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
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
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  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
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
Some of my own examples are shown below.
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
No Title. The Buffalo News 23 June 1985.
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
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’.
Alex and Edmund Harriss. Snowflake
Seashell Star. Canongate Books Ltd, 2015
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!
Hans-Günther. Reguläre Parkettierungen.
Mit Anwendungen in Kristallographie, Industrie, Baugewerbe, Design und Kunst.
BI Science Publisher, 1994 GB
Regular tilings: With applications in Crystallography, industry,
construction, design and art.
[*] 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
Day, Lewis, F. Pattern
Design. London, B. T. Batsford 1979. First
I include the reference from Day with a good degree of reservation, as this is
Elaine and John Sharp. Tiled Torus Quilt
with changing tiles. Bridges 2010
Sharp inspired parquet deformations.
Michael. Structure and Form in Design:
Critical Ideas for Creative Practice. Bloomsbury Academic 2012
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).
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
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
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
[*] 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
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),
[*] ————. 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
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
interest as to ‘all things’ Huff. Parquet deformation is only mentioned in
passing, as a reference.
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
discussions on parquet deformations (Beecombing Blossoms, p. 198, Fylfot Flipflop
p. 194, albeit at a most abstruse level.
L. Christine, Teresa E. Moore. Symmetry,
Shape and Space: An Introduction to Mathematics Through Geometry
discussion on Escher parquet deformations within the context of his prints.
Craig S. ‘Metamorphosis in Escher’s art’. In Bridges 2008: Mathematical Connections in Art, Music and Science, pp.
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
impressive, highly advanced parquet deformations.
Jay. Connections. The Geometric Bridge Between Art and Science.
McGraw-Hill Inc. 1991
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
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
Ed. ‘Designed To Be Different’. The Pittsburgh Press, Sunday, February 27, 1972, pp.
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.
design, not that I get any credit!
deformation java program.
of my design without credit.
reference to ‘Grasshopper’ program by ‘Josclag’ (no proper name given)
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.
in Five, Parquet Deformations, and POW!’ 21 November 2013
Illustrated with one of my designs without credit, albeit with a link.
[*] Sharp, John
hints and tips
[*] Tuğrul, Yazar
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
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.