Parquet Deformations

First, what exactly is a ‘parquet deformation’, and what is its relation to tessellation? At first glance it has no obvious reference or connection to tessellation matters, with the obvious thought of it referring to flooring per se. Indeed, the term and concept may be wholly unfamiliar to most people, even with an interest of tessellations, which I shortly detail. Such an aspect of tessellation has received relatively scant coverage, and in my opinion disproportionately so as to its inherent worth; the effect is most gratifying. Therefore, to clarify such matters, I now define a parquet deformation as a ‘geometrical tessellating metamorphosis’, usually shown as a ‘long strip’ of a horizontal presentation (although other formats are possible), of which the tiling changes, subtly so at each alternate stage, almost inconsequentially so, but drastically so when seen from beginning to end; the beginning and end tiles bearing no relation to each other.

As such, in contrast to any ‘orthodox' picture, for example a portrait or landscape, or indeed a tessellation in the ‘normal’ sense, with the eye wandering ‘at whim’, a parquet deformation is not intended to be viewed in a generalised sense, but is instead intended to be read in a specific way. For the commonly to be seen strip, as detailed above, this is then ‘scanned' by the eye, in a left to right or vice versa direction, as in reading a line of text. Whereupon having so done, a given tiling thus can be seen to gradually metamorphosis in outline, until upon ‘completion' of its metamorphosis it is unrecognisable from its beginning.

Concerning the history and background to parquet deformation, only in relatively recent years has such an idea come to the fore. Indeed, despite tessellations per se having been of interest for centuries, such a apparently simple metamorphic concept was apparently not thought of. As such, to all intents and purposes, the originator, and the describer of the term, is William S. Huff, a Professor of Architecture at the
State University at New York (SUNY) with examples dating back to the 1960s. Although earlier, isolated examples do indeed occur, such as in parts of M. C. Escher's Metamorphosis, and also some isolated examples ‘of sorts' in the book Pattern Design by Lewis Day, these cannot be considered in any way equal to Huff's more thorough and systematic work in this field, albeit he himself does not compose these in the normal sense of the word. Indeed, Huff has, in his role as Professor of Architecture at the State University at New York (SUNY) not only brought the concept of such a thing to the fore, but has also inspired numerous students of his to compose many new examples, many of notable skill and ingenuity. Such studies eventually led to the first popular account, with the publication (1983) of his students’ work in the Scientific American article ‘Parquet Deformations: Patterns of Tiles That Shift Gradually in One Dimension' by Douglas Hofstadter. Further examples by Huff's students are to be seen in Intersight One 1990, although this is rather an obscure publication. Of note is that all these predate the age of the computer; all the examples above are drawn by hand. Subsequently, the computer has been used in the undertaking, of which this offers up new possibilities that are impractical by hand.  However, despite these initial, promising beginnings, subsequent interest has been lukewarm, to say the least, with no books or papers appearing, at least to my knowledge. Indeed, it can be said the subject went into a slumber, and has only relatively recently been revived, at least in published form, with new examples in 1998, comparable in quality to the students of Huff's, most notably with examples by Craig S. Kaplan and John Sharp, details of which appear on the links page. Some other names to look out for are Andrew Cooke, Daniel Piker and Norman Courtney. Also see Scott Kim pp. 14-15 for a uncredited instance. 

As alluded to above, different formats are indeed possible, and not just the ‘usual’ portrayal of the long strip. Indeed, the only limit is one’s imagination. Frequently seen are square blocks, and for the sake of variety, I show a long strip; albeit this wasn’t thought through at the time; it is lacking in aesthetics. I also show a rectangular loop, of which in principle this is merely four parquet deformations butted together and then bent to form the loop. Furthermore, one is not necessarily confined to a square based; equilateral triangles can also be used.

What I term as aesthetics is a most important part of parquet deformations, in that this determines the quality. This is a vital aspect of parquet deformations, and is too often neglected. As alluded to above, since Huff’s day, the computer has in effect stormed the world of design, and is being used more and more, to the exclusion of hand drawings. Indeed, with the advent of this tool, it opens up new possibilities that simply are impractical by hand; different formats, more complex examples. it also has the potential of time saving; drawing by hand can indeed be laborious Be that as it may, what still counts is the inherent quality, or aesthetics of the design. What makes for a ‘good’, or indeed ‘bad’ parquet deformation? As a basic statement, those by Huff’s students serve as this model; they possess a degree of ‘elegance’ that others in this field do not possess. And I might add that mine do too, but differently subtly from Huffs. Huff’s instances can be broadly divided into two types (a) those that evolve into simple geometric tiles: e.g. Crossover, Dizzy Bee, Consternation, and (b) those that more involved final tile, such as Crazy Cogs, Arabesque, Curracha. In contrast, mine are all ‘simple’ geometrical tiles, such as a right angled triangle to a rectangle (No. 1). Another important aspect is the tempo. By this, I mean that the changing of the tiles should be neither too ‘quick’ nor ‘slow. A quick example would be where the change is too drastic, of just a cycle of say, two or three stages. Here, the change would be too quick for the eye to view. Slow would be say, one hundred, where change is imperceptible at each stage, and so would make for tedious viewing.

A feature of this way of designing is that the finished deformation is not predictable, resulting in a pleasing surprise upon completion. As a rule, I favour a simple beginning, typically of a square, as this acts as a fundamental aspect, in contrast to an arbitrary geometrical shape. A different type of approach is to begin and end a deformation with a certain polygon or geometrical shape in mind. Some examples of this include the Laves tilings, where Craig Kaplan [5] show this. Note that mostly in contrast to the Huff examples, mine can in a sense be continued; for example No.4 can be butted on to No.1, which can in turn be butted with No.5, and so on. Mine are broadly of a self contained nature, with a beginning, middle and end, where typically they begin quietly, ‘came to life’ in the middle, before fading towards the end. In essence, each of these is a one-off, albeit as alluded to above, can indeed be joined if so desired. In contrast, Craig Kaplan’s are more of a ‘all-encompassing’ approach. For instance, he set himself the challenge of combining all suitable Laves tilings as parquet deformations, and met with success. 

However, despite the above ‘revival', such as it is, somewhat disappointingly I still find so few people undertaking new examples or who are at least interested in this aspect of tessellation, and so I would be more than interested in hearing from fellow enthusiasts on this matter, especially from the ex-students of Huff's. Of especial interest would be to make contact with the designers of the parquet deformations in the Scientific American article. Where are you: Fred Watts (Fylfot Flipflop), Richard Lane (Crossover), Richard Mesnik (Dizzy Bee), Scott Grady (Consternation), Francis O’Donnell (Oddity out of Old Oriental Ornament), Leonard Chan (Y Knot), Arne Larson (Crazy Cogs), Glen Paris (Trifoliate), Joel Napach (Arabesque), unknown (Razor Blades), Jorge Guttierrez, (Curacha), Laird Pylkas (Beecombing Blossoms), Vincent Marlowe (Clearing the Thicket). And latterly in Intersight One; Jacqueline Damino (Right Right Left Right), Rodney Wadkins (In Two Movements) Darren Moritz (Enlarging on Four Points), Alexander Gelenscer (Hex-baton), Maurizio Sabini (Venetian Net), Robert Johnson (untitled). Did you do any others or simply left these behind upon completion of your student days? I would love to hear from you! Or indeed, from anyone who knows these people.

The reference by Francis O’Donnell to ‘Oddity out of Old Oriental Ornament’, allude to a tiling (or more accurately lattice design) that can be seen in Chinese Lattice Designs. 1200 Designs by Daniel Sheets Dye. p. 420 , example b.

Some of my own examples are shown below.

No. 2
No. 3

No. 4

No. 5

No. 6

No. 7

No. 8

No. 9

No. 10

No. 11

No. 12

No. 13

No. 14

No. 15


First, note that although the listing below is believed to be the best bibliography available, this is by no means exhaustive. Many of the references I have found concerning parquet deformation are somewhat obscure, and of limited interest and value. In short, to list ‘everything’ would result in confusion rather than clarity; given that some references are of a single sentence, and of no particular significance. Note that I do not have all of the books listed in possession (judged not worth pursuing, with time constraints, and the effort/cost involved in obtaining). A useful reference was Google Books, which I shorten to GB below and can be found on Google. The more important, regarded as essential reading on the subject, is signified with the author’s name in bold. Other books/articles are also ‘highly desirable’, and indeed, there is no firm dividing line at times between the references. Are there other references of note? Is so, do let me know. The listing below is separated in to two parts; (i) print, with books, articles and newspapers and (ii) web. The listing is to be considered as a work in progress, and so is subject to revision/addition.

Print: Books, Articles, Newspapers

[*] Anon. No Title. The Buffalo News 23 June 1985.
On parquet deformation, Huff. One of only two newspaper references I am aware of. The whole article is not available  to me, seen only by chance  as a clipping next to a story on dancing that showed up when searching! Shows ‘Fylfot Flipflop’, with a discussion in general sense.

[*] Bellos, Alex. ‘Crazy paving: the twisted world of parquet deformations’. The Guardian 9 September 2014
Essentially a feature on Craig Kaplan’s work rather than a general discussion on the subject, and all mightily impressive. Of a cross section of his work, in which he brings his full range of mathematical abilities to the premise, leaving lesser mortals far behind: ‘fractal’ tiles, ‘organic labyrinthine curves’, ‘Islamic parquet deformation’, ‘One that goes in two dimensions’, ‘And one that is shaped in a circle’.

[*] Bellos, Alex and Edmund Harriss. Snowflake Seashell Star. Canongate Books Ltd, 2015
One Harriss parquet deformation (unpaginated, with likely Bellos title) ‘De-four-mation’, of four non-periodic tilings positioned in a corner, which morph left to right and top and bottom. Beat that!

[*] Bigalke, Hans-Günther. Reguläre Parkettierungen. Mit Anwendungen in Kristallographie, Industrie, Baugewerbe, Design und Kunst. BI Science Publisher, 1994 GB
Translated: Regular tilings: With applications in Crystallography, industry, construction, design and art.
P. 232 illustration.

[*] Crowell, Robert A. (editor). Intersight One. State University of New York at Buffalo 1990.
See 10. William S. Huff What is Basic Design?: 76-85 and in particular generic problems of synthetic design Students' work from the Basic Design Studios of William S. Huff 80-85.

With works by: Jacqueline Damino (Right Right Left Right), Rodney Wadkins (In Two Movements), Darren Moritz (Enlarging on Four Points), Alexander Gelenscer (Hex-baton), Maurizio Sabini (Venetian Net), Robert Johnson (untitled).
A most impressive collection, second only to the Hofstadter article, and highly recommended.

[*] Day, Lewis, F. Pattern Design. London, B. T. Batsford 1979. First published?
I include the reference from Day with a good degree of reservation, as this is weak indeed.

[*] Ellison, Elaine and John Sharp. Tiled Torus Quilt with changing tiles. Bridges 2010
Mostly John Sharp inspired parquet deformations.

[*] Hann, Michael. Structure and Form in Design: Critical Ideas for Creative Practice. Bloomsbury Academic 2012
P. 104 Craig Kaplan ‘fractal’.

[*] Hofstadter, Douglas. 'Parquet Deformations: Patterns of Tiles That Shift Gradually in One Dimension'. Scientific American 1983 14-20
The classic account. The importance of this article can hardly be overstated; the one article that overrides everything else. This is the first popular account of parquet deformations, with William Huff’s student-inspired works, of which Hofstadter does it full justice, with 12 stunning examples, with works by:
Fred Watts (Fylfot Flipflop), Richard Lane (Crossover), Richard Mesnik (Dizzy Bee), Scott Grady (Consternation), Francis O’Donnell (Oddity out of Old Oriental Ornament), Leonard Chan (Y Knot), Arne Larson (Crazy Cogs), Glen Paris (Trifoliate), Joel Napach (Arabesque), unknown (Razor Blades), Jorge Guttierrez, (Curacha), Laird Pylkas (Beecombing Blossoms), Vincent Marlowe (Clearing the Thicket).
And the titles are most amusing too! To pick a favourite is invidious. However, if pressed ‘Fylfot Flipflop’. Of note is that these are all linear. Absolutely delightful!
However, Hofstadter seems to have erred somewhat in his commentary; Branko Grunbaum, in Tilings and Patterns takes him to task, p. 170, correcting his comment on p. 14 ‘Despite the claims to the contrary most of the tilings shown in Hofstadter’s article include tiles which are not prototiles of any monohedral tilings’.

[*] ————. ‘Parquet Deformations: A Subtle, Intricate Art Form’. July, 1983 190-199. In Metamagical Themas: Questing for the Essence of Mind and Pattern. Basic Books; New edition 1996, First Printing edition 1985
This essentially repeats Hofstader’s original July 1983 column in Scientific American (his last), with extra, minor text, but also, more importantly, a ‘post scriptum’, in which a parquet deformation of David Oleson’s ‘I at the Center’ is illustrated and discussed, and much praised.

[*] ————. I Am A Strange Loop. Basic Books 2008. GB
David Oleson discussion.

[*] Huff, William. ‘The Landscape Handscroll and the Parquet Deformation’, 307-314. In Katachi U Symmetry. Tohru Ogawa, ‎Koryo Miura, ‎and Takashi Masunari. Tokyo: Springer-Verlag 1996
This has four parquet deformations from Huff’s studio, with works by: Alexander Gelenscer (Swizzle Stick Twirl), 1986; Pamela McCracken (Cloisonné), 1990; Loretta Fontaine (Seven of One Make Three), 1991; Bryce Bixby (They Come, They Go), 1991
Highly recommended.

[*] ————. Best problems from Basic Design 20 February 1978
Three parquet deformations, one of which, on page 31, made it into the Scientific American article.

[*] ————. ‘Simulacra in non-reorientable surfaces-experienced in timing’. In Spatial Lines, Patricia Muñoz, compiler
See Chapter 4. Minor reference to parquet deformations.

[*] ————. ‘Defining Basic Design as a Discipline’. In Symmetry: Art and Science, Vol 2 (new series) Numbers 1-4, 2012
Only of interest as to ‘all things’ Huff. Parquet deformation is only mentioned in passing, as a reference.

[*] Joseph, M. and R. Shyamasundar. Foundations of Software Technology and Theoretical Computer Science: Fourth Conference, Bangalore, India December 13-15, 1984. Proceedings: v. 181 (Lecture Notes in Computer Science) 2008. GB
Occasional discussions on parquet deformations (Beecombing Blossoms, p. 198, Fylfot Flipflop p. 194, albeit at a most abstruse level.

[*] Kinsey, L. Christine, Teresa E. Moore. Symmetry, Shape and Space: An Introduction to Mathematics Through Geometry
A discussion on Escher parquet deformations within the context of his prints.

[*] Kaplan, Craig S. ‘Metamorphosis in Escher’s art’. In Bridges 2008: Mathematical Connections in Art, Music and Science, pp. 39-46
Within an Escher framework of transitions, has much on parquet deformations, 42-45

[*] ————. ‘Curve Evolution Schemes’. In Bridges 2010 Mathematical Connections in Art, Music and Science, pp. 95-102
Some most impressive, highly advanced parquet deformations.

[*] Kappraff, Jay. Connections. The Geometric Bridge Between Art and Science. McGraw-Hill Inc. 1991
Parquet deformation 190-194, within Chapter 5, ‘Tilings with Polygons’, 5. 10. 5 ‘One-dimensional parquet deformations’, albeit this is mostly merely excepted from Huff’s article (1983), as the author credits. ‘Consternation’ is shown.

[*] Kim, Scott. Inversions. W. H. Freeman and Company New York 1989.
Parquet deformation, unaccredited, pp. 14-15.

[*] Schwartz, Jordan. Art of LEGO Design: Creative Ways to Build Amazing Models. No Starch Press, 2014 71-72. GB
Although nominally of parquet deformation, I have considerable reservation here; the transitions are far too abrupt.

[*] Wang, P. S. P. (editor). Array Grammars, Patterns and Recognizers. World Scientific Series in Computer Science. 1989. GB
Technical. Quotes Hofstadter’s article

[*] Wintermantel, Ed. ‘Designed To Be Different’. The Pittsburgh Press, Sunday, February 27, 1972, pp. 10-11.
One of only two newspaper references on William Huff and parquet deformations I am aware of. Contains a new work and new name not seen before, Roland Findlay (the designer and work is oddly not titled or discussed in the text), and a photo of Huff, the earliest one of him I have seen. Also of significance as the first popular reference to parquet deformations in print form.


[*] Anon
Uses my design, not that I get any credit!

[*] Cooke, Andrew
Parquet deformation java program.

[*] Henn Company
Again, use of my design without credit.

[*] Josclag’
With reference to ‘Grasshopper’ program by ‘Josclag’ (no proper name given)

[*] Kaplan, Craig
Craig Kaplan talk at Berlin School. A video of the BMS Friday lecture by Craig S. Kaplan (U Waterloo) entitled "Metamorphoses and deformations of tilings". The talk was given on June 06, 2014 at the BMS Loft at Urania.

[*] MathMunch.
‘Isomorphism in Five, Parquet Deformations, and POW!’ 21 November 2013
Illustrated with one of my designs without credit, albeit with a link.

[*] Sharp, John
Some useful hints and tips

[*] Tuğrul, Yazar

A most impassive analysis of other people’s (including my own) parquet deformations, as well as ideas of his own. Use of the program Grasshopper. Especially see: I at the Center, Crossover, Trifoliolate, Parquet deformation with 10 components, Parquet deformation manually (my designs), Parquet deformation of Islamic patterns.

Created 17 April 2014. Previously (29 September 2009), the material was over three pages, of which I now show as a single page, complete with further, additional text

30 November 2016:
References greatly expanded, replacing the previous six entries with minor commentary text with thirty-two entries, and all with further, extensive commentary and detail.