First, note that although the listing below is believed to be the best bibliography on parquet deformation available, this is by no means exhaustive, although it is indeed decidedly thorough as is practically possible. The quality of the references varies tremendously. In truth, only a few are of any real substance and worth. The more important references, regarded as essential reading on the subject, are signified by the author’s name in bold. Most of the references I have found are somewhat obscure and of limited interest and value. In short, if a reference concerns parquet deformations, it is included, no matter what, of a considered piece of writing to a mere mention in passing. However, I have not been judgmental in what to include. Accompanying the bare bibliographic detail are some comments of my own. These vary in extent as to depth, largely at whim. On occasion, I add further detail about the author or journal, for the sake of general interest. In the fullness of time, I may add more detail in the round. Note that I do not have all of the books/articles listed in my possession, although all have been viewed to some extent. A useful reference was Google Books. This typically gives a snippet preview and on occasion a more extensive preview. Although not always ideal, this thus permitted an assessment of sorts as to intrinsic worth; many were judged simply not worth pursuing, with time constraints, and/or the cost involved in obtaining.

Of the various outlets, citing from ‘Issuu’, an electronic publishing platform (detailed below) is in particular problematic; there appears to be no clear-cut way of doing so. Consistency of references is lacking. Authorship, or credit to the parquet deformation is not always clear. Sometimes people’s names are stated, sometimes a pseudonym is given, and others not at all. Titles, even when given, are most vague. None of the publications are downloadable, all have seemingly chosen not to permit this possibility. In all aspects, it’s all most trying. It's a pity really, in that there would appear to be some good ideas, but the format is not conducive to study. As such, I have assembled as best I can in the circumstances, at least of an initial attempt. Wikipedia: Issuu is an electronic publishing platform founded in 2006, enabling creators of publications to share their content digitally. Issuu converts PDFs into digital publications that can be shared via links or embedded into websites. Users can edit their publications by customizing the design, using templates, or adding links and multimedia to the pages of their documents. Issuu also provides tools for measuring and monetization of content.

Most of the references spring from the related fields of architecture and design. Strict mathematical references, despite the obvious affinity with tessellation, are relatively rare. Curiously, references to parquet deformations are also to be found in a variety of unlikely places. For instance, they are discussed in the context of ‘terminal weighted array grammars’, and ‘terminals’, of which I remain ignorant of. 

The listing below is separated into two parts; (i) print, with books, articles and newspapers and (soon) (ii) web. Some people appear in both categories. The listing is to be considered as a work in progress, and so is subject to revision/addition.

Are there other references of note? If so, do let me know.

Anceschi, Giovanni. New Basic Design a Venezia e Basic Design a Ulm, ISIA Urbino, Self Published on Issuu January 19, 2011, 72 pp.

Conference by Giovanni Anceschi, Reference teacher Nunzia Coco.

Three untitled parquet deformations, by Fred Watts, Peter Hotz (Holtz in original) and Richard Lane, p. 58; David Oleson’s The I at the Center, p. 64. The piece also repeats illustrations from Huff’s two ‘An Argument for Basic Design’ articles. There is no discussion on parquet deformation, just illustrations.

Of note is that Aneceshi is associated with 19 Rassegna (and I think I have seen his name elsewhere). He is on Facebook but his email is not readily found.


Giovanni Anceschi (b.1939) is an Italian artist who is considered by many to be the founding member of kinetic and programmed art in that country. Anceschi studied theoretical philosophy at Milan University and became a founding member of Gruppo T, as well as being a fundamental participant in the Nouvelle Tendance movement of the 1960s.

Anon. ‘In Brief. Awards and Announcements’. B/a+p. News from the School of Architecture and Planning University at Buffalo, Spring 2014

Text, p. 4:

Basic Design: An Exhibition of Works by Students of William S. Huff

The works of students of William S. Huff, professor emeritus of architecture, were featured in

“Basic Design,” a recent exhibition at the Ulm Museum in Ulm, Germany. An internationally

noted scholar, Huff studied at the Ulm School of Design and Yale University and then taught

at Carnegie-Mellon University before joining the faculty of the Buffalo School in 1974. Over the

years, Huff has amassed a collection of material documenting design theory, from the Bauhaus

to the HfG/Ulm to the latest methods in design education. Huff has gradually donated much

of this material, including the results of many Buffalo School student design assignments, to

the HfG/Ulm Archive. “Basic Design” features 40 graphic works and 20 study models from this

collection, highlighting Huff’s experimentation with symmetry (programmed design), black

& white and color rasters (grid manipulation), congruent sectioning of space, effecting color

in pigments as color in light, and the deformation of parquet patterns. Huff’s fundamental

doctrine has impacted basic design teaching around the world.

Nicholas Bruscia is also mentioned, but not in the context of parquet deformation.

Akira, Ito*, ‎S. P. Patrick, P. Wang, and ‎K. G. Subramanian. Array Grammars, Patterns and Recognizers. World Scientific Publishing, 1989, p. 69.
Snippet view on Google Books:
SMG AND PARQUET DEFORMATION Yet another interesting application of the indexed SMG is in the description of parquet deformation. A parquet deformation is one in which a regular tessellation of the plane gets deformed progressively in ...Recognizers.
* Trove gives the editor as P. S. P. Wang.

Alpert, Richard. ‘Tracks of Motion in an Enclosed Space: Connections between Performance and Visual Imagery’. Leonardo, 1984, Vol. 17, No. 3 (1984), pp. 167–171.
Inconsequential. P. 171, references, credits Hofstadter, and Alpert briefly alludes to them, p. 170.

Anon. No Title. The Buffalo News Sunday, 23 June 1985. Page number/s not known
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 a general sense.

American Drawings and Watercolors in the Collection of the Museum of Art, Carnegie Institute. Publisher: Carnegie Museum Store; 1st edition, 1985,  P. 276.
Snippet view on Google Books:
Parquet Deformation - Variant II, 1965 - 66 pen and ink on paper 39 516 X 28 in. ... Bibliography: Douglas R. Hofstadter, “Metamagical Themas, Parquet Deformations: Patterns of Tiles that Shift Gradually in One Dimension,” Scientific …

Annual Report of the Director Issues 83-84, Carnegie Institute 1980, p. 51.
Snippet view on Google Books:
...and Robert Skolnik Parquet Deformation
I cannot find any more detail on this.

Artist’s Page. Mutahir Arif. Crossing Disciplines – Scope: (Art & Design), 9, 2014
P. 86. Untitled (2014) was inspired by Islamic parquet ‘deformations’ created by Craig Kaplan. Kaplan’s work was based on research by William Huff, and was later popularised by Douglas Hofstadter who in turn had been influenced by M.C. Escher’s Metamorphosis series.²
I extended Kaplan’s spatial animation work by applying the animation principles, known to most professional animators, to the star forms of Islamic patterns. Although Kaplan was successful in rendering his Islamic star patterns by way of an inference algorithm written as a standalone and executable Java application, he conceded that “(t)he construction of Islamic parquet deformations requires many separate invocations of the inference algorithm, and [was] currently too slow to run interactively.”³ To solve this problem, I adopted a more traditional approach by constructing the parquet deformations animation the old-fashioned way by using a more basic technique, frame-by frame animation. The drive to realise Kaplan’s vision of “a gently changing geometric design that is still recognizably Islamic” was inspired by the fact that nobody had attempted this approach before, owing to the painstaking nature of the undertaking. Changing shapes by hand and then executing the application was very time-consuming.⁴ Even in today’s digital age, achieving such a result is no mean feat…
Inconsequential. Oddly, the work Untitled, Fig. 3, although said to be parquet deformation related, is not a parquet deformation as such, and is merely a typical Islamic repeat tiling! Further, Figure 1 and Figure 2 are not (or appear to be) parquet deformations!
I am not too sure as to what exactly ‘Crossing Disciplines’ is. It may be an off-shoot of the New Zealand-based ‘Scope’: Scope: Contemporary Research Topics is peer-reviewed and published annually in November by Otago Polytechnic/Te Kura Matatini ki Otago, Dunedin, New Zealand.
It's unclear as to how to cite this. I will place it under ‘Artist’s Page’.

‘BAD’ (Built by Associative Data). By ‘MUQ’?, ‘Computation Coding/Recoding Islamic Patterns’. Self Published on Issuu March 13, 2015, 102 pp.

See p. 91. Three Islamic-style parquet deformations, of the more ‘intricate’ type. Quite who the designer is for this is unclear - ‘Muq’?

Bellos, Alex. ‘Crazy paving: the twisted world of parquet deformations’. The Guardian, 9 September 2014
Essentially a substantial feature on Craig Kaplan’s work rather than a general discussion on the subject, and all mightily impressive and is of required reading. Of a cross-section of his work, in which he brings his full range of computer scientist/mathematical abilities to the premise, leaving lesser mortals far behind. Most of these are simply impractical without the aid of a computer. Bellos captions these as: Grid-based parquet deformation, Funky tiles, Iteration deformations, Organic labyrinth growth, Islamic tiling, 2D parquet deformation, and Circular deformation.
Unusually, these are shown coloured. Kaplan is one of the few who use colour (others include de Villiers), of both flat and graduated colours.

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

Translated: Regular tilings: With applications in Crystallography, industry, construction, design and art.

P. 232 illustration and Huff mention. A ‘Square to double basketweave to Cairo deform’, with an Alhambra transition not seen before (in ‘Square to double basketweave’ section), with the designer not credited. Seems to be only a single page study.


Snippet view on Google Books:

2Ähnliche "parquet deformations" sind z . B . von S . HUFF , Department of Architecture , Carnegie Institute of Technology , Pittsburgh / Pennsylvania , oder von M . C . ESCHER , Z . B . in seinen " Metamorphosen " von 1939 und 1967


Hans-Günther Bigalke (born February 23, 1933 in Celle ; † April 19, 2019 there ) was a German mathematician and university professor. He was one of the pioneers of didactics of mathematics in Germany and co-founder of the Society for Didactics of Mathematics.

Bonner, Jay, with contributions by Craig Kaplan. Islamic Geometric Patterns: Their Historical Development and Traditional Methods of Construction. Springer, 2017

Kapan’s (extensive) contribution to the book is Chapter 4, ‘Computer Algorithms for Star Pattern Construction’, pp. 549–-572, and of parquet deformation relevance Chapter 4.6 ‘Extensions’, with Chapter 4.6.1 ‘Parquet Deformations’, pp. 569–570. Chapter 4.6.1 is a small subchapter over two pages in the context of Islamic tilings. This reuses the top two illustrations in his article ‘Islamic Star Patterns from Polygons in Contact‘. 

The contact angle used to construct motifs can be varied smoothly, producing a continuum of possible designs (though for some template tilings, certain angles are more canonical than others). There is no reason why the contact angle cannot also be varied spatially within a single design, producing a pattern that undergoes a slow, graceful metamorphosis (Kaplan, 2005). I have experimented with these “spatial animations.” I generate a long, narrow strip of the template tiling. Then, for each edge midpoint, I choose a contact angle for its rays based on the position of the midpoint as a fraction of the way from the start to the end of the strip. The contact angle might be different for every edge midpoint in a given tile, but the motif generation algorithm can still operate as before. Two examples of this process are shown in Fig. 536. Note that these designs are able to transition gradually between acute, median, and obtuse pattern families.

[Two images]

Fig. 536 Two examples of parquet deformations. The top diagram shows isolated motifs constructed with continuously varying contact angles, which are then elaborated into a complete design in the middle

The bottom drawing is a parquet deformation based on the tiling of Fig. 503, with colors added manually in Adobe Illustrator

I name such designs “Islamic Parquet Deformations,” after the design style pioneered by Huff and described by Hofstadter (1986). They are also inspired by, and share aesthetic qualities with, Escher’s use of metamorphosis (Kaplan, 2008).


Front Matter work, designs with diminishing scale, parquet deformations as per the work of Craig Kaplan…

Bosch, Robert. Opt Art: From Mathematical Optimization to Visual Design. Princeton University Press, 2019.
See pp. 149–150, 187 (index)
From Mathematical Optimization to Visual Design. Figure 10.6 displays four versions—with squares, fans, square rings, and a parquet deformation
A square to Shepherd’s Check parquet deformation, in his series of Frankenstein-themed ideas. Also, Vermeer’s ‘Girl with a Pink Earring’, p. 154.

Bosch, Robert and Andrew Pike. Map-Colored Mosaics. Proceedings of the 2009 Bridges Banff Conference, held in Banff, Canada. Edited by Craig S. Kaplan and Reza Sarhangi, pp. 139–146

P. 142: Once we have created a map-colored mosaic that pleases us, we can modify it by replacing its square tiles with other tiles that behave like squares. By doing this, we obtain images that are reminiscent of Escher-like tessellations [6,10] or Huff-like parquet deformations [7,9] when viewed from up close, yet still look like familiar images when viewed from a distance. See Figures 5 and 6.

Andrew Pike (LinkedIn)

Experienced Researcher with a demonstrated history of working in the e-learning industry. Skilled in Cell Culture, Science, Western Blotting, Laboratory Skills, and Protein Expression. Strong human resources professional with a PhD focused in Molecular Microbiology and Immunology from Johns Hopkins Bloomberg School of Public Health.

Brandstetter, Gabriele and Marta Ulvaeus. ‘Defigurative Choreography: From Marcel Duchamp to William Forsythe’. The Drama Review, Winter, 1998, Vol. 42, No. 4 (Winter, 1998), The MIT Press, pp. 37–55.

Inconsequential. Brief mention (not illustrated) of parquet deformations (in Hofstadter's Metamagical Themas) on pp. 48–49 in the context on dance.

Pp. 48–49. In terms of the relation of figure and space, the patterns of such choreography reveal a similarity with the designs that are known as "parquet deformations" (Hofstadter 1985:195-218): gradually developing transformations of divisions of the plane, or tessellations, which, through the lengthening or rotating of a line or through the introduction of a hinge, result in a complete distortion or regrouping-like a type of ornamental morphing.


TDR traces the broad spectrum of performances, studying performances in their aesthetic, social, economic, and political contexts. With an emphasis on experimental, avant-garde, intercultural, and interdisciplinary performance, TDR covers performance art, theatre, dance, music, visual art, popular entertainments, media, sports, rituals, and the performance in and of politics and everyday life.

Comptes Rendus - Interface Graphique. National Research Council of Canada, 2005 


Snippet view on Google Books:

P. 177. We show how this method can be adapted to construct Islamic designs reminiscent of Huff's parquet deformations. Finally, we introduce a geometric transformation on tilings that expands the range of patterns accessible using our method.

Bibliographic detail is scanty here. Although not stated, the text is taken from Kaplan’s

‘Islamic star patterns from polygons in contact’.

I have not been able to find out more on this Canadian journal.

Crowell, Robert A. (editor). Intersight One. State University of New York at Buffalo 1990.
See (chapter)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.

Dawson, Robert J. MacG (probably). ‘Crooked Wallpaper’. Journal of Graphics Tools. Volumes 8–9, A. K. Peters, 2003, pp. 33–46
Snippet view on Google Books:
p. 43 seen in the “parquet deformation" of Figure 10, which shows seven tilings (a–g), all with the same periodicity. Applying stretch to any tiling (except the right - most) yields the one immediately to its right; the shaded fundamental domain …
The Journal of Graphics Tools (JGT) was a quarterly peer-reviewed scientific journal covering computer graphics. It was established in 1996 and published by A K Peters, Ltd., now part of Taylor & Francis. From 2009-2011 the journal was published as the Journal of Graphics, GPU, & Game Tools. In 2012, a large part of the editorial board resigned to form the open access Journal of Computer Graphics Techniques (JCGT). The Journal of Graphics Tools continued with a new editorial board. The last editor-in-chief is Francesco Banterle (Istituto di Scienza e Tecnologie dell'Informazione). Previous editors-in-chief have been Andrew Glassner, Ronen Barzel, Doug Roble, and Morgan McGuire. The final volume was released in 2013 and the journal formally ceased with its final issue in 2015.
Can’t find with ease. Available at Taylor & Francis?

Day, Lewis, F. Pattern Design: a book for students, treating in a practical way of the anatomy, planning and evolution of repeated ornament. London. First published 1903, Batsford 2nd Edition Hardcover, 1933 Amor Fenn (revised by), B. T. Batsford 1979.
I include the reference from Day with a good degree of reservation, as this is not strictly a Huff-style instance. However, a ‘proto parquet deformation’ is indeed implied here, and so is thus included. The relevant pages are, with captions repeated: p. 27 (Simple and more complicated trellis lines), p. 29 (Relation of Octagon to circle diaper), and p. 35 (Wavy lines, ogee diaper, and interlacing ogees, giving hexagonal shapes).
Lewis Foreman Day (29 January 1845 – 18 April 1910) was a British decorative artist and industrial designer and an important figure in the Arts and Crafts movement... He was an influential educator and wrote widely on design and pattern. His Cantor Lectures on Ornamental Design for the RSA[2] (1886) led to a series of publications, including The Anatomy of Pattern (1887), The Planning of Ornament (1887), Pattern Design (1903), Ornament and its Application (1904), and Nature and Ornament (1908–9). He published in many journals, including the Magazine of Art, the Art Journal and the Journal of Decorative Art. Other books were Windows (1897),[3] Stained Glass (1903), Alphabets Old and New (1898) and Lettering in Ornament (1902).[4] (1903)

de Villiers, Michael (Facebook). Whirly-gig, 24 September 2013 posting (image 2003)

Here's another geometry doodle I did using the idea of a 'parquet deformation' of a basic rectangular tiling. Read more and experience a dynamic version of this parquet deformation at:

I find the transitions here a little too abrupt (especially of the orange and red tiles). A rare instance of a colour, in a rainbow style, but without map colouring rules. Ideally, this would have been observed.

Documentation Abstracts. American Chemical Society. Division of Chemical Literature. American Documentation Institute. Volume 20, Issues 7–12, 1985, p. 818
Snippet view on Google Books:
Motivated by the idea of describing parquet deformations using grammarsdescribing parquet deformations using grammars and also of describing an infinite number of terminals starting with only a finite set, this paper defines a terminal weighted grammar, where the terminal generated at any step of a derivation…

Durant, Stuart. Ornament: A Survey of Decoration Since 1830, 1986, p. 81
Snippet view on Google Books:
Parquet deformations', 1961 – 3. ULM 12 / 13. Zeitschrift der Hochschule für Gestaltung, March 1965. These exercises are reminiscent of Bauhaus methods, in which simple tessellated shapes are transformed sequentially, by deformation …

Durant, Stuart. Ornament, from the Industrial Revolution to Today. Woodstock, N. Y. : Overlook Press 1986, p. 81
Snippet view on Google Books:
56 Left François MORELLET: Random design of isosceles deformation, into more complex forms. simple tessellated shapes are transformed sequentially, by... These exercises are reminiscent of Bauhaus ... Parquet deformations', 1961 –

Ellison, Elaine K. and John Sharp. ‘Tiled Torus Quilt with changing tiles’. [Sic] Bridges Pécs: Mathematics, Music, Art, Architecture, Culture. Conference Proceedings,  2010, Tessellations Publishing, pp. 67–74
In short, parquet deformation applied to quilts, by the quiltmaker Elaine Ellison, with advice and input from John Sharp. As such, I am not at all impressed by the standard of the paper. Her writing (excluding Sharp’s contribution) is all over the place, of which I lack the will to document. The quilt uses Cairo tiles, a likely Sharp influence. It is not entirely clear if Ellison is striving for one- or two- dimensions. Fig. 2, although at first glance 2-dimensional, is not. References Huff and Kaplan in the discussion. Select paragraph from article: P. 68. Huff used the idea which he called "parquet deformation" as an exercise to develop visual thinking in his students. He was inspired by Escher's woodcut Night and Day. In July 1983 Douglas Hofstadter, writing about these ideas in his Scientific American column [3], explained the basics of one-dimensional parquet deformation.

Erland, Jonathon. ‘Front Projection: Tessellating the Screen’. SMPTE Journals, Volume: 95, Issue 3 March 1986) 1985, pp. 278286
Quotes Hofstadter’s article, p. 286, from superscript in the text, p. 280, but does not discuss parquet deformations as such. As an aside, discusses ‘apogee hexagons’.
D. R. Hofstadter, "Metamagical Themas: Parquet Deformations Patterns of Tiles that Shift Gradually in One Dimension", Scientific American, July 1983.
From SMPTE (Society of Motion Picture and Television Engineers)
SMPTE people form a global professional society of individuals and corporations collaborating for the advancement of all things technical in the motion picture, television and digital media industries. The Society fosters a diverse and engaged membership from both the technology and creative communities, delivering vast educational offerings, technical conferences and exhibitions, informational blog posts, and the renowned SMPTE Motion Imaging Journal…
The Society of Motion Picture and Television Engineers (SMPTE), founded in 1916 as the Society of Motion Picture Engineers or SMPE, is a global professional association of engineers, technologists, and executives working in the media and entertainment industry...

Fathauer, Robert. Tessellations: Mathematics, Art, and Recreation. A K Peters/CRC Press, 2020

Chapter 19, Tessellation Metamorphoses and Dissections pp. 301307. Parquet deformation amid a chapter on tiling metamorphosis in general, including Escher-like art. ‘Morphs’ is his preferred title. Regarding parquet deformation, includes a discussion on Metamorphosis II. Also shows an Islamic-type isometric instance of his own. Has a brief discussion of Huff’s work, p. 302:

… Linear geometric morphs were explored by Williams S. Huff in the 1960s. He called them “parquet deformations” and had his architecture students design them [Hofstadter1986]. Craig Kaplan has developed schemes to allow parquet deformations to be generated by computer [Kaplan 2010].

Greenberg, Bob. Handbook of Practical Geometry. CDM Business Services, 1982, p. 177
Snippet view on Google Books:
HOFSTADTER, D. R. Parquet deformations, Scientific American, July, 1983, p. 14 - 20 Demonstrates cross - deformation of field symmetry patterns.
Robert Greenberg was a Professor of Architecture at Ryerson University. Greenberg graduated Magna Cum Laude with a Bachelor of Architecture from Syracuse University. He taught at Ryerson for 28 years in Design Theory, Architectural History and Studio. He lectured widely throughout North America and the UK, particularly on the subject of Descriptive Geometry. He was Founder and Studio Master of the Ad Quadratum Studio of Geometry in Art and Architecture, which mounted exhibitions in Canada, the US, and the UK. He is the author of the Handbook of Practical Geometry (2nd ed., 1982) and retired to Emeritus status in 2000. He passed away May 28, 2007.

Grünbaum, Branko and ‎G. C. Shephard. Tilings and Patterns. W. H. Freeman, 1987
P. 170: A concept akin to isotopy but distinct from it has been used by some artists. M. C. Escher utilized it in his famous woodcut “Metamorphosis III” and other works (see Escher [1971, [1982]). The tiling is represented on a long strip of paper and it changes gradually as one moves from one end to the other. For other examples of such tilings see Hofstadter [1983], Huff [1983]. It is quite challenging to design deformations of this kind in such a way that every tile can serve as a prototile of a monohedral tiling. Despite the claim to the contrary (see Hoftstadter [1983, p. 14]), most of the tilings shown in Hofstadter’s article include tiles which are not prototiles of any monohedral tilings.
A brief discussion of Hofstadter and Huff, p. 170. In matters of mathematical contention, I thus defer to Grünbaum and Shephard.

Hann, Michael. Structure and Form in Design: Critical Ideas for Creative Practice. Bloomsbury Academic, 2012


Three references, p. xviii, p. 104, 179 (index)

P. 104 Craig Kaplan ‘fractal’.

————. The Grammar of Pattern. CRC Press, 2019, 229 pp.


Three minor references. Page number is odd on Google Books, with page 4–17, 6–163, 6–164 (index)! Quotes Kappraff. Seems inconsequential.

...and one-dimensional parquet deformation (Kappraff, 1991, p. 184).


The Grammar of Pattern describes characteristics of textile and other surface patterns, and identifies, illustrates, and reviews a wide range of pattern types including spotted, striped, checked, tessellating and other types of all-over patterns with original drawings and images.


Professor Michael Hann (BA, MPhil, PhD, FRSA, FRAS, FTI) holds the Chair of Design Theory at the University of Leeds. He is also Director of ULITA – an Archive of International Textiles, an important international archive (and, in the context of this book, a source of illustrative material). He has published across a wide range of subject areas, has made numerous keynote addresses at international conferences, and is an acknowledged international authority on the geometry of design. Recent book publications include: Hann, M (2012). Structure and Form in Design (London: Berg); Hann, M. (2013), Symbol, Pattern and Symmetry (London: Bloomsbury) and Hann, M. (2015), Stripes, Grids and Checks (London: Bloomsbury). He has held adjunct, visiting or invited professorships at institutions in Belgium, Taiwan, Hong Kong, Korea and the Peoples’ Republic of China.

Herbst, Michel. ‘Art Concret, Basic Design and meta-design’. TUGboat, Volume 38 (2017), No. 3, pp. 324328.

Minor discussions on parquet deformations among other matters in the article. Likely uses my deformation, retitled ‘Transformation’, fig. 9, p. 328 without credit. Quotes Hofstadter’s book Metamagical Themas: Questing for the Essence of Mind and Pattern. Oddly, he seems to credit Transformation to Hofstadter!

14 mentions of Huff, but without parquet deformation! Herbst prefers the term ‘transformation’.

Much talk of Huff and Ulm in the round, including the years of Huff at Ulm (1962–66), Carnegie-Mellon (1960–1972), and Buffalo (1974–98).

TUGboat describes themselves as: The TeX Users Group (TUG) is a membership-based not-for-profit organization founded in 1980, for those who are interested in typography and font design, and/or are users of the TeX typesetting system created by Donald Knuth.

Herrero, R, S. Askins, M. Victoria, C. Domínguez, I. Antón. ‘Thermal Effects and Other Interesting Issues with CPV Lenses’. PV module Reliability Workshop-CPV March 1st 

Document is unclear; it does not appear to be an article, and is not paginated.

Lower effect than lens parquet deformation

Inconsequential. A single mention, of uncertain connection with Huff-style parquet deformations. Arguably could be omitted.

Hoeydonck, Werner Van. ‘William Huff’s Parquet Deformations: A Viennese Experiment’.

Conference: Symmetry: Art and Science, 2019 – 11th Congress and Exhibition of SIS Special Theme: Tradition and Innovation in Symmetry – Katachi 形 Kanazawa, Japan, November 25-30, 2019, pp. 286–289.


My presentation in Kanazawa aims to bring a renewed interest and a new nexus of activities around William Huff’s work, especially as an exercise in transformational thinking in the field of architectural education. After his retirement, professor Huff donated a beautiful collection of his students’ work to the HfG-Archiv in Ulm, Germany, which inspired us at the Institute of Art and Design in Vienna to conduct a semester assignment for 450 students around the topic of Parquet

Deformations (P.D.). The open search for strategies to transfer the spatiotemporal idea

of a planar P.D. into 3D led to fascinating results. Hereby we discovered a field of

formal research that opened the students’ eyes for two and three-dimensional

relationships and made them enthusiastic and sensitive for spatial transformations. This

is a brief report on how we conducted the exercise, as a fruitful exercise for the basic

design education of young architects and designers.

Includes a recreation of ‘Leather of the Lesser Gator’, by Thomas C. Davies, 1964. Also includes a modern-day work by Kim Ye-ryum, 2017, but no others. It is not a tutorial as such. My website gets a mention.

————. ‘William Huff’s Parquet Deformations: Two Viennese Experiments’. Bridges 2020 Conference Proceedings, pp. 383386.


This paper aims to bring a new nexus of activities around architect and educator William Huff’s work, presented here as an experimental design assignment in architectural education. After his retirement, professor Huff donated a collection of his students’ work to the HfG Archive in Ulm, Germany, which inspired us, at the Institute of Art and Design in Vienna to give a semester assignment focusing on parquet deformations (PD). The open search for strategies to transfer the idea of planar PDs into 3D led to fascinating results. We discovered a field of formal research that broadened the students’ horizons concerning 2 and 3-dimensional relationships enthusing them considerably by making them aware of the unlimited possibilities of spatial transformation. This is a brief report of a fruitful project in basic design education for architects and designers.

Includes a work by Tobias Dirsch, 2017 and an anonymous designer but no others. Mostly the concern is 3D. My website gets a mention.

Hofstadter, Douglas. 'Parquet Deformations: Patterns of Tiles That Shift Gradually in One Dimension'. ‘Metamagical Themas’, Scientific American 1983, pp. 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), and 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 Grünbaum and G. C. Shephard, in Tilings and Patterns, p. 170, takes him to task, concerning his comment on monohedral tiles, 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 pp. 190–199. In Metamagical Themas: Questing for the Essence of Mind and Pattern. Basic Books; First printing 1985, New edition 1996
This essentially repeats Hofstadter's original July 1983 column in Scientific American (his last), with extra, minor text, but also, more importantly, a ‘post scriptum’, in which a new (not previously shown) parquet deformation of David Oleson’s ‘I at the Center’ is illustrated and discussed, and much praised.
“… One of my favorite parquet deformations is called “I at the Center”, and was done by David Oleson at Carnegie-Mellon in 1964. This one violates the premise with which I began my article: one-dimensionality. It develops its central theme-the uppercase letter`I’-along two perpendicular dimensions at once. The result is one of the most lyrical and graceful compositions that I have seen in this form. I am also pleased by the metaphorical quality it has. At the very center of a mesh is an I-an ego; touching it are other things-other I’s-very much like the central I, but not quite the same and not quite as simple; then as one goes further and further out, the variety of I’s multiplies. To me this symbolizes a web of human interconnections. Each of us is at the very center of our own personal web, and each one of us thinks, “I am the most normal, sensible, comprehensible individual.” And our identity-our “shape” in personality space-springs largely from the way we are embedded in that network-which is to say, from the identities (shapes) of the people we are closest to. This means that we help to define others’ identities even as they help to define our own. And very simply but effectively, this parquet deformation conveys all that, and more, to me…” 

————. Fluid Concepts and Creative Analogies. Computer Models of the Fundamental Mechanisms of Thought. Harvester Wheatsheaf 1995, and Allen Lane The Penguin Press 1997, 501 pp.
A single page discussion on parquet deformations, albeit without diagrams, p. 477. A heavyweight tome, of largely of an academic nature, although readable, but obscure, with numerous essays, albeit invariably of limited interest, way beyond my understanding. And an interesting tidbit on Amazon, below the main discussion!

P. 477. For me, what Lenat and Chamberlain did for their programs is strongly reminiscent of the role that is played by William Huff, an architecture professor, with respect to students in his design courses. Huff has a long-standing tradition of assigning his design students the challenge of creating "parquet deformations" - tilings of the plane that gradually metamorphose in an Escher-like manner as they move across the plane (many examples are given and discussed in Chapter 10 of Hofstadter, 1985). To get the idea across to the students in each successive class, Huff shows a portfolio consisting of what he considers to be the best examples from previous years. Thereby inspired, the current crop of students then produces a large set of new parquet deformations, most of which are not great, but usually at least a few of which are novel and exciting. As one would expect, Huff applies his own keen artistic judgment to the latest harvest, pruning the weak ones out and adding his favorites to the growing portfolio to be shown to subsequent classes. In this way, a process of evolution takes place, with Huff playing the role of natural selection, letting artistically weak specimens die and strong ones survive, and then propagating the "most fit genes" by exhibiting the survivors to his class the next year. Over a period of some twenty or more years, Huff has managed to direct the course of evolution of parquet deformations in a very interesting way.

The question naturally arises as to the authorship of all these pieces. Huff has a practice of labeling each piece, when they are exhibited in a museum or gallery, "from the studio of William Huff", with no further information. However, when I decided to publish a small selection of these beautiful studies, I felt that Huff's labeling practice was too one-sided, and so for each piece I listed both Huff's name and the student's name. I felt this was fairer. But I certainly could see two sides of this question. There was no doubt in my mind that Huff deserved a large portion of the credit. Whether it was less or more than 50 percent remains an unresolved but fascinating question in my mind.

Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought is a 1995 book by Douglas Hofstadter and other members of the Fluid Analogies Research Group exploring the mechanisms of intelligence through computer modeling. It contends that the notions of analogy and fluidity are fundamental to explain how the human mind solves problems and to create computer programs that show intelligent behavior. It analyzes several computer programs that members of the group have created over the years to solve problems that require intelligence. It was the first book ever sold by


————. I Am a Strange Loop. Basic Books, 2007 Hardback, 2008 Paperback, 412 pp.

David Oleson I at the Center parquet deformation discussion in Chapter 1, ‘How We Live in Each Other’, pp. 241–258. The premise of the book, detailed below, thus makes David Oleson's work here an obvious choice for inclusion. Note that Hofstadter here is making a point in his writing, and is not a discussion as to the parquet deformation in general, as with his Scientific American 1983 piece.

Page 252. All of this suggests that each of us is a bundle of fragments of other people’s souls, simply put together in a new way. But of course not all contributors are represented equally. Those whom we love and who love us are the most strongly represented inside us, and our “I” is formed by a

complex collusion of all their influences echoing down the many years. A marvelous pen-and-ink “parquet deformation” drawn in 1964 by David Oleson (below) illustrates this idea not only graphically but also via a pun, for it is entitled “I at the Center”:


Here one sees a metaphorical individual at the center, whose shape (the letter “I”) is a consequence of the shapes of all its neighbors. Their shapes, likewise, are consequences of the shapes of their neighbors, and so on. As one drifts out toward the periphery of the design, the shapes gradually become more and more different from each other. What a wonderful visual metaphor for how we are all determined by the people to whom we are close, especially those to whom we are closest!


I Am a Strange Loop is a 2007 book by Douglas Hofstadter, examining in depth the concept of a strange loop to explain the sense of "I". The concept of a strange loop was originally developed in his 1979 book Gödel, Escher, Bach...

In the end, we are self-perceiving, self-inventing, locked-in mirages that are little miracles of self-reference.

— Douglas Hofstadter, I Am a Strange Loop p. 363

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. Picture of Huff, p. 25. 

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, Hotz uses a Cairo tiling here, and of a transition from a square to basketweave, in a way (I believe) that I have not seen previously!

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. ? 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.

Note that somewhat confusingly, Huff also wrote another article under the same title, in urban structure, of 1968, with effectively reuse of the text of the entire article here, indeed almost word for word, with only minor occasional changes, and he also reused some of the diagrams, albeit to a lesser degree. In short, urban continues his ideas.

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

In short, the text here is largely a rehash of his earlier (1965) article of the same title. Indeed, he reuses the text (and bibliography) up to and including p. 274 almost word for word, with only minor occasional changes, with the new text beginning ‘Descriptions of four major projects’. Again, oddly, there is no discussion on parquet deformation in the text. Two new parquet deformations are shown, albeit without a title, year, or designer. These are simply captioned:

Top Parquet deformation

A development on a square grid


Centre Parquet deformation

A development on the special rhombic grid

Interestingly, Top uses a Cairo tiling.

Of note is the historical significance here, the second appearance in print of parquet deformations, of text and images (the first instance, by Huff, was of 1965).

————. ‘Symmetry’. Oppositions. Issue 3, p. 23, 1974. Published for The Institute for Architecture and Urban Studies by The MIT Press


Snippet view on Google Books:

Parquet deformation by Richard Lane. Basic Design course, 1963. Teacher: William S. Huff. departure from the Bauhaus tradition found clear expression in three sets of academic courses that were common to all four departments. First, in the …


Oppositions was an architectural journal produced by the Institute for Architecture and Urban Studies from 1973 to 1984. Many of its articles contributed to advancing architectural theory and many of its contributors became distinguished practitioners in the field of architecture. Twenty-six issues were produced during its eleven years of existence.

————. “Best Problems” from Basic Design - - 20 Feb. 1979. REVISED 20 Feb. 1979. THE PARQUET DEFORMATION (text and capitalization as given) N.B. Appears in Tim McGinty’s Best Beginning Design Projects (q.v)
There is some uncertainty as to what exactly this partial(?) document is. It is subtitled ‘The Parquet Deformation’. It appears to be a ‘study guide for students’, used in Huff’s classroom. Further, I only have two sheets (hand) numbered pp. 30–31 and 33. It is ordered in three sections as ‘The task’, ‘The principle’, and ‘The pedagogic goal’. Three parquet deformations, one of which, p. 31, made it into the Hofstadter Scientific American article.
Uncertainties aside, the document from Buffalo in 1979, discusses parquet deformation concepts. However, it is not a tutorial as such. Of interest is that it is stated a ruling pen was used to execute the designs.
Another like ‘guide’ (without reference to parquet deformations) is titled ‘The Mirror-Rotation Symmetry’, p. 34, with the same sections as given above. Are there more? (scroll to the end of the document)

 ————. ‘The Landscape Handscroll and the Parquet Deformation’, In Katachi U Symmetry. Tohru Ogawa, ‎Koryo Miura, ‎and Takashi Masunari. Tokyo: Springer-Verlag, 1996, pp. 307314.
Of fundamental interest, and a must-read in the field. Huff compares the Landscape Handscroll and the Parquet Deformation with the Sino-Japanese right-left viewing and the Western left to right, and more, including temporality, off-shoots of D'Arcy Thompson and Escher. An excellent all-round overview, with all aspects considered. This being so, I thus detail this article more extensively than others. The main substance is Chapter 2. The article includes:

1 The Aesthetics of the Parquet Deformation: Canons and their Afterimage

1.1 Do East and West Share the Same Sense of Drama?

1.2 Spatial versus Temporal Art Forms

1.3 Temporal Visual Art, Experienced through Channeled "Serial Images"

1.4 How Are Handscrolls and Parquet Deformations Composed?

2 The Mathematics of the Parquet Deformation: Constraints of Symmetry and


2.1 Parquet Patterns, a Recent Diversion in Geometry

2.2 Designing Novel Parquet Patterns and Deforming One into Another

2.3 Influence of D'Arcy Thompson; Comparisons with M. C. Escher

The article has seven parquet deformations from Huff’s studio, with works by, in sequential order, Liou Jiunn-liang (Romeo and Juliet, 1993), Fred Watts (Fylfot Flipflop, 1963), Pamela McCracken (Cloisonné, 1990), Loretta Fontaine (Seven of One Make Three, 1991), Vincent Marlowe (Clearing the Thicket, 1979), Alexandria Gelencser (Swizzle Stick Twirl, 1986), Bryce Bixby (They Come, They Go, 1991). All are from Buffalo, save for the Watts (Carnegie-Mellon) instance. However, perhaps a little oddly, there is no discussion (or even a reference) of these in the text. Likely, these serve for generic illustrative purposes.

Of note, in detail:

2.1 Parquet Patterns, a Recent Diversion in Geometry

Essentially, the background to tiling, titled ‘parquet’ by Huff. Mentions of Thomás Maldonado (HfG), Martin Gardner (Scientific American column), and Branko Grünbaum (definition of monohedral tiling). Mentions ‘improper’ parquets.

2.2 Designing Novel Parquet Patterns and Deforming One into Another

The title here suggests a tutorial, but this is not so. Rather, it discusses matters of Bravais lattices and rotational symmetry.

Chapter 2.3 Influence of D’Arcy Thompson; Comparisons with M. C. Escher.

Of note here in the first line is: 

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.

The ‘unnamed student’ is Peter Hotz, derived from Huff’s notes for a SEMA talk (2003). Mentions D’Arcy Thompson’s, On Growth and Form and his chapter ‘On the Theory of Transformations’, and Escher's comparable work. Also detailed is, in so many words, ‘permissible’ and ‘non-permissible’ parquets (not illustrated), the intricacies of which (without visual aid) I am at a loss to understand. Much here is taken from Hofstadter's 1983 article.
In the references, Huff quotes four people, including Walther Lietzmann, Anschauliche Topologie and P. A. MacMahon, New Mathematical Pastimes. However, I’m not sure why. In particular, Lietzmann only has a small four-page chapter on tilings, and these are the basics of a school-level introduction. He mentions the MacMahon book elsewhere in general for his tiling interest.

————. "About Parquet Deformations" in transforma, Book of Abstracts of the 2° Congreso Internacional y 4° Nacional de la Sociedad de Estudios Morfólogicos de la Argentina (SEMA), 9. Córdoba, Argentina, 2003, p. 9.
The impression here is of a dedicated article, of perhaps some substantive nature. However, this is not so. Rather, it is, as the title suggests, an abstract, of a single page, without parquet deformation diagrams. More exactly, this is the ‘official’ account of a slide show Huff gave to SEMA, of which the talk was presented based upon a series of ‘notes’ not in the public domain. These notes are arguably the best source of his interest in parquet deformation in the round. The document is invaluable in so many ways.

The reference to this little-quoted article was in Patricia Muñoz’s Spatial Lines.

From SEMA website:
SEMA is the acronym for the Society of Morphological Studies of Argentina. Our society summons those who investigate, teach and produce forms in different disciplinary fields in order to build a common territory. SEMA tries to be a meeting space for architects, artists, designers, mathematicians, musicians, poets, philosophers, biologists ... and for all those genuinely interested in the territory of Form.
We founded SEMA in December 1996 in Buenos Aires, Argentina. Since then, we have held eleven biannual congresses (eight of them international), published publications and organized numerous artistic and scientific conferences and meetings throughout the Argentine territory... SEMA has become a rich space for meeting, working and exchanging knowledge between people and institutions from all over the country and related groups from all over the world.
The SEMA website does not appear to have archives, and I could not find this publication on Bookfinder.

————. ‘Simulacra of Nonorientable Surfaces—Experienced through Timing’. In Spatial Lines, (Líneas espaciales) Patricia Muñoz, compiler. Buenos Aires: De la Forma, 2010, 128 pp., 42–49.
I have Huff’s paper (and the book, as a pdf) in different styles. First the book, of a (free) English translation, but this is without illustrations (and page numbers), seemingly purposefully so, as a preview pending purchase, or so I surmise. This gives Huff’s article, without illustrations, and of perhaps by the title is not of obvious parquet deformation connection. However, this is in the context of time, but is of little significance in itself, as well as being a brief account; see below.   

I also have Huff’s chapter from the book (kindly supplied by Claudio Guerri), in Spanish, with illustrations, but no parquet deformations are shown. Interestingly, in the references, p. 49, he mentions the SEMA 2003 conference, relating to the text, which is where this obscure text first came to my attention.

See Chapter 4. One paragraph of reference to parquet deformations amid a Möbius Band premise:
The Experience of Timing
On previous occasions, I gave oral and written accounts of a type of design, regularly assigned in my basic design studio—the parquet deformation—which disposes time to participate as an integral third dimension, thus dynamizing the two-dimensional spatial content of the design. Commentary on the aesthetic potential of the parquet deformation was presented at the Katachi 2 conference (Huff 1994: 219-222), and commentary on its geometric requisites was presented at the SEMA 4 conference (Huff 2003: 9). I liken the parquet deformation to a remarkable art form, the Chinese handscroll, which, in its most exceptional, but younger genre, the landscape handscroll, goes back a thousand years. Time unfolds as the scroll is synchronously unrolled and rolled—pleasurable frame by pleasurable frame—not dissimilarly to how music flows. Time is engaged, however, in a different manner in respect to compositions whose three dimensions are all spatial.
Background of the compiler: Professor Patricia Muñoz, Industrial Designer, Doctor UBA
This book has several authors. Some are part of a research group at IEHU, Laboratory of Morphology, FADU, University of Buenos Aires. Others are part of the teaching team that brought this topic to the classroom at the Industrial Design Career, FADU, UBA. In addition, three guests make a significant contribution from other areas. Roberto Doberti refers to the interdisciplinary nature of this work, saying: It is no less important that the book be a product of various hands, none of which loses its particular tenderness in the caress of these forms. In turn, Claudio Guerri points out that this production:unites temporal extremes turning them into spatial neighborhoods, by exploring a theme that links ancient Greek developments with contemporary design practice.

————. ‘Defining Basic Design as a Discipline’. In Symmetry: Art and Science, Vol. 2 (new series) Numbers 1–4, 2012, pp. 91–98
Parquet deformation is only mentioned once, in passing, as a reference in his list of publications, on p. 91 (re Katachi U Symmetry).

Jablokov, Alexander. ‘Living Will’. Isaac Asimov's Science Fiction Magazine, Davis Publications, Dozois, Gardner (ed). June 1991, Vol. 15, Issues 7–9, p. 64.
Full text at Internet Archive:
The bathroom was clean tile with a wonderful claw-footed bathtub. The floor was tiled in a colored parquet-deformation pattern that started with ordinary bathroom-floor hexagons near the toilet, slowly modified itself into complex knotted shapes in the middle and then, by another deformation, returned to hexagons under the sink. It had cost him a small fortune and months of work to create this complex mathematical tessellation. It was a dizzying thing to contemplate from the throne and it now turned the ordinarily safe bathroom into a place of nightmare. Why couldn’t he have picked something more comforting?

Joseph, M. and R. Shyamasundar. Foundations of Software Technology and Theoretical Computer Science: Fourth Conference, Bangalore, India December 13–15, 1984.
Snippet view on Google Books:
Terminal weighted matrix grammars are used to describe parquet deformations. The hierarchy of families generated by putting various restrictions on the functions is studied. 1. INTRODUCTION It has been of interest to generate various .
Occasional discussions on parquet deformations Terminal
I am at a loss to explain ‘weighted matrix grammars’. From what I have seen, it is abstruse.
Beecombing Blossoms, p. 198, Fylfot Flipflop p. 194, albeit at a most abstruse level.

Kaplan, Craig S. and David H. Salesin. ‘Escherization’. SIGGRAPH '00: Proceedings of the 27th annual conference on Computer graphics and interactive techniques, July 2000 pp. 499–510
A brief mention on parquet deformations in passing (not illustrated), p. 510:
...This research suggests many future directions, including generalizing our algorithms to handle multihedral and aperiodic tilings, parquet deformations [13, Chap. 10], or tilings over non-Euclidean domains, such as the hyperbolic plane [7]....
The reference quotes Hofstadter’s Metamagical Themas.
Note a similar titled paper by both authors, ‘Dihedral Escherization’, of 2004, does not contain any references to parquet deformation or Huff.

Kaplan, Craig S. Computer Graphics and Geometric Ornamental Design. A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 2002
A major writing, of 18 references to parquet deformation. Many references to parquet deformations in Chapter 3.4.1 ‘Islamic parquet deformations’ pp. 57–58 and Chapter 5.4, ‘Deformations and metamorphoses’, pp. 187–194. Also see p. 212. A select few references:
P. 57
Parquet deformations are a style of ornamental design created by William Huff, a professor of architectural design, and later popularized by Hofstadter in Scientific American [83, Chapter 10]. They are a kind of “spatial animation,” a geometric drawing that makes a smooth transition in space rather than time. Parquet deformations are certainly closely related to Escher’s Metamorphosis prints, though unlike Escher’s work they are purely abstract, geometric compositions. They will be discussed in more detail in Section 5.4. Hankin’s method can be used as the basis for a simple but highly effective method of constructing Islamic patterns in the spirit of parquet deformations. I lay out a strip of the template tiling and then run a modified inference algorithm where the contact angle at every contact point is determined by the location of that point in the strip. Smoothly varying the contact angle results in a gently changing geometric design that is still recognizably Islamic. The construction process is illustrated in Figure 3.7; two more examples appear in Figure 3.8. These parquet deformations occupy an interesting place in the world of Islamic geometric art. They have enough overall structure and balance to satisfy the Islamic aesthetic, but they would not have been produced historically because very little repetition is involved. The effort of working out the constantly changing shapes by hand would have tested the patience of any ancient designer.

P. 58
Figure 3.7 The construction of an Islamic parquet deformation based on Hankin’s method. The top rows shows the effect of continuously varying the contact angle of a ray depending on the x position of the ray’s starting point in the design. When the process is carried to all other tiles, the design in the second row emerges.
Figure 3.8 More examples of Islamic parquet deformations based on Hankin’s method. 

P. 190
Figure 5.2 Examples of parquet deformations.
Presumably, a one-to-one correspondence is established between the tiles of T1 and T2, and as a parameter t moves from 0 to 1, each individual tile gradually deforms from its T1 shape to its T2 shape. The transition might be carried out spatially as in Escher’s art, or even temporally as a smooth animation from T1 to T2.
As was mentioned in Section 3.4.1, the parquet deformations of William Huff are a kind of spatial animation. Huff was inspired directly by Escher’s Metamorphoses. He distilled the style down to an abstract core, considering only interpolation transitions, and favouring abstract geometry rendered as simple line art to Escher’s decorated animal forms. As reported by Hofstadter [83, Chapter 10], Huff decided further to focus on the case where T1 and T2 are “directly monohedral,” in the sense that every tile is congruent to every other through translation and rotation only. We may also assume he had only periodic tilings in mind. Finally, he asked that in the intermediate stages of the deformation the tile shapes created could each be the prototile of a monohedral tiling (Hofstadter amends this rule, pointing out that some deformation might be necessary to make the intermediate shapes tile). Inspired by parquet deformations and by Escher’s interpolation transitions, we may pose the

P. 191
related problem of finding a smooth transition between any pair of isohedral tilings. A solution to this problem might then be expanded to encompass Escher’s work (by considering a k-isohedral extension) or parquet deformations (by introducing the restrictions mentioned above). In any case, the isohedral problem is sufficiently interesting, and the results sufficiently attractive, that it can be fruitfully studied in isolation....

P. 193
As long as any pair of Laves tilings is joined via a path of base cases, we should be able to move between any two isohedral types. I have found topological transitions that obey all the restrictions of parquet deformations and that unify all the Laves tilings except for (4.6.12).
P. 194
Figure 5.3 A collection of parquet deformations between the Laves tilings

————. ‘Islamic Star Patterns from Polygons in Contact’. Proceedings of Graphics Interface 2005, pp. 177–185

Islamic designs reminiscent of Huff's parquet deformations. ... in the style of Huff's parquet deformations [16, Chap- ... book also appear in a recent paper [4].

————. ‘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, pp. 42–45.

 ————. ‘Curve Evolution Schemes’. In Bridges 2010 Mathematical Connections in Art, Music and Science, pp. 95–102.
Some most impressive, highly advanced (in concept) parquet deformations.

————. Introductory Tiling Theory for Computer Graphics. Morgan and Claypool Publishers, 2009

A brief reference in passing.

P. 53. 13. Write a program to create parquet deformations: patches of tiles that slowly evolve…Parquet deformations were devised by Huff… My Bridges 2008 paper discusses methods for drawing parquet deformations based on isohedral tilings.

Brief mention of Huff and parquet deformation.

————. ‘Animated Isohedral Tilings’. Bridges 2019 Conference Proceedings, pp. 99–106

A brief discussion. There are other implied parquet deformations in the context of the animations.
Escher was a master of this form; as I have explained elsewhere [3], he used a number of visual “metamorphosis” devices to draw tilings that change spatially. Inspired by Escher, the architect and designer William Huff developed “parquet deformations” [2], which were designed to be more abstract and geometric exercises. I have explored several techniques for drawing parquet deformations [3, 5], some of which are relevant here. Temporal animations of tilings are arguably easier to construct than spatial animations, because in the latter case the tiles are changing their shapes in the same dimension in which they are trying to interlock.

Kalay, Yehuda E (ed.). Computability of Design (Principles of Computer-Aided Design), John Wiley & Sons, 1987

P. 30 ... highly similar and the intention of the drawing activity can be captured in a single phrase, “put windows in the facade”. Another example is in the creation of tilings of the plane, a design exercise originated by William Huff (Hofstadter 1983) 

Implies parquet deformation.

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

Kheybari, Abolfazl Ganji, Dr. Hamed Mazaherian; Mohammad Amin Farahbakhsh; Setare Bitaraf. ‘Parametric Development of Star-shaped Motifs in Islamic Geometry’. Privately published as a Word Doc?

Inconsequential. Craig Kaplan-inspired studies, essentially repeating his 2000 work, even using his diagrams!

P. 6. Islamic parquet deformations are a style of ornamental design and a geometric drawing that makes a smooth transition in space. In a strip of the template tiling the contact angle at every contact point is determined by the location of that point in the strip. Varying the contact angle results a gently changing geometric design.

Figure (7) - The construction of an Islamic parquet deformation based on Hankin’s method. [9-P58]

Quotes Kaplan in the references:

[12]. Craig S. Kaplan, Computer Generated Islamic Star Patterns, 2000

Kim, Scott. Inversions. W. H. Freeman and Company, New York, 1989. Originally published Peterborough, N. H.: Byte Books, 1981 (the latter not seen)
A single ‘square to arrow’ parquet deformation and discussion by Kim, pp. 14–15. However, this is not discussed in terms of a parquet deformation/metamorphosis or in an otherwise related style, but rather as of figure and ground (just one of his many fields of interest). The work is not titled nor dated. Below I repeat the text. A slightly more detailed description is on his website (and of which gives the date), shown below the book text.
P. 15. My first inversion was in fact one of the most unusual. I was quite lucky to have started with such a challenging theme. In 1975 I attended a class in graphic design. One of the assignments was the following:
Produce a flat design in two or more colors that has no background: that is, one in which the spaces between forms are as positive as the forms themselves (as in a checkerboard). The objective is to make all of the parts of your composition interrelate — use all of the space and make it all work.
Foreground and background are also known as figure and ground, respectively. One way (but by no means the only way) to interrelate figure and ground is to make them exactly the same shape. In the figure on the left, for instance, the spaces between the black arrows form white arrows. If we repeat this pattern, we can cover the whole plane with alternating black and white arrows.
A slightly more detailed description, essentially based on the same text, is on his website:
Figure 1975
SYMMETRY. Figure/Ground tessellation by translation.
INSPIRATION. Created in response to an assignment in a graphic design class. First reproduced in Godel, Escher, Bach by Douglas Hofstadter (Not so; I looked but could not find!)
STORY. The following story is excerpted from the introduction to my book Inversions.
My first inversion was in fact one of the most unusual. I was quite lucky to have started with such a challenging theme. In 1975 I attended Matt Kahn’s Basic Design course in the art department at Stanford University. One of the assignments was the following:
The following text is then repeated as above in the June 23 2020 entry.
A pleasing parquet deformation, in black and white, with broken symmetry. I have not seen this instance reproduced outside of the book, nor discussed elsewhere, likely due to the lack of reference as a parquet deformation. An open question here is Hofstadter's influence, if any; he is close friends with Kim, and wrote the foreword. The work predates Hofstadter’s ‘official’ interest in print (of 1983).

Kinsey, L. Christine and ‎Teresa E. Moore. Symmetry, Shape and Space: An Introduction to Mathematics. Key College Publishing, 2002. Hardcover Wiley 2008, 494 pp.
NOT SEEN, GOOGLE BOOKS REFERENCE N.B. I have a like titled book by the same authors Symmetry, Shape and Space with the Geometer's Sketchpad, 169 pp., which is notably different. I think I thought I was getting the original material and the new sketchpad material when I ordered the latter!  P. 108. Chapter 4 Tesselations [sic] Parquet Deformations In 1937 the Dutch graphic artist M. C. Escher began to experiment with the metamorphosis of his tiling patterns… P. 113. A leter development of the idea of transforming tiling took place in the architecture studios of William Huff, at Carnegie Mellon and SUNY-Buffalo. He coined the phrase parquet deformation and dictated two rules (quoted from Hosftadter) These rules are far more formal than the transitions used by Escher, but they lead to some beautiful work, some examples of which are owned by the Museum of Modern Art. The interested reader is encouraged to investigate the Hofstadter book from the suggested readings for this chapter. From what I have seen on Google Books, there are no Huff-style parquet deformations, but rather a discussion on Escher’s usage, in connection with his life-like tessellations in prints. Quotes the Hofstadter book in the suggested reading at the chapter end. Wikipedia Laura Christine Kinsey is an American mathematician specializing in topology. She is a professor of mathematics at Canisius College.

Kitchen, Paul. Portfolio. Student Architectural Portfolio. Self Published on Issuu March 5, 2018, 23 pp.

See pp. 8–9, 18–19

Goes from 1-dimension to 2-dimensions and then (ostensibly) to 3-dimensions.

N. B. I looked for Paul Kitchen, parquet deformations separately, but without success.

KPMG Peat Marwick Collection of American Craft: A Gift to the Renwick Gallery of the National Museum of American Art. Published by Smithsonian Institute. 1994. Foreword, Jon C. Madonna; introduction, Michael W. Monroe; essays, Jeremy Adamson
Snippet view on Google Books:
In scientific terms, such a progressive, step-by-step alteration is known as a parquet deformation. But in Studstill's case, the principle of incrementally graduating tones was not soley [sic] based on scientific or color theory, but also intuited …
HathiTrust: Published on occasion of the exhibition held at the Renwick Gallery, National Museum of American Art, Smithsonian Institution, Washington, D.C., Feb. 25-April 17, 1994.
Introduction briefly describes the history of the Renwick Gallery of the National Museum of American Art which specializes in American crafts and decorative arts. The rest of the book concerns the donation of the American craft collection from KPMG Peat Marwick. Includes a background of corporate collecting in America in the past century, reproductions of many works in the collection, and biographies on the artists.
As the eye moves from top to bottom, Quilt #17 undergoes a subtle but definitive transformation in color and light. Gradually, almost imperceptibly, lighter color values change into darker ones. ​“Each of my quilts,” the artist writes; ​“is a study in light.” Surface paint helps ease the transition from one color to another. In scientific terms, such a progressive, step-by-step alteration is known as a parquet deformation. But in Studstill’s case, the principle of incrementally graduating tones was not soley [sic] based on scientific or color theory, but also intuited from the experience of her Texas Hill Country surroundings. ​“I am inspired,” she writes; ​“by landscape views and vistas, fields of [all kinds] … and changes in my local landscape.”
N.B. Quilt #17, 1982, is not a parquet deformation as such!

Kreutzer, Wolfgang and ‎Bruce McKenzie. Programming for Artificial Intelligence: Methods, Tools, and Applications. Addison-Wesley, 1991
Snippet view on Google Books:
P. 650 Parquet deformations: a subtle, intricate art form. In: [Hofstadter (1985), 191 - 212 ] …
Appears to be merely a bibliographic reference.

Krithivasan, Kamala and Anindya Das. ‘Terminal weighted grammars and picture description’. Computer Vision, Graphics, and Image Processing, Volume 30, Issue 1, April 1985, pp. 13-31.
Motivated by the idea of describing parquet deformations using grammars and also of describing an infinite number of terminals starting with only a finite set, this paper defines a terminal weighted grammar, where the terminal generated at any step of a derivation is defined as a function of time. It is seen that terminal weighted regular grammars generate exactly the class of recursively enumerable sets. Terminal weighted matrix grammars are used to describe parquet deformations.
What exactly is a ‘terminal weighted grammars’ is unclear. From Ricardo Wandre Dias Pedro et al:
Grammars are widely used to describe string languages such as programming and natural languages and, more recently, biosequences. Moreover, since the 1980s grammars have been used in computer vision and related areas. Some factors accountable for this increasing use regard its relatively simple understanding and its ability to represent some semantic pattern models found in images, both spatially and temporally.

Lamm, Dan. Material Systems, MIT Media Lab, 2015. Self Published on Issuu October 31, 2015, 169 pp.

See pp. 80-81.

Prepared for admission into the Mediated Matter research group in the Media Lab at MIT.

The first part of this project was to select and analyze a hand-drawn parquet deformation with no computational ...

N.B. I Looked for ‘Dan Lamm parquet deformations’ separately, but without success.


Laparidis, Stavros. ‘The Role of Allusion in Ligeti's Piano Music’. Dissertation, 2012, p. 22. 

19 Example 5. Étude 9: Vertige, opening seemingly static but constantly changing type

of music as “parquet deformation,” a very insightful term to describe this compositional design…


Only a part-preview is available on ProQuest, of which just the first 13 pages are viewable.

Although likely of a mention in passing, of interest due to one of the few music links.

Lawrence, Cindy. ‘Adding it all Up: Building the National Museum of Mathematics’. Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture, pp. 548-550
A brief mention essentially in passing of a fabricated(?) Craig Kaplan parquet deformation upon the opening of the museum.
P. 545. Art and math are interwoven within the very fabric of the Museum. A parquet deformation designed by Craig Kaplan surrounds the front façade (Fig. 5).
Figure 5: Parquet deformation (C. Kaplan)
Disappointingly, Fig. 5 is in close up, rather than the parquet deformation in context with its surroundings.

Lee, Kevin. ‘Algorithms for Morphing Escher-Like Tessellations’. Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture pp. 483–48
Inspired by the way M.C. Escher combined metamorphosis and regular division in his art, I explore linear and nonlinear algorithms that automatically morph tiles from the base polygon to a final shape. The morphing can be visualized as an animation or as a parquet deformation...
[2] Craig S. Kaplan. Curve Evolutions Schemes for Parquet Deformations. In Bridges 2010: Mathematical Connections in Art, Music and Science, pages 95-102, 2010.
Inconsequential (as good as the paper may be in morphing aspects), of just two mentions in passing. Mostly on morphing Bruce Bilney’s elephant tessellation.


Leone, Francesca. Portfolio Progettazione Grafica. Self Published on Issuu July 14, 2019, 68 pp. 

In Italian. Ostensibly on parquet deformation, but in reality not. Gives Escher’s Day and Night. Mentions Huff, p. 14.


Leopold, Cornelie. ‘Structures and Geometry in Design Processes’. Journal title is not given

A brief mention of (implied?) parquet deformation P. 7:

The students at Ulm of Design worked on tessellations and patterns and developed one

pattern in another, called net transformations or metamorphosis.

Figure 12: Net transformations by student by Arno Caprez at Ulm of Design 1965/66, teacher William S. Huff (17)

The work is untitled.

Some of those methods are presented in this paper which had been one of the background for the DAAD Summer School;Structure – sculpture in Buenos Aires, where students worked on the design task analyzing and redesigning the Ulm Pavilion by Max Bill.


Lindinger, Herbert. Ulm Design. The Morality of Objects, MIT Press, 1991.

I am given to understanding that parquet deformation is featured here, but have lost the reference!


Llonardi, Giulia. Portfolio. Self Published on Issuu May 19, 2017, 16 pp.

See p. 13 (no text)

N. B. I Looked for ‘Giulia Llonardi, parquet deformations’ separately, but without success.

Maldonado, Tomás. Il futuro della modernità. Feltrinelli, 1987, p. 52
Snippet view on Google Books:
Molto importanti sono anche gli studi di W. Huff nel campo delle “parquet deformations”. Vedasi al riguardo D. HOFSTADTER, Parquet deformations a subtle intricate art form, in Metamagical Themas: Questing for the Essence of Mind and...

Maldonado, Tomás. ‘Il contributo della scuola di Ulm’ = The Legacy of the School of Ulm. Rassegna, 19, September 1984, pp. 5, 36–39. In Italian, with an English version (not seen)
A major piece on Willam Huff on basic design in the round, with seemingly his own words, including parquet deformation illustrations, albeit somewhat peripherally to the article; they are not discussed or even referred to as such! Includes a picture of Huff, and five parquet deformations, titled and designer given: Toccata, by Robert Glenn, Five to Four and Two Halves, by Robert Nagel, Proboscidean by Ralf Glasmeir, Venetian Net, by Maurizio Sabini, and Crackle by David Goldman. Note that both Proboscidean and Venetian Net feature the Cairo tiling. Although I do not have the English translation issue, I have a PDF of the Italian original kindly provided by Maurizio Sabini and a translation kindly provided by Werner Van Hoeydonck.

Mathematical Reviews. American Mathematical Society, Vol. 87, 1987
Snippet view on Google Books:
P. 433. The algorithms discussed in weights 'to matrix grammars for describing parquet deformations this paper have proved to be successful in delivering sequences of is extended to Ex - TMGs . ”pass directions of minimal length for several ..

Miles, Thomas H. Critical Thinking and Writing for Science and Technology. Heinle & Heinle Publishers Inc., U.S. 1989, 1990 p. 232.
Snippet view on Google Books:
...fair amount and understand only a little, but there is something in those articles in the Times and Science News and Scientific American — something compelling in the various terminologies (“dark matter," "flavor," "parquet deformation”, lepton. ..
Of an initial look, I couldn’t find any biographical or contact details on Miles.

Moradzadeh, Sam and Ahad Nejad Ebrahimi. ‘Islamic Geometric Patterns in Higher Dimensions’. Nexus Network Journal Vol. 22, 11 May 2020, pp. 777–798 


Springer snippet: William Huff, an American architecture professor, used the term “parquet deformation” in the 1960s and later Douglas Hofstadter developed this…

Neves, Isabel Clara et al. ‘The Legacy of the Hochschule für Gestaltung of Ulm for Computational Design Research in Architecture’, 2013. Open Systems: Proceedings of the 18th International

Conference on Computer-Aided Architectural Design Research in Asia (CAADRIA 2013), 293–302

From a reference by Tuğrul Yazar in his ‘Revisiting Parquet Deformations…‘, 2017 paper. Skim read. Only of background interest in regards to parquet deformation as to the Hochschule in itself. However, that is it; there is nothing of parquet deformation or Huff, or indeed, anything connected directly to the subject itself. Only of peripheral interest at best.

Isabel Clara Neves’s name came to my attention in a single reference (The contribution of Tomas Maldonado... of 2013) in Tugral Yazar’s (2017) paper. Consequently, I then investigated her further. In the light of this, I found four other papers of hers, mostly on HfG matters, rather than parquet deformation. However, the paper ‘The Emergence…’ does contain three parquet deformations, by James Eisemann.

Neves, Isabel Clara, João Rocha and José Pinto Duarte. ‘Computational Design Research in Architecture: The Legacy of the Hochschule für Gestaltung, Ulm’. International Journal of Architectural Computing, Vol. 12, No. 1 March 2014.

Only of background interest in regards to parquet deformation as to the Hochschule in itself. Nothing on parquet deformation. Two mentions of Huff (re Maldonaldo), including one in the references. Mostly on Maldonado, with 63 mentions!

However, of note is the space-filling curve and Sierpinski triangle, pp. 10-11. Also see below for the same diagram.

The use of computational processes in architecture is a widespread practice which draws on a set of theories of computer science developed in the 60s and 70s. With the advent of computers, many of these methodologies were developed in research centres in the USA and the UK. Focussing on this period, this paper investigates the importance of the German Hochschule fur Gestaltung, Ulm (HfG) design school in the early stages of computation in design and architecture. Even though there were no computers in the school, it may be argued that its innovative pedagogy and distinguished faculty members launched analogical computational design methods that can be seen as the basis for further computational approaches in architecture

Neves, Isabel Clara and João Rocha. ‘The contribution of Tomas [sic] Maldonado to the scientific approach to design at the beginning of computational era. The case of the HfG of Ulm. Porto:FAUP, 2014, pp. 39–49.

Only of background interest in regards to parquet deformation as to the Hochschule in itself. Nothing on parquet deformation. Four mentions of Huff, only one of which is a single sentence.

However, of note is the space-filling curve and Sierpinski triangle, p. 43.

ABSTRACT: Nowadays the use of computational design processes in architecture is a common practice which is currently recovering a set of theories connected to computer science developed in the 60`s and 70`s. Back then, such pioneering experiences were carried out by an interest in employing scientific principles and methodologies in architectural design, which, with the help of computers, were developed in Research Centres mainly located in the USA and the UK. Looking into this period, this paper investigates the relevance of the German design school of the Hochschule für Gestaltung of Ulm to the birth of computation in architecture. Even though there were no computers in the school, this paper argues that the innovative pedagogies introduced by a group of distinct professors built clear foundations that can be understood as being at the basis of further computational approaches in architecture. This paper focuses on the remarkable work done by Tomas Maldonado. His contribution was paramount in the emergence of analogical ways of computer design thinking. This analysis ultimately wants to emphasize how the HfG Ulm’s role and its scientific approach have paved the way for the onset of the computational era in architecture.

Neves, Isabel Clara. ‘The Emergence of Computational Design Research and Education. A Technical-Scientific Approach, 1950–1970’, 2018, pp. 88–102. In Proceedings of the Sixth Conference on Computation, Communication, Aesthetics & X Madrid, Spain. Edited by André Rangel, Luísa Ribas Mario Verdicchio, Miguel Carvalhais. Published by Universidade do Porto, Praça Gomes Teixeira Huff pp. 92–95, 100–101. 27 Mentions on Huff, but none on parquet deformation! Huff is extensively mentioned on pp. 93-94. P. 96 shows three parquet deformations by James Eisenman, of 1966, at Carnegie-Mellon. The other non-Huff aspects are of no real interest. Abstract A convenient framework of computational design research and education history in architecture is fundamental to formulate the possibilities of a “digital continuity” or “revolution in the discipline” (Oxman 2006). Contributing to this framework, this article presents an analysis of the cultural and technological context that led to the emergence of Computational Research and Design Education — HfG-Ulm and its American counterpoint — focusing specifically on the way teaching and architecture design approached science in the period 1950-1970. This is based in educational programs and places where a remarkable set of teachers, ideas and work converged.

Nirma, N, and R. Rama. ‘Terminal Weighted L-Systems’. International Journal of Pattern Recognition and Artificial Intelligence. World Scientific, Vol. 4, No. 1, 1990 pp. 95-112.


Terminal weights are attached to L-systems by replacing each terminal generated by an OL-system by fa(i) in the ith step of a derivation...

…Now we give an example to illustrate how the parquet deformation of Ref. 17 can be generated

...Fig. 4. The parquet deformation


An L-system or Lindenmayer system is a parallel rewriting system and a type of formal grammar. An L-system consists of an alphabet of symbols that can be used to make strings, a collection of production rules that expand each symbol into some larger string of symbols, an initial "axiom" string from which to begin construction, and a mechanism for translating the generated strings into geometric structures.

Obviously advanced. Unlikely to be of any real interest.

Olmsted, Zachary T., Tim D. Riehlman, Carmen Branca, Andrew G. Colliver, Adam M. Winnie, Janet L. Paluh. ‘Metamorphic Pattern Formation and Deformation: In Vivo and In Vitro Mechanisms’. Biophysical Journal, January 2013, p. 142a
...Our goal is to apply insights on patterning biological polymers in vivo to development of hybrid biosynthetic systems capable of utilizing microtubules in self-assembling metamorphic patterns including parquet deformation behavior.
I’m not too sure what the brief text is; an article, a note or...? As such, it is more of a note. The text is academic. Whatever, a single mention in passing.

Pitici, Mircea. The Best Writing on Mathematics 2011. Princeton University Press, 2012


P. 148 D. Hofstadter, Parquet deformations: Patterns of tiles that shift gradually in one dimension, Scientific American (July 1983): 14–20. Also in Metamagical Themas: Questing for the Essence of Mind and Pattern, Basic Books, New York, 1985.

Princeton University Press.

This annual anthology brings together the year’s finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, The Best Writing on Mathematics 2020 makes available to a wide audience many articles not easily found anywhere else—and you don’t need to be a mathematician to enjoy them. These writings offer surprising insights into the nature, meaning, and practice of mathematics today. They delve into the history, philosophy, teaching, and everyday aspects of math, and take readers behind the scenes of today’s hottest mathematical debates.

Seemingly the series began in 2010.

Uncertain author credit. DR. SEFIK MEMIS YRD. DOÇ.DR. MURAT SENTÜRK (Author) I. Çekmeköy sempozyumu: Sehir, tarih, toplum, gelecek. Tebligler kitabi: 22-23 Ekim 2016. Publisher: Çekmeköy Belediyesi, 2017
Şehir Çekmeköy, Tarih, Toplum ve Gelecek
In short, a Turkish symposium, on, surmising on the translated text, ‘city history society future’?
P. 28. Two parquets illustrated:
Şekil 11. William Huff, 1979. The Parquet Deformation-iki örnek
Trans. According to Huff, it is a harmonious change of shapes (Figure 11).
P. 29 Quotes Huff in a list of references:
Huff, William. 1979. The Parquet Deformation. McGinty, T. (der). Best Beginning Design Projects Volume 1. Milwaukee: University of Wisconsin. S.30-33.

Plender, Richard (ed). Dora Kostakopoulou. Chapter 5. ‘The Capricious Games of Snakes and Ladders: The Nexus of Migration and Integration in Light of Human Rights Norms’, pp. 91-110. See pp. 108-109 (p. 109 Oleson illustration). Issues in International Migration Law Brill - Nijhoff, first edition 2015, p. 108. Preview available on Google Books. In a sociology type article:

I have not seen a better depiction of the co-operative model of society mentioned above than in David Oleson's 1964 'parquet deformation' picture featuring below. Entitled the 'I at the Center', it shows how our personal identities are shaped by the myriad of influences of neighbouring others, whose shapes, in turn, become increasingly different as they drift away from the centre. The same would hold true if The ‘I at the Center’ was substituted by ‘Community’. By going beyond the ‘I at the Center’ and Hofstadter's insightful remark that in Oleson’s pen and ink design we see that ‘each of us is a bundle of fragments or other people's souls, simply put together in a new way’…


Reddy, Hasitha. Architecture, Interiors and Urban Design Portfolio. Self Published on Issuu February 21, 2019. 58 pp.

See pp. 50-51

Robots and Architecture, Deformation, Art with Kuka Robot

Weak premise.

N. B. I looked for ‘Hasitha Reddy, parquet deformations’ separately, but without success.

Rozenberg G. and ‎A. Salomaa. The Book of L. Springer-Verlag 1985 and 1986, p. 415.
Snippet view on Google Books:
Weights are attached to terminals in the sense that terminals are treated as functions of time and this interesting idea to describe parquet deformations [ 23 ] is extended to the model in [ 90 ] …
Likely, but not certainly, referring to a Huff-style deformation.

Sakkal, Mamoun. ‘Intersecting squares: applied geometry in the architecture of Timurid Samarkand’. Journal of Mathematics and the Arts, 2018, Vol 12, Nos. 2–3, pp. 65–95.

Saputra, R. A., C. S. Kaplan, P. Asente. AnimationPak: Packing Elements with Scripted Animations’. Proceedings of Graphics?, Proceedings of the 2019 on Creativity and Cognition?, pp. 173-186?, 2019

Pages 95–102

In this paper, I consider the question of how to carry out aesthetically pleasing evolution of the curves

that make up the edges in a parquet deformation. Within …
Various uncertainties as to this article. Oddly, I cannot find the quote, from Google Scholar, in the pdf and no parquet deformations are shown!

Schaffer, Karl. ‘Dancing Deformations’. Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture, pp. 253–260
On the analogies of dance with parquet deformation. Not illustrated with Huff-type examples. Also see a like later paper of his, ‘Dichromatic Dances’, of 2017. Such comparisons are rare. Also see Gabriele Brandstetter and Marta Ulvaeus on the same theme.
P. 253:
The performing art of dance employs symmetry in a variety of ways. Often choreographers blur the lines between symmetries or seamlessly morph from one symmetry type to another. This may be seen to be similar to parquet deformations, visual images in which one tiling deforms seamlessly into another…
Parquet Deformations. The artist M.C. Escher created a number of works in which one tessellation morphs into another. Later in the 1960s the architect William Huff investigated these designs with his students, and received wider attention when explored and written about by Douglas Hoffstadter [4]. Recently Craig S. Kaplan has presented his investigations of them at Bridges [6]. These visual designs usually change seamlessly in a horizontal direction through several tiling patterns. Dance choreographers often utilize similar deformations, in both time and space.
P. 260:
[4] Douglas R. Hofstadter, “Parquet Deformations: Patterns of Tiles that Shift Gradually in One Dimension,” Scientific American, 1983.
[6] Craig S. Kaplan, “Curve Evolution Schemes for Parquet Deformations,” Bridges Proceedings, 2010, pp 95-102.

Schaffer, Karl. ‘Dichromatic Dances’. Proceedings of Bridges 2017: Mathematics, Art, Music, Architecture, Education, Culture, pp. 291–298
On the analogies of dance with parquet deformation. Not illustrated with Huff-type examples.
P. 291
...This paper investigates danced two-colored or dichromatic symmetry patterns, and continues an earlier investigation on how such danced symmetry patterns may be seamlessly morphed from one symmetry type to another, in a manner similar to visual parquet deformations...
....In this paper, I extend to two-colored or “dichromatic” patterns an exploration into danced parquet deformations [9], in which symmetric patterns of dancers morph from one pattern to another without breaking symmetry.
P. 292
… The earlier paper [9] examined how this may allow parquet like deformations from one symmetric dance formation to another.
P. 293
Figure 3 shows a “parquet deformation” sequence of positions for 4 dancers from a recent dance by the author titled “Blacks and Whites,” using possible two colorings of the one-color designs from Figure 2…

Schattschneider, Doris. M. C. Escher: Visions of Symmetry W. H. Freeman, 2004
A brief discussion (not illustrated):
P. 281: At least one professor of design, W. S. Huff, found inspiration in Escher continuous deformations of interlocked motifs in his metamorphosis works, and explored various possibilities for these with his students. Some of this work is reported in the article, “Parquet Deformations: Patterns of Tiles That Shift Gradually in One Dimension" by Douglas Hofstadter.
P. 362: (bibliography, Hofstadter article, as above).

 ————. ‘The Mathematical Side of M. C. Escher’. Notices of the American Mathematical Society. Volume 57, Number 6, June/July 2010, pp. 706–718
A brief discussion in the context of the overall mathematical side of Escher:
P. 716: Metamorphosis, or topological change, was one of Escher’s key devices in his prints. His interlocked creatures often began as parallelograms, squares, triangles, or hexagons, then seamlessly morphed into recognizable shapes, preserving an underlying lattice, as in his visual demonstration in Plate I in [19]. At other times the metamorphosis of creatures changed that lattice, as occurs in his Metamorphosis III. William Huff’s design studio produced some intriguing examples of “parquet deformations” that preserve lattice structure [30], and, more recently, Craig Kaplan has investigated the varieties of deformation employed by Escher [34].

Schorr, Natalie (Facebook) 11 May 2019

I finished a piece this morning that I had been working on for a long time. It's the first in a series called "On the Street Where We Live," and this one is "Exotic." The thought is that there are all kinds of people in our community, and the community is always changing and moving and fluid....It starts with a drypoint street scene, includes a partially cut out parquet deformation pattern, along with a portrait of someone holding or wearing something very colorful.

An (mixed media?) artwork, titled ‘Exotic’, with a parquet deformation backdrop. However, the parquet deformation is not particularly clear, but as the artist is explicit in its description, I will accept this at face value.

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

Science Digest, 1984, Vol. 92, p. 25.
On Fylfot Flipflop.
Science Digest was a monthly American magazine published by the Hearst Corporation from 1937 through 1988. No known archive online.

SILTA - Volume 16, Studi italiani di linguistica teorica ed applicata (Italian studies of theoretical and apple linguistics) Liviana Publishing, 1987


P. 281. Con “deformazione di parquet” Hofstadter indica un regolare tassellamento del piano, idealmente disegnato con segmenti e curve di spessore zero. Le trasformazioni che intervengono a modificare tale tassellamento devono rispondere a …


Italian Studies of Theoretical and Applied Linguistics (SILTA) is an international magazine, published since 1972 under the direction of Luigi Heilmann and Enrico Arcaini, sole director from 1987 to 2015. The magazine acts as an international comparison point between the theoretical and methodological approaches to different analyzes in the linguistic field. The magazine publishes articles in Italian and foreign languages ​​(French, English, German, Spanish) and also intends to outline an important cross-cultural exchange project.

Simmi, Simone (ed?). Eredità, 19 Nov 2012. 16 pp.

Inconsequential. Article by Anceschi?

Bill Huff svilupperà questa tematica intitolandola “Parquet deformations” 

= Bill Huff will develop this theme entitled "Parquet deformations"

eredità = heredity

An obscure publication in Italian, with much uncertainty. Minor mention of Huff and parquet deformations, p. 9.

Sousa, J. P. ‘Calculated Geometries. Experiments in Architectural Education and Research’. In: Viana V., Murtinho V., Xavier J. (eds) Thinking, Drawing, Modelling. Geometrias 2017. Springer Proceedings in Mathematics & Statistics, Vol. 326. Springer, Cham. (2020)


This work resonates to the “Parquet Deformation” studio taught by William Huff at Carnegie Mellon in 1966, when, without using computers, such adaptive design concepts were already thought of and exercised [13] …

Quotes Huff.

13. Huff, W.S.: What is basic design? In: Crowell, R.A. (ed.) Intersight One. State University of New York, Buffalo (1990)

Talley, Charles (Editor). Surface Design Journal - Volumes 16-17. United States: Surface Design Association, pp. 8–10, 1991. Neither author nor article title is given.
Snippet view on Google Books:
Quotes (with spelling mistake) Hofstadter's Metamagical Themas Questing for the Essence of MInd and Matter.
P. 8. Another off-beat book with fresh ideas that apply to quilts is Metamagical Themes [sic] Questing for the Essence of Mind and Matter based on a series of columns written for Scientific American by Douglas R. Hofstadter. One particular relevant chapter is devoted to “Parquet Deformations,” a “subtle, intricate art form” developed by William Huff, a professor of architecture through assigned student studies)
P. 10. The incremental pace of change in a parquet deformation is rather like that of days moving one season to the next. Hofstadter notes its temporal character, equating a parquet deformation to visible music. The underlying principle seems to be …

Teo, Sebastian? Ytsproject. Portfolio 2017, Self Published on Issuu March 19, 2017, 28 pp.

See p. 12. Uses my ‘France’ parquet deformation in a square configuration, l-r, u-d. Oddly, I cannot find a single mention of ‘parquet deformation’ on the page, although it showed up upon the initial generic search!

Thompson, D’Arcy. On Growth and Form. Cambridge University Press, 1917
Especially see the chapter ‘On the Theory of Transformations’, but does not specifically refer to his parquet deformation instance, Fig. 133, first edition 1917,The first parquet deformation? In appearance, yes, but I’m not entirely sure this is what Thompson is intending. The caption and text do not imply a parquet deformation.
P. 335. 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.
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.”)
In Chapter VII ‘The Forms of Tissues or Cell-Aggregates’. Wikipedia On Growth and Form is a book by the Scottish mathematical biologist D'Arcy Wentworth Thompson (1860–1948). The book is long – 793 pages in the first edition of 1917, 1116 pages in the second edition of 1942. The book covers many topics including the effects of scale on the shape of animals and plants, large ones necessarily being relatively thick in shape; the effects of surface tension in shaping soap films and similar structures such as cells; the logarithmic spiral as seen in mollusc shells and ruminant horns; the arrangement of leaves and other plant parts (phyllotaxis); and Thompson's own method of transformations, showing the changes in shape of animal skulls and other structures on a Cartesian grid.

Tuğrul, Yazar. 'Revisiting Parquet Deformations from a computational perspective: A novel method for design and analysis'. In International Journal of Architectural Computing. Volume: 15 issue: 4, pp. 250267, 2017.
Broadly written from an architecture viewpoint. Tuğral (who I have corresponded with previously on parquet deformations, and who has studied these extensively on his Design Coding website), here presents his (first?) paper on them. However, I don’t quite know what to make of this, as the writing and explanations are somewhat technical. It is essentially a study of Huff student-inspired works and in particular ‘Trifoliate’ by Glen Paris. Other Huff-related works include ‘Crossover’, by Richard Long and ‘I at the Center’ by David Oleson. Further, his own students' works are included as well. Extensive use is made of the computer plug-in Grasshopper, much beyond my understanding, or indeed interest, as good as it may be in the right hands.
Mentions of myself, p. 254, and in the acknowledgements, p. 265. Gives a good and extensive bibliography, although many of the references are peripheral, and/or not readily found to those without academic access.
Whatever, my perceived views, an article of substance, and required reading.
Click on download, Tugral.

Wang, Patrick Shen-Pei (editor). Array Grammars, Patterns and Recognizers. World Scientific Series in Computer Science. 1989.
Technical. Quotes Hofstadter’s article.

Google Snippets:
Subramanian, K. G. (article) ‘Siromoney Array Grammars’.

Pp. 69 347 : 6 SMG and Parquet Deformation Yet another interesting application of the indexed SMG is in the description of parquet deformation. A parquet deformation is one in which a regular tessellation of the plane gets deformed progressively in one dimension and at each stage is a unit cell that combines with itself so that it covers an infinite plane exactly. In Ref 6 some parquet deformations have been described using the concept of attacking a weight function to the terminals of an array. Here we illstre how indices in the vertical grammar help us to describe interesting parquet deformations in which the pattern shifts at varying speeds in the upper half and lower half of the picture. This is a feature that cannot be described by the earlier technique

P. 71. .. a parquet deformation known as Consternation... 

P. 334. Finally the application of SMG with indices in describing parquet deformation is brought out

P. 350. Quotes Hofstadter's 1983 paper in references.

I must say I very much like the definition of a parquet deformation given here! I will suitably adopt and adapt it.

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 are 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.

Yao, Szu-An. Portfolio 2015. ‘Deformative Space Space Frame Structure by Adaptation of Parquet Patterns’. Self Published on Issuu January 7, 2016, pp. 56
See pp. 54–55. Arguably the best on Issuu. A discussion, with 16 references to parquet deformation. Mention of Huff and Hofstadter. Uses Escher's Sky and Water I.

Ying, Fu. Portfolio. ‘Landscape Mosque’. Self Published on Issuu April 8, 2015, 37 pp. 

See p. 28. 

...This idea is similar to the idea of parquet deformation that the roof is a single piece, but the geometries are varied.

Alludes to parquet deformation, rather than showing any.

Created 22 December 2020. Continually revised subsequently, too many times to list. Last updated from working document 22 July 2021.