Donald W. CroweDepartment of MathematicsUniversity of Wisconsin Madison, WI 53705 USA
Though it is entirely possible for the “player” to while away the hours covering the computer screen with more or less random selections of tiles, for more systematic constructions it is helpful to print out the available “prototiles” for reference. There are three pages of these for OpTiles (one Black-white, two Colored), two pages for SpaceTiles, and one for OrnTiles. (To call up the pages for SpaceTiles and OrnTiles, click on one of the numbers 1.1-10.2 or 1a-4d respectively on the introductory pages for those sections.) However, there are fewer prototiles than these printouts suggest: In the Black-white version of OpTiles there are really only five prototiles, each appearing in four versions as it is rotated repeatedly by 90° (except for the first two, which already have rotational symmetry, so that only one new version of each appears when they are rotated). But from the point of view of “antisymmetry” there are only three prototiles, since 2a and 2b are obtained from 1a and 1b by simply interchanging black and white, and 5a-5d are obtained in the same way from 4a-4d. Thus the enormous variety of “Op Art” patterns which you can create from the Black-white OpTiles is generated by only three basic building blocks. In the case of OrnTiles there are only two basic tiles; all the others are obtained by repeated rotation, or reflection, from 1a and 3a. The OrnTiles have an especially interesting ancestry, for
they derive from the “potato prints” that the artist
M. C. Escher devised for his children. His son, George Escher,
described in a 1986 talk entitled “Potato-Printing: A Game
For Winter Evenings” how his father cut a potato in half,
then carefully shaped the two flat faces into squares of the
same size. In one square he carved a few lines, making sure that
they met all four sides of the square in exactly corresponding
points, then cut away the potato to use as a stamp with an ink
pad. He then stamped the other potato with the first as a stamp,
to transfer the pattern, then carved the second to match the
first, except that of course it is the reflected image. Judging
by the illustrations in Doris Schattschneider’s From the layman’s point of view much of architectural construction is modular: bricks, blocks of stone, strips of siding, decorative tiles, etc. It is only by working with modular elements and experimenting with their combinations that we learn - as children do with their Lego blocks - to feel the extent and variety of the visual and conceptual possibilities inherent in even very small sets of structural elements. Samuel Beckett has alluded to these possibilities in an entertaining
scene involving the main character of his comic novel In contrast to the sad ending of Murphy’s combinatorial
adventure, with Slavik Jablan’s program no dog can eat your
modules. You may contemplate your constructions at your leisure,
and if you have a simple inkjet printer you can print them out
to use in any way you like. The cross-reference to “Modularity
in Art” in each of the four games clicks immediately to
Jablan’s article in the electronic journal
Excursions into Mathematics.
The Millennium Edition (A K Peters Ltd, 2000). To order this book from Amazon.com,
click
here.Donald W. Crowe and Dorothy
K. Washburn,
Totally Tessellated Science U: Tilings and Tessellations Tilings Plain and Fancy
Excursions Into Mathematics (with
A. Beck and M. N. Bleicher). At the instigation of Claudia Zaslavsky
(Africa Counts) he began to investigate the geometric
symmetries in real-world patterns. For the past 20 years he has
worked with Dorothy Washburn to present this material in a form
usable by archaeologists and anthropologists (Washburn and Crowe,
Symmetries of Culture, Univ. of Washington press, 1988).
He has collaborated with Joe Malkevitch, Dénes Nagy and
Paulus Gerdes on mathematics education, Fijian patterns and ethnomathematics.
Copyright ©2000 Kim Williams top of
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