Polymorphic prototiles

JOURNAL

OF COMBINATORIAL

THEORY,

Series A 34, 119-121

(1983)

Note Polymorphic ANNE FONTAINE

State

Prototiles

Department of Mathematics University of New York at Albany, Communicated Received

E. MARTIN

AND GEORGE

and Statistics, Albany, New

by the Managing April

POLYMORPHIC

York

12222

Editors

7, 1982

PROTOTILES

A tiling of the plane is monohedral if each tile is congruent to one particular tile, called a prototile of the tiling.. A prototile admitting, up to congruence, precisely r monohedral tilings is r-morphic. Previously, r-

FIG. 1. (A) Hexamorphic prototile; and (D) Decamorphic

prototile; prototile.

(B)

Heptamorphic

prototile;

(C)

Octamorphic

119 0097-3165/83/010119-039603.00/O Copyright 0 1983 by Academic Press, Inc. All rights of reproduction in any form reserved.

FIG.

2.

(ak(e).

I

I I I FIG.

2.

(f)-(i).

POLYMORPHIC

121

PROTOTILES

FIG. 2. (j).

morphic prototiles have been given for r = 2 and r = 3 by Griinbaum Shephard [ 21 and for r = 4 and r = 5 by the authors [ 11. THEOREM.

and

There exist r-morphic prototiles for r = 6, I, 8, and 10.

By the simple but tedious process of trying all possible combinations, the prototiles in Fig. 1 are seen to satisfy the theorem. The decamorphic prototile admits the ten tilings in Fig. 2. The octamorphic prototile of Fig. 1 admits tilings suggested by a, b, c, d, e, f, h, and j of Figure 2. Both the hexamorphic and heptamorphic prototiles of Fig. I admit tilings suggested by a, b, c, d, e, and f of Fig. 2, and the heptamorphic prototile also admits a tiling like that of g in Fig. 2.

REFERENCES G. E. MARTIN, Tetramorphic and pentamorphic prototiles, J. Combin. (1983), 115-l 18. Some problems on plane tilings, in “The 2. B. GR~~NBAUM AND G. C. SHEPHARD, Mathematical Gardner” (D. A. Klarner, Ed.), pp. 167-196, Wadsworth, Belmont, California, 198 1. 1. A.

FONTAINE

Theory

AND

Ser. A 34

Primed

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