A miscellany compilation of brief aspects of my parquet deformation studies collected en masse, of which due to the brevity only (many are of the same intrinsic worth as to more in-depth entries) are deemed not deserving of a dedicated page. As such, very much a beginning. No claim is made to exhaustiveness.


1. Some Notes on William Huff

1. Some Notes on William Huff

With the 2021 article for Werner Van Hoeydonck in mind, given the scattered (and sometimes obscure) literature, it is not always easy to find any specific aspect for inclusion. To this end, I here compile a ‘miscellaneous listing’, in chronological order,  in which such details are readily to hand.

Terms include Proper and improper parquet deformations, Aesthetics, Music, and Peter Hotz. The relevant words are highlighted in red

Proper and improper parquet deformations 


The Landscape Handscroll…, p. 311

A parquet pattern (or monohedral tiling-Grunbaum) is defined as a space-filling array of congruent pieces-precluding, on that account, any gaps between pieces or any overlapping of them. A puristic approach is to admit only pieces that are superposable. Congruence does not, however, stipulate handedness; and there are, indeed, patterns whose pieces conform to the three strictures congruence, no gaps, and no overlaps-but are also enantiomorphic (or nonsuperposably left and right-handed). Might these be called improper parquet patterns or tilings?

The Landscape Handscroll…, p. 312

 On the other hand, Escher is quite apt to employ two different interlocking pieces (A and B pieces-fishes and birds). These designs are clever, but not as difficult to achieve as might be assumed: The two different pieces, taken together, merge into one proper parquet piece; the dividing contours between the two subpieces are under no mathematical constraints and are, therefore, completely pliant to the command of the artist


In ‘About Parquet Deformations’ (SEMA 2003). Unpublished notes, p. 1. Examples by Morris Kreitz

Here are two parquet deformations designed by Morris Kreitz at Carnegie-Mellon in 1966.  

According to terminology borrowed from crystallography, the upper design may be called a proper parquet deformation; the lower design, an improper deformation.  

The word improper indicates congruent elements that are both right-handed and left-handed. 

The parquet deformations from my basic design course are, by reason, mostly of the proper type; a few are improper.


Spatial Lines

Commentary on the aesthetic potential of the parquet deformation was presented at Katachi 2 conference… 


The Landscape Handscroll…, p. 307

[Abstract] Parquet Deformations, designs produced for over three decades in a curriculum of formative studies, are rooted in two analytical disciplines: monohedral tiling from geometry; continuous deformation from biological morphology. As designs, outcomes of deliberate choice, they are inevitably subject to aesthetic assessment.

The Landscape Handscroll…, p. 307

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

The Landscape Handscroll…, p. 310

In the end, it should not be surprising that I found in my expeditious library investigation the same aesthetic modalities in Eastern art that are characteristic of Western art

The Landscape Handscroll…, p. 310

Question: How, then, is the Sino-Japanese handscroll aesthetically structured? Answer: Within the culture and age-according to the individual artist's own sense of drama. 

The Landscape Handscroll…, p. 311

Among the variety of teaching assignments developed in my design studio, the parquet deformation offers, as I have suggested, an uncommon experience in the realm of aesthetics--the challenge to deal with a visual art that derives its essential vigor from the factor of time, but that is fitted, nonetheless, with a full compliment of planar tangibles.

The Landscape Handscroll…, p. 312

While an Escher design and a parquet deformation both exploit the trickery of interlocking congruent pieces, a significant difference between the two concoctions is evident in the aesthetics of their pieces' respective modes of rendition.


In ‘About Parquet Deformations’ (SEMA 2003). Unpublished notes, p. 1. (Referring to the ‘The Landscape Handscroll…’ article, rather than a discussion a such here

I have addressed at length the aesthetic qualities of the parquet deformation—as more a temporal art than a spatial art —in my paper, “The Landscape Handscroll and the Parquet Deformation.”   

I now discuss the origin of the parquet deformation and the rules of geometry that permit and limit their variety.



The Landscape Handscroll…, p. 307

Though they occupy graphic formats, these designs are, by virtue of their disproportionate horizontal dimensions, appreciated more as temporal than as spatial compositions - manifesting a distinct affinity with the nonplastic arts of music, dance, poetry, and theater. 

The Landscape Handscroll…, p. 307

(at their right ends) and a thinning of pictorial elements drained the final passages (at their left ends). This, I reflected, is contrary to classical Western dramatic structure, characteristic of theater, music, and dance. That is to say: Act One--commencement, introducing motifs and statement; Act Two-- development, building to a climax; Act Three-finale, often incorporating a refrain.

The Landscape Handscroll…, p. 308

The different arts are frequently classified as either spatial or temporal. The spatial arts (painting, sculpture, architecture) occupy two-dimensional or three-dimensional real space. The temporal arts (music, dance, poetry, theater) occupy the dimensional space of the instantaneous present, continuously advancing through a dimension that unfurls in time-regularly regarded as "the fourth dimension of the space-time continuum."

The Landscape Handscroll…, p. 309

Music, linearly regulated in respect to time, does have spatial elements (notes and chords). Even as entities with evident periods, these spatial elements, separated by intervals of time, occupy no more than the zero-dimensional moment of the present. 

The Landscape Handscroll…, p. 310

"What is the difference between a Chinese painting and a Western painting?" another corroborating authority, art historian Nelson Wu, has written: The student will quickly recognize that all pictorial expressions have the same building blocks: line, area, color, space, movement, and all the other privileges and limitations that are, part and parcel, the birthright of a two-dimensional art. These components in their analyzed form, simple and pure, are universalities, behaving like musical tones, favoring no particular culture or tradition and belonging to all.

The Landscape Handscroll…, p. 312

In the case of our [parquet] deformations, the contours are the object; they make the music. Fill between these contours with colors, textures, or figurative subjects-the music crashes. 

Peter Hotz


When I made my board-to-board design studio criticism of my original iling assignment, Peter Hotz, a gifted student, made an observation:

that incremental changes could be affected from one parquet variation to another.

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

At the time that Peter put together his design, the first parquet deformation, 

I considered it to be a one-time variant of my assignment.  

However, after reading D’Arcy Thompson’s chapter, “The Theory of Transformations, or the Comparison of Related Forms,” 

I had second thoughts and assigned to my whole class, in the third year of my teaching, the assignment for all to produce parquet deformations.

Huff-Bailey email 25 January 2003

Anyhow, when I came back to the States and eventually began to teach Basic Design, I presented first an assignment in making intricate parquets and eventually we developed the deformation.

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

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

Non Huff References


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…

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