*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.