History

Introduction

Concerning the history and background to parquet deformation of the William Huff model1 (all further references to the ‘Huff model’ here are based on this description), this is not a straightforward task to document as there are certain grey areas. Although there is indeed a ‘big bang’ moment, in 1960, by Peter Hotz/William Huff, there are indeed a few precursors, from as far back as 1886, some more tenuous than others, intended or otherwise, of which I document, on the grounds of including ‘all’, no matter how tenuous, given that there are so few. Indeed, some are so tenuous that others may disagree as to their inclusion. So be it. I would be very much indebted to anyone bringing to my attention any more, no matter how tenuous. The concept may very well appear in some obscure pattern/design book or other fields that is not necessarily well documented in the mathematical literature.

What is perhaps surprising, to me, is that no one had effectively thought of the concept before 1960. Despite many tiling studies, especially since the late 19th century, when the subject per se essentially took off, no one had apparently thought to have such a simple, and obvious, next-stage concept of a ‘dynamic’ tiling. And even after the 1960s ‘true’ beginning, it was still largely unknown, confined to obscure design journals, and certainly not the more likely field of mathematics. Only in 1983, with the publication of Douglas Hofstadter’s article in the more mainstream Scientific American [*], did it attract popular and mathematical study. Since then interest in the subject has exploded, in relative terms, with over 200+ practitioners, albeit this is still relatively low compared to the interest in tiling as such.

Note that this is very much an initial investigation, and like most of such investigations, will likely be improved upon a return after a period of reflection. Even of a first draft, it could still be improved further. However, the ‘dreaded drift’ is occurring, a term I use when my studies become effectively becalmed, and therefore I bring proceedings to a halt for now. Nonetheless, it remains an excellent piece. As far as I am aware, there is no previously written history of any kind in the field.

1 Defined by myself as a geometrical tiling metamorphosis, typically in 1-dimension as in a strip form, as an abstract entity (i.e. without any Escher-like element). Huff’s definition, in Scientific American, is more restrictive, with conditions of handedness).


Contents

1. 1886, Lewis Foreman Day, forerunners in The Anatomy of Pattern
2. 1903, Lewis Foreman Day, forerunners in Pattern Design
3. 1917, D’Arcy Wentworth Thompson, first true parquet deformation, by accident or design
4. 1937+, Maurits Cornelis Escher (1), forerunners
5. 1960, Peter Hotz/William Huff, the first (1-dimensional) true parquet deformation, by intention
6. 1964, Arthur Carlson/Thomas C. Davies/David Oleson, the first 2-dimensional instance
7. 1965, William Huff, the first reference to parquet deformation in print
8. 1967-1968, Maurits Cornelis Escher (2), parquet deformation
9. 1968, William Huff, the first definition in print
10. 1983, Douglas Hofstadter, the first popular account in print
11. 2003+, Craig Kaplan, advanced concepts
12. 2015, Edmund Harriss, the first non-periodic (and incidentally 2-dimensional) parquet deformation


1. 1886, Lewis Foreman Day The Anatomy of Pattern (1)

Some early forerunners, perhaps generously stated (as others may disagree) of parquet deformation can be seen in the work of Lewis F. Day (1845–1910), a prominent exponent of the Arts and Craft movement in the UK. However, it is not entirely clear what he is striving for. For certain, what we consider parquet deformation is not explicit, either in the diagrams or commentary. These appear in two books, in slightly different forms. The first appears in Text Books of Ornamental Design I, The Anatomy of Pattern, of 1887 [*] and the second Pattern Design, of 1904 [*], which repeats two of the diagrams, and of which for the sake of clarity I discuss each book individually. 

Plate 11, of The Anatomy of Pattern shows two precursors to parquet deformation, with the upper of less connection than the lower. As the diagram was apparently drawn in 1886, I use this date rather than when the book was published, in 1897.  

Upper, described by Day on the plate as ‘zigzag’ and ‘wave’ are somewhat tenuous, even within a generous definition of parquet deformation. A repeat zigzag line of a sharp angularity steadily progresses into a shallower wavy line. This (obviously) lacks a tiling element and so could easily (and fairly?) be summarily dismissed. However, as it is a rudimentary ‘development’ I have included for interest's sake.

Lower, described by Day on the plate as (i) ‘Wave-lines opposed’ (ii) ‘Joined to form ogee’ and (iii) ‘Inter-laced to form a net. An open repeat lattice based on ogee lines abruptly closes to form a tiling in the centre, of which this then again abruptly closes so the lines overlap. In effect, this is of three elements butted together; there is no smooth transformation. Prima facie, this is indeed of a better resemblance to the Huff model than the upper. But again, does this qualify as a parquet deformation? Cases for and against can be made. I am ambivalent about this. Day is not explicit in what he is doing. The commentary text in the book gives:

One may very easily deduce many of the common curvilinear patterns directly from angular motives. (Plate 11.) The wave, for example, is a zigzag, just blunted at the points. Soften the lines of the hexagon, and you have the ogee. Interlace straight rods, and you get waved lines, as may be seen in the perspective view of the common hurdle. Round the corners of the hexagon or octagon, and you arrive at a rude circle. The relation of the hexagon or octagon diaper to the diaper of circles is obvious. Presumably, the busy bee, if one may suggest such a thing without irreverence, only works in a circle, and the hexagonal form of the cells of the honeycomb is simply the result of gravitation; just as you find that cylinders crowded all become hexagonal prisms.

Metamorphosis, or development, is not mentioned. It could even be argued that he is simply concatenating his ideas as a single diagram. I am ambivalent overall on how to assess this.

Fig. 1. Plate 11. The Anatomy of Pattern


2. 1903, Lewis Foreman Day

As is his wont, Day then reuses/modifies diagrams in earlier publications, of both the upper and lower  The Anatomy of Pattern, in Pattern Design, to which he adds another diagram. Given that two of these are effectively the same as in The Anatomy of Pattern, I refer the reader to the commentary above, although for the sake of clarity I show all three diagrams. The new diagram, p. 26, is captioned by Day, p. 26, ‘Simple and More Complicated Trellis Lines’, in a chapter on ‘The Octagon’. Oddly, it is not discussed in the text! Again, it is by far from clear what Day is intended. The first half begins with a square tiling, then shows a series of overlapping squares, and then again overlapping rhombs. The transitions are abrupt. As such, best described as a forerunner ‘of sorts’.


Fig. 1a. P. 26 Pattern Design. Simple and More Complicated Trellis Lines

Fig. 1b. P. 27, Pattern Design. Zig-Zags developing into Wave Lines



Fig. 1c. P. 34. Pattern Design. Wave Lines, Ogee Diaper and Interlaced Ogees, Giving Hexagonal Shapes

3. 1917, D’Arcy Wentworth Thompson

In his famed book On Growth and Form, D’Arcy Thompson (1860–1948) a biologist, classical scholar, mathematician, and man of letters, shows the first known unambiguous parquet deformation, p. 335, fig. 133. However, it is far by clear he had in mind or was intending of what is now known as parquet deformation.


Fig. 2. On Growth and Form, Fig. 133, p. 335

This is the only instance of its type in the book (or so far as I am aware in any of his scientific publications; albeit I have only studied Growth and Form).

This is captioned: Fig. 133. Diagram of development of “stellate cells,” in pith of Juncus. (The dark, or shaded, areas represent the cells; the light areas being the gradually enlarging “intercellular spaces.”

Of note is Thompson’s use of ‘development’ in the caption, which suggests he was aware of the latter-day parquet deformation nature. The accompanying text to the image is rather long-winded (shown in full in the references), and quite what Thompson is presenting here is unclear. In short, it appears to be a natural process viewed through a binocular microscope. This is also discussed with the same illustration in F. T. Lewis, but again, it's all rather long-winded, and in any case, I am hindered in that it is not my field. From (lightweight) research as to stellate cells and Juncus, I suspect that it is a greatly simplified, perhaps idealised, diagram (can any authority confirm this)?

And what are stellate cells? As the title suggests, there is a starlike element. Wikipedia (re the central nervous system):

Stellate cells are any neuron in the central nervous system that have a star-like shape formed by dendritic processes radiating from the cell body...

Also of note here is the book and Thompson himself, of whom Huff was a great admirer, and had read his books. Natural speculation is that Huff must surely have been inspired in his parquet deformations by this diagram (he explicitly mentions Growth and Form). However, this does not appear to be so. He does not refer to it in his writings and clearly states the instigation, in 1960, came from Peter Hotz.


4. 1937+, Maurits Cornelis Escher

The work of M. C. Escher, the famed graphic artist known for his work in tessellations, also has elements of forerunners, although he did not strictly adhere to the Huff model, and in any case, he was not specifically striving for a parquet deformation. That said, many of his tessellation Escher-like prints from 1937 onwards nonetheless contain an implied parquet deformation. For example, in the first instance, there is Metamorphosis I, of a hexagon deforming (or developing) to a Chinese man. Like prints followed in quick succession, such as Development I (square to lizards), and Cycle (rhombs to a human figure). However, throughout, this is with a representational aspect to the composition, and so does not strictly adhere to the Huff model. Perhaps the nearest adherence to the Huff model is that of a latter-day print, Metamorphosis III, of 1967, which I discuss under that date.

Interestingly, in his notebook sketches, Escher shows some ‘forerunner’ parquet deformations in the form of overlapping as in Lewis Day’s Pattern Design, above. However, whether this is coincidental or not is unclear; it is not known if he studied this book. Certainly, the book is not obscure, and I could readily have imagined Escher may have seen this. 


5. 1960, Peter Hotz/William Huff
Although credit as to the originator is overwhelmingly given to William Huff, this is not strictly so; rather, it is a student of his, Peter Hotz, arising as an offshoot in a tiling assignment given by Huff. This is made explicit, or implied, by Huff himself, in print/correspondence no less than six times. However, without the initial input from Huff, Hotz would not have devised this. Therefore, although Hotz is strictly the originator, I have given joint credit, with Hotz leading. For the sake of clarity, I show the quotes in full, with context, below. Below that I show the first published instance, in ulm 12-13, p. 28, 1965 (Fig. 3).


1996, in The Landscape Handscroll and the Parquet Deformation. In Chapter 2.3 ‘Influence of D’Arcy Thompson; Comparisons with M. C. Escher’, p. 312.

The intriguing possibility of the incremental deformability of one parquet pattern into another came to our attention in 1960 when it was recognized in one student's designs of several very different

looking patterns that there were underlying, but far from obvious morphological relationships

between them.

Hotz is not directly named, but the ‘unnamed student’ is undoubtedly Peter Hotz, derived from Huff’s notes for a SEMA talk (2003), below, and others.


2003, in ‘About Parquet Deformations’ unpublished notes for SEMA
When I made my board-to-board design studio criticism of my original tiling assignment, Peter Hotz, a gifted student, made an observation: that incremental changes could be affected from one parquet variation to another.

In fact, he had sketched, not three variants of the original parquet, but five or six.

At the time that Peter put together his design, the first parquet deformation, I considered it to be a one-time variant of my assignment.

However, after reading D’Arcy Thompson’s chapter, “The Theory of Transformations, or the Comparison of Related Forms,” I had second thoughts and assigned to my whole class, in the third year of my teaching, the assignment for all to produce parquet deformations.

Hotz is directly credited as the originator for the first time.


2003, Huff-Bailey email 25 January
Anyhow, when I came back to the States and eventually began to teach Basic Design, I presented first an assignment in making intricate parquets and eventually we developed the deformation.

(I will digress here, because my assignment was stated like this:  A student was to develop an interesting parquetry with an interesting figure (out of the various possible lattices) and then to make two more variations on that.

When I was working with one student, it was suggested (I no longer remember by whom) that transitions (i.e., deformations) could be made between the different parquets, so the first deformation was born; but I did not try to do these with all students in a class until two years later.)

Hotz is not directly named.


2009. Source? BETWEEN THE INSTRUCTORS AND THE STUDENTS:

One student, Peter Hotz, who had sketched a number of distinctly differently shaped tiles, showed me that through a carefully selected number of steps there would accrue to the perception of a flow of one shape deforming into another. I encouraged him to develop the potential of this intriguing visual effect and suggested that he look up Escher (not yet the property of pop) in the library. In fact, he interlinked five notably distinct noticeable tile shapes—bracketed at each end by square tiles that conform to the design’s underlying lattice. The following year, I assigned the original parquetry exercise with some amendments—for one, orientation was to be featured through coloration; for another, three different tilings that had evident relationships to one another were to be imparted. 

It was not until my third academic year that it occurred to me to elevate the brief Parquet Exercise to that of an extended project, the Parquet Deformation: all the class should make continuous deformations in the mode of Hotz, which I had been considering a one time chance occurrence. The idea/revived study was reinforced by D’Arcy Thompson’s article “On the Theory of Transformations.”

Hotz is directly named.


2015. Email Huff-?
Several years after the "Parquet Study" (The Study was usually, but not always, three variations), I assigned the 'Parquet Deformation." In my 1960 class at Carnegie, Peter Hotz, invented the first "Parquet Deformation."  I did not make that an assignment for all the class until 1963.  In my class at the HfG, I listed it as "Parkettierungsdeformation" (1963).

Hotz is directly named.


2018. Email Huff-Hoeydonck,

I was not inspired by Escher.  I picked up the parquet exercise from Tomás Maldonado at Ulm in 1956-57; our text was a German mathematical book-------with no Eschers.  The deformation occurred when working on pure parquets, Peter Hotz, a student in my first design class pointed out that there could be transitions from one parquet to another and then went on to make the first P. D.

Hotz is directly named.


6. 1964, Arthur Carlson/Thomas C. Davies/ David Oleson. The First 2-Dimensional Instance

Following on from the 1-dimensional instances, in 1960, it (seemingly) took a few more years to extend the concept to 2-dimensions, these emanating from 1964. Certainly, it is the next obvious step, albeit, in relative terms, is more complex in idea and execution. However, it is not known as to how this arose. It may have been a Huff suggestion, or it may not. Further, it is not possible to give the exact first work; in the Ulm archive [*] there are three works, by Arthur Carlson, Leather of the Greater Gator (Rectangle), Thomas C. Davies, Leather of the Lessor Gator (Square), and David Oleson, The I at the Center (Square), all bearing the same season and year, namely Spring 1964. It seems very unlikely that we will ever know who was the originator, and so I thus give a joint credit.

 

7. 1965, William Huff; The first reference in print

Huff, William S. ‘An Argument for Basic Design’. ulm 12/13. Journal of the Ulm School for Design, 1965, pp. 25–38. 

In a general article on basic design (of both German and English), parquet deformations as a term and concept appear for the first time in print, p. 28, albeit essentially as illustrations only, with brief caption text, of both German and English. Oddly, there is no discussion in the main body of the text. This is not an outlier; such a presentation with other topics is throughout the article.

Three parquet deformations are shown, all credited, by Fred Watts, Peter Hotz, and Richard Lane, p., dated, but all untitled. That by Hotz is significant, being the first parquet deformation, although not stated or discussed as such here.

Interestingly, D'Arcy Thompson is mentioned extensively in the article, re On Growth and Form, which has potential significance as to the inspiration of the concept. Previously, I thought that Huff may have been influenced by the image of the book in p. 335, Fig. 133, first edition 1917, but this now seems unlikely, given his credit to Hotz elsewhere, in many places, directly and indirectly,  as the innovator of the concept.


Fig. 3. The first parquet deformation, 1960, centre image

 

8. 19671968, Maurits Cornelis Escher

As alluded to above, Escher use of parquet deformation was as a step in transforming a generic tile into a life-like motif, and not as an end in itself. Perhaps his nearest ‘true’ parquet deformation is that of the first part of the 19671968 print Metamorphosis III, in which there is a transition of squares to rhombs and squares, albeit still with an apparent intent of lifelike motifs (flowers and leaves) slightly further along.


9. 1968, William Huff, the first discussion in print

————. ‘An Argument for Basic Design’. In urban structure by David Lewis (ed). Architects' Year Book: Urban Structure, Elek Books, 1968, pp. 269–278.

The first discussion of the term, in ‘An Argument for Basic Design’ in urban structure [*], or at least can be interpreted as such, amid a general discussion on ‘Descriptions of four major projects’, one of which involves the parquet deformation. However, it is not particularly clear, with certain ambiguities. In his various writings, typically Huff by ‘parquets’ refers to what we now know as tilings or tessellations. It is not clear if Huff is referring to parquets (tilings) or parquet deformation, save for the last sentence. Perhaps he is implying it by the title alone? Whatever, this remains the first discussion.

Parquet Deformation

A parquet [tiling, tessellation] is an endless configuration of pieces that pack the plane. It appears that all parquets conform to lattices, sometimes rigidly, sometimes eccentrically. It is possible to construct a few parquets (e.g. the 1 x 2 brick) with relative randomness.

Each lattice has its mutation groups and subgroups and infinite variation possibilities within each group. Some of the most interesting parquets are developed on the square lattice and on the special  (60°-90°) rhombic lattice, the latter allowing both the equilateral triangular and the regular hexagonal tessellations; the more general lattices, especially the parallelogram lattice, are more limiting, since they afford less sub-division possibilities, in fact, only one - a two-fold rotation. The exercise, in which the lattice is always invariant, reveals some of the more interesting relationships of parquet variants  - the deformations being developed along katametric1 lines.

1 Huff delights, on occasion, by using obscure, non-standard terms relating to symmetry, one of which is here with ‘katametric’. Leopold [*] gives background details:

Maldonado invented “katametry” in the course based on the ideas of the German chemist K.L. Wolf. In a task of Maldonado, katametry is defined as rotation-dilation around a centre [Lindinger 1987: 57] (fig. 8). With this concept different levels of structures can be distinguished, and we become aware of the background of a theory of symmetrical structures. Huff later worked on these ideas in the U.S., which he explained in an essay entitled “Ordering Disorder after K. L. Wolf” [2000], where he compared the system of Wolf (fig. 9a) with the typology of mapping after March and Steadman (fig. 9b). “Katametry” (literally, low measure) is seen as the lowest level of symmetric structure.


10. 1983, Douglas Hofstadter 

In 1983, Douglas Hofstadter wrote a seminal piece in his ‘Metamagical Themas’ column in Scientific American 'Parquet Deformations: Patterns of Tiles That Shift Gradually in One Dimension'. The importance of this article can hardly be overstated; for the first time, parquet deformation was brought to a wider audience, aside from (relatively obscure) design journals, known but to few initiates. Further, the article is an exemplary piece of writing, and is one of the finest of all, if not the best. Who knows where the subject may have developed without this piece? It could conceivably still be languishing in obscure design journals.


Fig. 4. The first page of Douglas Hofstadter’s ‘Parquet Deformations: Patterns of Tiles That Shift Gradually in One Dimension’


11. 2002+, Craig Kaplan
Craig S. Kaplan (1972–), a Canadian computer scientist/mathematician, can be said to have progressed the science of parquet deformations, building on the effectively ‘simple’ design process of the Huff model with more advanced mathematical concepts that without the aid of the computer would simply otherwise be impractical. A typical example is shown below, “organic labyrinthine curves”, using an algorithm based on Brownian motion and developed by Hans Pedersen and Karan Singh.

Fig. 5. ‘Organic labyrinthine curves’ by Craig Kaplan 


12. 2015, Edmund Harriss

Edmund Harriss, an advanced mathematician, devised the first non-periodic (and incidentally 2-dimensional) parquet deformation, titled De-four-mation. This is created from four different non-periodic tilings, one in each corner, that morph into each other from left to right and from top to bottom, thus giving a two-dimensional deformation. These are based upon the Amman-Beeker, 10-fold Penrose tiling, 12-fold Socolar tiling and 14-fold substitution tiling. This is all admirably described and illustrated by Harriss in the video from his presentation at the 'Gathering 4 Gardner' (G4G), 2016. Also see the book with Alex Bellos Snowflake Seashell Star [*] where a single deformation of the above is shown, with likely Bellos wittily (as ever!) titled ‘De-four-mation’.

As might be imagined, this is of a different magnitude of order of drawing. One can only imagine the difficulty here in drawing this by hand; it is simply impractical.


Fig. 5. De-four-mation, by Edmund Harriss


References


[*] Bellos, Alex and Edmund Harriss. Snowflake Seashell Star. Canongate Books Ltd, 2015 


[*] Day, Lewis F. Text Books of Ornamental Design I. The Anatomy of Pattern. London B. T. Batsford 1887, 55 pp., with 35 plates

Minor references to forerunners Plate 11

 

[*] ————. Pattern Design. London, B. T. Batsford 1979. First impression 1903, and 1915 and 1923

Minor references to forerunners of parquet deformation pp. 26-27, and 34.

 

[*] HfG Ulm archive.

Individual parquet deformations are largely unpublished. Viewing is by appointment only. A few designs appear in various journals.


[*] Hofstadter, Douglas. 'Parquet Deformations: Patterns of Tiles That Shift Gradually in One Dimension'. ‘Metamagical Themas’, Scientific American 1983, pp. 14–20

The first popular account of parquet deformations, with 12 of William Huff’s student-inspired works.

 

[*] Huff, William S. ‘An Argument for Basic Design’. ulm 12/13. Journal of the Ulm School for Design, 1965, pp. 25–38. 

In a general article on basic design (of both German and English), parquet deformations for the first time appear in print, albeit essentially as illustrations only, with brief caption text, of both German and English). Oddly, there is no discussion in the main body of the text. This is not an outlier; such a presentation with other topics is throughout the article.

Three parquet deformations are shown, credited, by Fred Watts, Peter Hotz, and Richard Lane, p. 28, dated, but all untitled. That by Hotz is significant, being the first parquet deformation, although not stated or discussed as such here. 

Interestingly, D'Arcy Thompson is mentioned extensively in the article, re On Growth and Form, which has potential significance as to the inspiration of the concept. Previously, I thought that Huff may have been influenced by the image of the book in p. 335 Fig. 133, first edition 1917, but this now seems unlikely, given his credit to Hotz elsewhere, in many places, directly and indirectly,  as the innovator of the concept.


[*] ————. The Parquet Deformations from the Basic Design Studio of William S. Huff at Carnegie-Mellon University Hochschule für Gestaltung and State University of New York at Buffalo from 1960-1983.

Pages 1214 document the first parquet deformation albeit not entirely satisfactory, as the designer’s name is not given, but this is not an outlier; the whole book is without credit to the designers.

Page 12 is simply titled ‘the first parquet deformation’ without credit, p. 13 shows the deformation without any text, and p. 14 shows an exploded explanation of its devising.

 

[*] Leopold, Cornelie. ‘Didactics. Precise Experiments: Relations between Mathematics, Philosophy and Design at Ulm School of Design’. Presented at Nexus 2012: Relationships Between Architecture and Mathematics, Milan, 11-14 June 2012

‘Katametry’ explanation.

 

[*] Lewis, Frederic T. ‘A Further Study of the Polyhedral Shapes of Cells’. Proceedings of the American Academy of Arts and Sciences, Vol. 61, No. 1 (December 1925), pp. 1–34, 36.

On Stellate Juncus.


[*] Thompson, D’Arcy Wentworth. On Growth and Form. First published 1917, Cambridge University Press

Pp. 335336 text in full

A very beautiful hexagonal symmetry, as seen in section, or dodecahedral, as viewed in the solid, is presented by the cells which form the pith of certain rushes (e.g. Juncus effusus), and somewhat less dia­gram­ma­ti­cally by those which make the pith of the banana. These cells are stellate in form, and the tissue presents in section the appearance of a network of six-rayed stars (Fig.133, c), linked together by the tips of the rays, and separated by symmetrical, air-filled, intercellular spaces. In thick sections, the solid twelve-rayed stars may be very beautifully seen under the binocular microscope.

[Image, as above]

Shows the first known unambiguous parquet deformation, by accident or design, although it is far by clear what Thompson had in mind as what is now known as a parquet deformation. 

What has happened here is not difficult to understand. Imagine, as before, a system of equal spheres all in contact, each one therefore touching six others in an equatorial plane; and let the cells be not only in contact, but become attached at the points of contact. Then instead of each cell expanding, so as to encroach on and fill up the intercellular spaces, let each cell tend to contract or shrivel up, by the withdrawal of fluid from its interior. The {336} result will obviously be that the intercellular spaces will increase; the six equatorial attachments of each cell (Fig. 133, a) (or its twelve attachments in all, to adjacent cells) will remain fixed, and the portions of cell-wall between these points of attachment will be withdrawn in a symmetrical fashion (b) towards the centre. As the final result (c) we shall have a “dodecahedral star” or star-polygon, which appears in section as a six-rayed figure. It is obviously necessary that the pith-cells should not only be attached to one another, but that the outermost layer should be firmly attached to a boundary wall, so as to preserve the symmetry of the system. What actually occurs in the rush is tantamount to this, but not absolutely identical. Here it is not so much the pith-cells which tend to shrivel within a boundary of constant size, but rather the boundary wall (that is, the peripheral ring of woody and other tissues) which continues to expand after the pith-cells which it encloses have ceased to grow or to multiply. The twelve points of attachment on the spherical surface of each little pith-cell are uniformly drawn asunder; but the content, or volume, of the cell does not increase correspondingly; and the remaining portions of the surface, accordingly, shrink inwards and gradually constitute the complicated surface of a twelve-pointed star, which is still a symmetrical figure and is still also a surface of minimal area under the new conditions.


Page created 27 August 2021.

Previously, a short (24-line, non-illustrated) history was on the main page (now removed), but this was wholly inadequate as a history in itself. This 2021 piece leaves the old far behind.


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