G. C. Shephard

Pick's theorem

Some years ago, the Northwest Mathematics Conference was held in Eugene, Oregon. To add a bit of local flavor, a forester was included on the program, and those who attended his session were introduced to a variety of nice examples which illustrated the important role that mathematics plays in the forest industry. One of his problems was concerned with the calculation of the area inside a polygonal region drawn to scale from field data obtained for a stand of timber by a timber cruiser. The standard method is to overlay a scale drawing with a transparency on which a square dot pattern is printed. Except for a factor dependent on the relative sizes of the drawing and the square grid, the area inside the polygon is computed by counting all of the dots fully inside the polygon, and then adding half of the number of dots which fall on the bounding edges of the polygon. Although the speaker was not aware that he was essentially using Picks formula, I was delighted to see that one of my favorite mathematical results was not only beautiful, but even useful. (From DeTemple [1989].)

Some problems on polyhedra

Jacob Steiner asked, more than 150 years ago, whether every convex polyhedron in Euclidean 3-space is isomorphic to one all vertices of which lie on a sphere. It is well known that the answer to this question is negative, but many related problems are still unsolved.

The ninety-one types of isogonal tilings in the plane

ABSTRACr. A tiling of the plane by closed topological disks of isogonal if its symmetries act transitively on the vertices of the tiling. Two isogonal tilings are of the same type provided the symmetries of the tiling relate in the same way every vertex in each to its set of neighbors. Isogonal tilings were considered in 1916 by A. V. Subnikov and by others since then, without obtaining a complete classification. The isogonal tilings are vaguely dual to the isohedral (tile transitive) tilings, but the duality is not strict. In contrast to the existence of 81 isohedral types of planar tilings we prove the following result: There exist 91 types of isogonal tilings of the plane in which each tile has at least three neighbors.

Interlace Patterns in Islamic and Moorish Art

Repetitive interlace patterns are one of the hallmarks of Islamic and Moorish art. Through the study of various collections of such patterns, it is easy to verify that, despite the considerable complexity of the designs, most of the interlaces are formed by strands of a small number of shapes—often just a single shape stretching over many repeats of the design. This observation is described and documented by the authors, who present a simple explanation for this phenomenon.

Perfect colorings of transitive tilings and patterns in the plane

Abstract A k -coloring of a tiling is a partition of the set of tiles into k subsets (color). A coloring is called perfect if each symmetry of the tiling induces a permutation of the colors. The checkerboard is a familiar example of a perfect 2-coloring of the square tiling. The set of values k for which there exists a perfect k -coloring is determined for each of the three regular tilings of the plane (by squares, by regular hexagons, or by equilateral triangles). It isalso shown that the set of such k is infinite for every tile-transitive of the plane.