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Osborn, John A. L., "Amphography: The Art of Figurative Tiling", Leonardo, Vol.26, No.4, pp 289-291, 1993.
This article is copyrighted and does not extend to copying for general distribution, for promotion, for creating new works, or for resale.
Here is a link to the journal's homepage: http://www.mitpressjournals.org/loi/leon


Amphography: The Art of Figurative Tiling

John A. L. Osborn

Doing easily what others find difficult is talent; doing what is impossible for talent is genius.
--Frederic Amiel, 1821-1881


ABSTRACT

The author discusses his work in figurative tiling, or amphography. His tilings are based on various underlying geometries, including quadrilateral, hexagonal, rhombus based and quasiperiodic. The author argues that amphography is more closely ruled and limited by mathematics than any other art form and should be recognized as a distinct category of art.


The art of amphography (am-fog’-ra-fee) is the art of figurative tiling. A figurative tile, as I use the term, is a zoomorphic outline devised in such a way that multiple copies will interlock to tile the plane. Devising such an outline calls for drawing, with each side of one's line, a different part of the same figurative design. I call this double-duty line "amphographic," from the Greek amph" (“both") and graph ("to draw"). By so naming it, I emphasize its radical difference from the ordinary sort or outline that merely divides figure from ground. In the domain of the amphographic line, there is no ground- -only the same figure in multiple interlocking copies that tile the plane.

Figure 1

In my work I try to avoid the grotesque. My Bats (Fig. 1) shows realistic big-eared bats, (Plecotus townsendii). It is representative of much of my work. Though it is not always possible, (see Birds ,Fig. 2a), I also try to avoid what I refer to as the mere road-kill recognizability that afflicts much tiling work.

Generally I am willing to sacrifice the wirey thinness of the ideal tile-defining line in order to preserve a naturalistic appearance in my subjects. This sacrifice was avoided, for example, in Beetles (Fig. 2b) but can be seen in the black interstices in Moths (Fig. 2c). I also dislike allowing the organisms I depict to be arbitrarily cut off at any sort of framing border. I feel that such cutting is justified in certain cases, but generally I prefer to present patches of complete tiles.

In creating a tiling, I work with pencil and paper only (not a computer). Often I begin with no specific geometry in mind, but with only the intention of doing a tiling of specific subject matter. As the work proceeds, I may lose track entirely of my first choice of an underlying geometry and then be surprised when later I discover that the geometry I have ended up with is not at all the one I started with. For instance, on several occasions an initial quadrilateral basis became a hexagonal one in the finished tiling. This happened with the beetle in Fig.2b.

Figure 2

Many of my tilings are united by themes such as modern religions (Christs, Eco-Shiva Nataraja, Prophe, Hora, Birth of Zen, etc.), horse myths (Centaurs, Pegasus, Unicorns, Lady Godiva) or holiday themes. Others may be the result of my "going fishing" for whatever I can derive from such a familiar and interesting geometry as that of the rhombus-based tiles that Roger Penrose communicated to M.C. Escher [1]; these determine a specific tiling in which vertices shared by six, four, or three tiles periodically appear. This geometry was the basis for Escher's last tiling, his Ghosts [2] of 1972, and it underlies some 20 of my own figurative tilings. It also underlies the patch of birds in Fig. 2a, though here each rhomb forms two identical figures.

Despite the rosettes of figures that may occur in this latter geometry, most of my work (see Bats [Fig.1] and Beetles [Fig. 2b]) shows my strong preference for geometries that yield mutually upright figures in the tiling.

The challenge of quasiperiodic figurative tilings, however, overcame my lack of concern about specific geometry, as well as my preference for upright figures. My quasiperiodic tilings Birth of Zen, Buddhist Monks, Wasps, Pup-Dogs P2 (Fig. 3), Dragons, Dolphins, Snails and P2 Bats are all based on one or the other of Roger Penrose's two-tile quasiperiodic tile sets [3]. These tile sets, the “dart and kite" set [4] and the "Fat and Skinny Rhombuses" (also known as the P1 and P2 sets), were invented and patented by Roger Penrose [5] shortly after M. C. Escher's death. This was some years before they were discovered embodied in nature as the basis of quasicrystals [6], a discovery that recently has upset the paradigms of the science of crystallography. They are said to tile the plane nonperiodically or quasiperiodically because local combinations and orientations of tiles are not repeated at determinate intervals across the tiled plane. Figurative tilings based on these sets, of course, can tile the plane in the same strange semi-indeterminate manner. In these sets, only two different line segments form the eight sides of two differently shaped quadrilaterals, or four-sided figures. Thus, each side of the amphographic line must draw different parts of not just one, but two different figures. The  difficulty of this seems to me much more than double that of any non-quasiperiodic amphography.

Figure 3

Heretofore, figurative tilings have been most widely appreciated when presented as subject matter within a composition categorized as a print, a drawing or a painting. I feel strongly that amphographic art calls for a different sort of intellectual appreciation than do the other arts. Amphography is indeed an art in its own right, not merely design [7], for I frequently use it expressively, as is evident in Birth of Zen, Otter and Mudpuppy, Whooping Crane and other works. All of my work is, more subtly, either expressive of my sense of humor or of the respect and love I have for the multifarious forms of life on Earth.

Amphography has its own unique values, for it is far more closely ruled and limited by, though not determined by, mathematics than is any other art. A figurative tiling either conforms to the rules of tiling or it fails to be one and falls out of consideration altogether. A simple critical criterion arises from these values: the question "Is it a tiling?" A satisfactory answer can then be followed by the more subjective questions typical of art criticism, perhaps starting with the question "Can it reasonably be called figurative?" and going on from there.

I hope that, in the future, amphography will be recognized as a new and wholly distinct art form, one unbound by any of the categories of the past.


References

1. Roger Penrose, "Escher and the Visual Reprentation fo Mathematical Ideas.” in M. C. Escher, Art and Science, H.S.M. Coxeter et. Al. eds (Amsterdam: North-Holland, 1986) pp. 143-157

2. Penrose [1] and Doris Schattschneider, Visions of Symmetry, Notebooks, Periodic Drawings and Related Work of M.C.Escher (New York: Freeman, 1990) p. 229

3. M. Gardner, “Extraordinary Nonperiodic Tiling That Enriches the Theory of Tiling,” Scientific American 236, No. 1, 110-121 (January 1977).

4. B Grunbaum and G.C. Shephard, Tilings and Patterns (New York: Freeman, 1987) pp.531-548

5. U.S. Patent No 4,133,152; Issued 9 January 1979 to Roger Penrose

6. Hans C. Von Bayer, “Impossible Crystals.” Discover Magazine 11, No. 2, 69-78 (February 1990).

7. Doris Schattschneider, “The Fascinatino of Tiling.” In Visual Mathematics, Special Issue of Leonardo 25, No. 3/4/ 3410348 (1992)

8. Schattschneider [2] p. 230.

9. Gardner [3]




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