentioned first. But however natural this convention may be it is inconvenient, because the first two
C25/31 vectors are often so nearly equal in length that it is difficult to distinguish which is the shorter….
Uses a b c d not a0 b0 c0 d
C25/32 add. Para in p64: Delta may be called the 'leaf area', zeta the 'scalenity factor', and phi the lattice angle: psi is not of sufficient importance to be given a name.
C25/34 hcf of a and c not b and d
C25/35 interpolations in another hand
C25/36 has `This argument shows incidentally that' before `Every unimodular'
C25/42 has $2\pi(1-\omega)$ not $2\pi(1-\omega^{-1})$ and similarly the next
C25/76 end of MTP I draft
C25/77 MS .."Stability of 2nd order equations"
C25/78 continues C25/77
C25/79 Review of Toth by Coxeter (not in CSAC cat); printed March 1954
C25/80 Plate legend. 'Plate 1 shows diagrammatically a stem typical of the
regular leaf arrangement in plants or, as in _Pinus sylvestris_,
of the small lateral branches'. Sketch of some lattices and 'Dianthus'.
C25/81 Saunders p50. Diagram of a sunflower with the florets numbered.
Traced from C25/82. Blue and red ink and pencil. Photo entire sheet.
C25/82 Photograph of a sunflower head (Saunders p48).
Photo at same scale as C25/81.
The Saunders reproduction is upside down (relative to the diagram).C26
C26 contains Turing's typescript of MTPII. Used by Saunders for sections 4-6 of his MTPII.
C26/1 Note
'Another $4 should be included here even though incomplete'.
C26/6 has $S_{ms}$ not $S_{mrj}$ etc in II.1.10,11;
equation numbering differs slightly. Cancelled section may be clearer?
C26/24 'Noise effects' is numbered $4 not $5
(and cites Bartlett for epidemics!)
C26/36C27
C27/1
C27/2
C27/2v Discarded MUCML coding sheet for IBSEN 4
C27/3
C27/3v DISCARD MUCML coding sheet for Test crystal
C27/4 Table of sqrt(5k^4-1), k=0.8,1.3
C27/5 Plot of log(omega,k) vs sqrt(5k^4-1)-2k^2
C27/5v DISCARD coding
C27/6 Graph of l^2/Delta vs 2 pi rho /n for different parastichy numbers
C27/7 LATTICE `Pessimum compressed hexagonal golden lattice'
C27/8 LATTICE `Half way compressible golden lattice'
C27/9 LATTICE `Golden rectangular lattice'
C27/10
C27/10v (reverse 'Table 2')
C27/12
C27/13
C27/14
C27/14v DISCARD MUCML Check sheet for BURStypescriptB (2)
C27/15 EIGEN Draft of C24/13, C24/14
C27/16
C27/17 EIGEN Draft of C24/14
C27/17v EIGEN Discarded eigenvalue table X,Y,Z, X', Y', Z' vs 0,1,1',2,2',3
C27/18
C27/19
C27/20
C27/21
C27/22
C27/23 EIGEN Eigenvalue table similar to C27/17v; see C24/13.
C27/24 'FIRCONES. paper theory
C27/25 'KJELL Theory'
C27/26 discarded
C27/27
C27/28 LATDYN
C27/29
C27/30 KJELL
C27/31
C27/31v where is my copy of this?
C27/32
C27/33
C27/34
C27/35
C27/36 RICHARDS 'Morphogenesis of cellular structure' [B Richards]
C27/37 + RICHARDS
C27/38 + RICHARDS
C27/39 + RICHARDS
C27/40 + RICHARDS
C27/41
C27/41v is trace output from a FIR code (?)
C27/42
C27/43
C27/44 'Modifications to change OUTERFIR track'
C27/45 Double foolscap sheet of lattice diagrams
C27/46 MTPI. "If H. of G. P." cf Saunders p72
C27/46v MTPI rough (p,q) discussion
C27/47 LATDYN Dynamics of 3 mode system $\eta_1, \eta_2, \eta_3$.
C27/48 + LATDYN
C27/49 + LATDYN (typescript) (probably follows C27/48).
C27/50 LATDYN Defines $F$ and $G$ for $\eta_1=\ldots=\eta_6=\eta$.
C27/51
C27/52 LATTICE plot of `Equilateral high parastichy no.
lattice subjected to squaring (2 0 \\ 0 1/2). note poor packing'
C27/53 LATTICE plot of ideal lattice and second best lattice
C27/53v LATTICE Plot of square lattice
C27/54 LATTICE Plot of (3,5) lattice
C27/54v DISCARDED
C27/55 LATDYN: linear theory for small perturbations;
early draft of MTPI $10.
C27/56 MTPI discarded MS draft 'Naturally occuring
phyllotactic patterns' of $8.
C27/57 + MTPI
C27/58 MS list of symbols ('Delta leaf area')
C27/59 v rough MS
C27/60
C27/61 MS 'Amplitude with 1 dimensional waves'
C27/62 MS 'Rate of change of wavelength '
C27/63 MS Plot of a function of U_0
C27/64 MS working to select a value of G; 'Random coeff'
C27/65 Clipped with C25/66 and C25/67. 'Choice of values of G and H'
C27/66 MTPI Numbered 23; title '$7 Measurements taken on some specimens'
C27/67 + MTPI MS Table
C27/68 MS Details of computer calcs (compare 68 and 2)
C27/69
C27/70
C27/71
C27/72
C27/73
C27/74
C27/75
C27/76
C27/77
C27/78
C27/79
C27/80
C27/81
C27/82
C27/83
C27/84
C27/85
C27/86
C27/87 KJELL cf C27/25 derivative or eqbm?
C27/88 WARDLAW Typescript of Wardlaw paper,
'Evidence related to the diffusion reaction
theory of morphogenesis', published as Wardlaw (1955).
...
C27/106 WARDLAWD4
Letter from AMT to Gandy. November 23 1952.
'Had quite a jolly time lecturing on fir cones' [in Cambridge].
D13
Letter from AMT to Philip Hall describing morphogenesis work.
K1 78
Letter from AMT to JZ Young. Quoted in entirety Hodges p436-437 and EST p144-147.
K3
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.
List of Manchester material
Catalogued at the Manchester National Archive for the History of Computing. Boxes NAHC/TUR/C2; C3. The foliation given here is my own, and does not reflect the archive organisation, if any. This list was based on photocopies sent to me by Jon Agar in the mid 1990s. Warning: the NAHC material includes, among other things, one folder labelled C2, a second labelled C3 and a third labelled C2/C3; apparently that is the right thing to do archivally.
M1 PATEQ 'Kjell Carlsen'
Tree of pattern developments
States: Hom, (0)R, (1)S, (1)H, (0+1) alt,
(1+1) alt, (0+2) decuss[ate].
M2 PATEQ (0+1) pattern cos z/rho+ cos (1/2rho) (sqrt(3)x+y)+ cos
M3 MTPI (eg p74) looks like an early draft
M4 + MTPI/PATEQ continues M3 and another version of tree at M1
M5 LATTICE Lattice parameters for hexagonal golden lattice
M6 MTPI Early draft for MTP I (p56)
M7 + MTPI
M8 PATEQ List of likely pattern moves
M9 Offprint address list
M10 Letter accompanying Wardlaw draft dated 15/1/52
N1'May 24' Computer output Copy OS@. YI'=/HE/ 1 set of 3 HI'=////
N2 +
N3 + continues; illegible notation dated 23/11/79
N4 + Copy RS@ HI'=A////
N5 + May 24(2) A contour map.
N6 + May 24(3).
N7 + May 24(4) I'1/I1=(3/4)^2 3steps.
N8 + May 24(5) Copy OSE WI'=//E/ HI'=//// 4 steps 1 step HI'=@/// 1 set of 3
N9 + May 24(7) 1 steps 4 steps I1/I=(3/4)^n(?) and a contour map
N10 MTPI `Remarks to be incorporated'
N11 LATDYN 'Stability' (of hexagonal arrangement)
N12 PATEQ `Principle vectors remain p. vectors.
Varying $\rho, I_0, \sigma$, not $H$ or $R$.
Sequence of 2d stability plots
N13 PATEQ Better sequence of stability plots including
pictures of (0), (1) etc
(ie put before M1: (1)S = 1 strip; (1)H = 1helix.
N14 PATEQ more stab plots.There is also a draft of Wardlaw (1952).
Themes in the Turing morphogenesis papers
This section was a first attempt to group and interpret the looser papers.
Warning! The foliation of many of these images has now been changed as part of the Digital Archive Project. The foliation numbers listed below are now inaccurate. More than that, in the last 20 years I have changed my mind about a few of these themes…
DAISY
C24/1 (entitled `Outline of the development of the Daisy') to C24/11 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). There then follows some further TS 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'). C24/12 is indeed likely to continue C24/11 although the next page, C24/13, is part of EIGEN not DAISY. C24/21, C24/22 and C24/23 read as an alternate (perhaps earlier) draft of C24/7.
A confusing point is that Turing says at one point (XX) that $I_0$ near to asymptote at one-third circumference but later that says wavelength less than 1/3 circumference.
Saunders does not reproduce the final handwritten line on C24/10: "(This function calculated in `Subgroup [?] smooth' )".
C27/28: R is relative diffusivity, s is scale of cooking variability relative to k0. R small and s infinite.
F(eta)=eta(I0 + 2 eta - 6 H eta2), with I0 variable.
EIGEN
C24/13 and C24/14 provide a worked example of the theory in section 1 of MTP II, and presumably intended to be incorporated there. It is an example of how orthogonal eigenfunctions can be chosen for a linear model of diffusion between four discrete cells arranged as three satellites around a central cell. The example is based on rougher workings at C27/17 and C27/15. There is a comparable set of working at C27/23 and C27/17v.
LATDYN
A theme dealing with the stability of lattice solutions. (i) C27/47,C27/48,C27,49,C27/50; (ii) C24/56v,C24/57,C24/58,C24/59,C24/60 are a coherent series; also (ii) C24/49, C24/50, C24/51, C24/53, C24/56.
Also N11.
C24/24 precedes C24/54 (or C27/50?).
C27/28
C24/62, C24/63, C24/64, C24/65, C24/66, C24/67, C24/68v, C24/69v, C24/70.
LATTICE
Parameters for given lattice classes: C27/7, C27/8, C27/9, M5. Plots of lattices: C27/52, C27/53, C27/53v, C27/54.
FIBONACCI
Properties of Fibonacci numbers. C25/26 to C25/29, followed by Hoskin and Richards in C8.
PATSEQ
Pattern formation sequences N12, N13, N14, M4, M1.
MTPI
C27/66 is labelled $7 and seems a different draft of Saunders $8. It has been numbered 23; C27/67
C27/56 MS version of Saunders \S8 predates C27/66. C27/55 cf Saunders \S10.
C27/46v rough (p,q) discussion
C27/46 see class.
The $k$ after I.15.1 is $\left( \frac{2\pi}{J} \right)^\half \left(\frac{\rho}{\eta}\right)^\half$ which is not the same as that on p??: $\left( \frac{2\pi}{J} \right) \left(\frac{\rho}{\Delta^\half}\right)$.
Is he enumerating lattice solutions to see which are stable? but `the phyllotactic systems of botany do not arise in this way' C24/49
Note that sections 7 and 13-18 of the C24 ms are not numbered in the TS.
N10: "Remarks to be incorporated"; M6; M7; M3.
MTPII
Section II: Saunders has sections 1-7; but C24/15 is entitled '13 Stationary waves in continuous tissue and abstract space' ... and surely continues MTP II. Followed by C24/16, C24/17 and perhaps by C24/26v?
Drafts of material in Saunders MTPII C24/30 p101 (Effects of random disturbances) C24/31 probably follows C24/32 as an earlier draft of Saunders pp102-105 C24/33 Earlier draft of Saunders p99
C24/26v Refers to section 12.7 as general discussion of form of solution
C24/19; C24/27 then C24/29; C24/26; C24/34v to C24/39v.
C24/18
WARDLAW
C27/88 to C27/106: Typescript of Wardlaw (1955).
KJELL
C27/87; C27/30
DYN3
C24/41, C24/42, C24/46 (?8)
RICHARDS
The spherical case done by B Richards: C27/36 to C27/40.
DISCARDS
C27/54v
C24/56(?4)v
C24/19v discarded plot of $\half U_0+120 U_0^2 -3600 _0^3$
C24/20 discarded FIRSTART ?
C24/26 discarded draft of MTP section 12
C24/34..C24/39 C24/40 C24/47v C24/48v are drafts of article on `word problem' in Penguin Science News, early 1954.
C27/2v Discarded MUCML coding sheet for IBSEN 4
C27/3v Discarded MUCML coding sheet for Test crystal
C27/5v Discarded coding
C27/14v Discarded MUCML Check sheet for BURSTSB (2)
C27/17v Discarded eigenvalue table X,Y,Z, X', Y', Z' vs 0,1,1',2,2',3
C27/56 discarded MS draft 'Naturally occuring Phyllotactic patterns'
C27/57 continues C27/56