Correspondence about Cell-Aggregates


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It can be observed, that in the second edition of his book, On Growth and Form, D’Arcy greatly expands Chapter 7 and 8: on The Forms of Tissues, or Cell-Aggregates. Much of this chapter considers the different configurations cell groups can take and which of these are the most stable, with reference to many examples in nature. One of the most interesting additions to this chapter can be seen on pages 610-611 of the 2nd edition. Here, D’Arcy describes how a topologist may describe a group of cells, or a cell-aggregate. It is an arithmetical method of representing and enumerating the cell-aggregate forms.

This section is taken, practically verbatim, from a letter from G. T. Bennett (ms26070):

Method1
Method2
Method3

In following letters, Bennett asks D’Arcy what he thought of the method he presented in ms26070, with increasing frustration at D’Arcy’s lack of response:

“I gave you a brief account of an arithmetical method of representing the cell-groups with a complete catalogue up to n=8. In this last matter I am now rather specially interested; so let me know, would you, at your modified leisure, what you think of it?” (ms26071)

“I feel sure that you will revert faithfully to cell-geometry before long: and then I recommend to your attention the arithmetical notation I proposed. There seems to me a “rightness” about it with which I hope you find agreeing.” (ms26072)

“My arithmetical formulation still awaits your comments.” (ms26073)

“Some day I am going to abduct you and keep you prisoned on bread and water until you make some remark on my arithmetical mode of presenting and enumerating the different modes of cellular subdivision (the islands with maritime counties). A voluntary surrender would be still in order.” (ms26075)

“For what I have been inviting you to return to is the matter of the island with counties all maritime (and 3-way junction points). I made a purely arithmetical presentation of this, enabling the ‘count’ to proceed methodically, and you were to comment. (Or so say I.)” (ms26076)

However, D’Arcy does include Bennett’s arithmetical method in his On Growth and Form. Perhaps he liked it all along but enjoyed teasing Bennett!

SegmentingEggs
Examples from On Growth and Form (page 377) of aggregates of 8 cells occuring in segmenting eggs

Bennett and D’Arcy discussed cell-aggregates in some detail before Bennett sent D’Arcy his arithmetical method. One could say they deepened each others’ understanding: in determining the best way to represent the cell-aggregate problem and in determining the actual number of configurations for a certain number of cells, n.

The first main correspondence on this topic, in the collection, is a letter from Bennett to D’Arcy (ms26064), in which Bennett first gives his description of the cell-aggregate as an island with maritime counties and asks D’Arcy if this is indeed the correct way to view the problem. The enumeration, or the number of cell-aggregate configurations, then depends on the number of triangulations of the polygon into n-2 triangles. Bennett notices that n=8, corresponding to aggregates of 8 cells, gives 12 triangulations, that is 12 cell-aggregate forms, but there are 13 cell-aggregates of 8 cells according to D’Arcy: clearly there is some discrepancy. He encloses a page of 12 polygon figures he has drawn and indicates ones which correspond to figures in On Growth and Form:

Polygons
Polygon figures in ms26064

Bennett follows this up with another letter (ms26065) now including some neater polygon diagrams done on triangulated paper. These diagrams make it clearer what Bennett’s 12 drawings in the previous letter represent:

TwoColumns
First two columns
ThreeColumns
Last three columns

Here the two columns on the left relate to page 375 of On Growth and Form (1st edition): the red lines show the partition diagrams (as in Fig. 158) and the black lines are reciprocal to the red. The black lines, in fact, correspond with the 12 figures given by the chords of the triangulated regular octagon Bennett sent before! The last three columns show 12 arrangements of 8 hexagons, each giving an island with 8 maritime counties in cyclic sequence: the dotted black lines repeat the black lines from before. In this letter Bennett also makes first mention of his arithmetical method involving taking adjacent triangles of the triangulated n-gon, regarding them as a quadrilateral and then replacing the diagonal with another diagonal.

In the next letter from Bennett to D’Arcy (ms26066) it becomes evident that they, in fact, both agree at 12 configurations for 8 cells, the earlier mistake arising from D’Arcy using Plateau’s incorrect assertion of “au moins treize”: at least 13.

In ms26069, Bennett comments on D’Arcy’s proposal to use the two equations linear in the F’s to get a genesis of all the cell-configurations. Bennett does not see how the condition is included that all the counties must be maritime. We are not sure how D’Arcy addressed Bennett’s concern but we can see that D’Arcy did include an equation linear in the F’s in his 2nd edition, on page 599.

Bennett mentions again to D'Arcy an arithmetical method for finding and cataloguing all the possible cases. He has discarded the cumbrous one from before (ms26065) and says he will present a new one. It is this slicker version that he describes in ms26070 and that D’Arcy decides to include in his book.

Correspondents

Geoffrey Thomas Bennett

Introduction

   

Overview & D'Arcy's Life

   

On Growth and Form

   

Heilmann & Shufeldt

   

Maths of Transformations

   

Correspondence

   

D'Arcy and Mathematics

   

Coordinate Transformations

   

Logarithmic Spirals

   

Forms of Cells

   

Forms and Mechanical efficiency

   

Shrinkage

   

Wartime and D'Arcy

   

The Leg as a Pendulum

   

Recreational Maths

   

Fibonacci Sequence

   

Cell-Aggregates

   

All Correspondence Links

Claxton Fidler

   

Eric Harold Neville

   

John Marshall

   

Alfred North Whitehead

   

Charles Robert Darling

   

Peter Guthrie Tait

   

William Peddie

   

Geoffrey Thomas Bennett

   

Dorothy Wrinch

   


Main Index Biographies Index


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Cammy Sriram and Edward Smith © July 2019
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School of Mathematics and Statistics
University of St Andrews, Scotland

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