Doris Schattschneider

In Praise of Amateurs

One of the most appealing aspects of Martin Gardner’s column “Mathematical Games” is its presentation of mathematical problems designed to intrigue amateurs and encourage their personal efforts at solution. By his own insistence, Gardner is an amateur mathematician and gives no special deference to formal mathematical education—his column pays tribute to the efforts of the mathematical “great” and the mathematically unknown, names often appearing side by side with no titles to distinguish one from the other. Amateurs are his most avid followers and enjoy the challenge of matching their wits against others in solving problems. Amazingly, their lack of formal mathematical education is often an advantage rather than a hindrance and their ingenious solutions to problems sometimes top the efforts of the professionals.

Enumeration of Polyominoes, Polyiamonds and Polyhexes for Isohedral Tilings with Rotational Symmetry

We describe computer algorithms that can enumerate and display, for a given n > 0 (in theory, of any size), all n -ominoes, n -iamonds, and n -hexes that can tile the plane using only rotations; these sets necessarily contain all such tiles that are fundamental domains for p4, p3, and p6 isohedral tilings. We display the outputs for small values of n . This expands on earlier work [3].

IN BLACK AND WHITE: HOW TO CREATE PERFECTLY COLORED SYMMETRIC PATTERNS

Abstract The use of isometries to create and “perfectly color” symmetric tilings and patterns is explained. The reader is not presumed to have knowledge of isometries, group theory, or computer science. A student, designer, teacher, or any other person interested in the interplay of geometry and art (particularly geometric symmetry), and the possibility of implementation using computer graphics, can learn from this paper.

The Fascination of Tiling

Mosaics that cover surfaces have long been of interest to designers and artists. Recently, however, mathematicians have turned their attention to these visual displays and found them a fascinating source of interesting problems, many of which are still unsolved.

Lessons in Duality and Symmetry from M.C. Escher

The Dutch graphic artist M. C. Escher (1898–1972) carried out mathematical investigations that led to symmetry drawings of three distinct kinds of tilings with two colors, capturing the essence of duality. He used several of these drawings as key elements in his prints that further expressed ideas of duality. One of the most complex of his “duality” tilings was realized in Delft ceramic tile, wrapped around a large column for a school in Baarn, Holland. Recently, a Salish artist in Victoria, BC, Canada, has independently produced a tiling that contains many of the same elements as Escher’s complex duality tilings.

Tiling the pentagon

Abstract Finite edge-to-edge tilings of a convex pentagon with convex pentagonal tiles are discussed. Such tilings that are also cubic are shown to be impossible in several cases. A finite tiling of a polygon P is equiangular if there is a 1-1 correspondence between the angles of P and the angles of each tile (both taken in clockwise cyclic order) so that corresponding angles are equal. It is shown that there is no cubic equiangular tiling of a convex pentagon and hence it is impossible to dissect a convex pentagon into pentagons directly similar to it.

Combinatorial enumeration of 2 x 2 ribbon patterns

An algorithm for creating repeating patterns from a single decorated square gives rise to an obvious combinatorial question: How many different patterns can be created, following the rules? Answers vary according to the definition of equivalence of patterns, and computer sorting programs can provide numerical answers. But algebraic techniques give insight into the answers and provide general formulas for similar problems. Group actions on signatures assigned to patterns can also determine which patterns have symmetry.

Unilateral and Equitransitive Tilings by Squares

Abstract. Recent results obtained by Martini et al. [4] facilitate the proof that there are exactly eight unilateral and equitransitive tilings of the plane by squares of three sizes. This refutes a conjecture by Schattschneider that appears in the book Tilings and Patterns by Grünbaum and Shephard [2].

Marjorie Rice and the MAA tiling

ABSTRACT The tiling in the lobby of the headquarters building of the Mathematical Association of America (MAA) in Washington, DC, is unusual. Congruent pentagonal tiles fit together to fill the space, with no readily discernable pattern. Also unusual is the history of how the tiling came to be, and who discovered it. Not a mathematician, Marjorie Rice discovered this tiling while pursuing for years her self-assigned task of finding all types of convex pentagons that can tile the plane. GRAPHICAL ABSTRACT

Dylan Thomas: Coast Salish artist

The young Coast Salish artist Dylan Thomas produces unique artwork strongly influenced by his cultural background, historical sources, other contemporary Coast Salish artists and his love of symmetry. His works are not just carefully composed compositions of Salish figures, but each pays homage to traditions and stories of the Coast Salish. Here, the artist recounts how he came to make his artwork and mathematician Doris Schattschneider points out particular geometric elements in his work.