Perplexing Pentagons

by Doris Schattschneider, Moravian College, Bethlehem, Pennsylvania
(from the Discovering Geometry NEWSLETTER, vol. 7, no.,1, Spring 1996)

It is easy to show that every triangle tessellates and every quadrilateral tessellates. One way is to simply rotate the polygon by 180 degrees about the midpoints of each of the figure's sides and repeat this procedure again and again.

But the regular pentagon does not tessellate, so not every pentagon will tessellate.

Is there at least one pentagon that will tessellate? Yes, there are many. But how many? Prior to 1968, it was thought that all tessellating pentagons could be classified into five types. But in that year R. Kershner found three more types and thought that the problem had been solved. No further discoveries were made until 1975, when Martin Gardner wrote a column in Scientific American based on Kershner’s article.

Soon Gardner reported the discovery of another type found by one of his readers, Richard James III. James had cleverly taken the familiar tiling by octagons and squares, separated the rows of octagons and discarded the squares, divided the octagons into four pentagons, and filled the remaining space with copies of these same pentagons. This new discovery sparked the curiosity of another reader, Marjorie Rice, who quickly began her own investigations.

With no formal training in mathematics beyond high school, she soon uncovered a tenth type of pentagon that tessellates. Her method of search was completely methodical, beginning with an analysis of what was already known. She drew little pentagons to represent each of the nine types known to tessellate and for each, wrote down the equations on angles and constraints on the sides that had to be satisfied. Then she devised a way to mark on each pentagon the angle relationships. Here is one example:

This notation was the key to all her investigations, allowing her to easily consider all the possible cases of markings and then decide (by experimental construction) which markings might lead to a pentagon that tessellated.

By 1977, Marjorie Rice had discovered three more new types of tessellating pentagons and more than 60 distinct tessellations by pentagons. Mathematics professor Doris Schattschneider of Moravian College brought Rice's research to the attention of the mathematics community and confirmed that Rice had indeed discovered what professional mathematicians had overlooked. In her article in Mathematics Magazine Schattschneider also reported that a high school class in New South Wales, Australia, had made a project to discover equilateral pentagons that tessellate and had discovered many different types.

In 1985, a fourteenth type of tessellating pentagon was discovered by Rolf Stein, a German graduate student. Are all the types of convex pentagons that tessellate now known? The tessellating pentagon problem remains unsolved.

Further information on the pentagon problem can be found in the following references:
Doris Schattschneider, "Tiling the plane with congruent pentagons," Mathematics Magazine, 51 (1978)
"A new pentagon tiler,"Mathematics Magazine, 58 (1985)
Doris Schattschneider, "In Praise of Amateurs,"The Mathematical Gardner, D. Klarner (ed.). Belmont, CA: Wadsworth, 1981
B. Grünbaum and G.C. Shephard, Tilings and Patterns. New York: W.H. Freeman, 1987.