How do quadrilaterals tessellate

Mathematical truth. Truth of propositions. If-then statements. Proportional reasoning. Deductive reasoning. Halving a rectangle. Proof: The foundation of mathematics. Reasoning for all students. Challenging all students. Effective questioning. The W questions.

Promoting deep thinking. A routine element of every class. Student-to-student dialogue. A social activity. Making mathematical connections. What do connectionist teachers do? In fact, there are pentagons which do not tessellate the plane. Attempting to fit regular polygons together leads to one of the two pictures below:. Both situations have wedge shaped gaps that are too narrow to fit another regular pentagon. Thus, not every pentagon tessellates.

On the other hand, some pentagons do tessellate, for example this house shaped pentagon:. The house pentagon has two right angles. Thus, some pentagons tessellate and some do not. The situation is the same for hexagons, but for polygons with more than six sides there is the following:. This remarkable fact is difficult to prove, but just within the scope of this book.

However, the proof must wait until we develop a counting formula called the Euler characteristic, which will arise in our chapter on Non-Euclidean Geometry. Nobody has seriously attempted to classify non-convex polygons which tessellate, because the list is quite likely to be too long and messy to describe by hand.

However, there has been quite a lot of work towards classifying convex polygons which tessellate. Because we understand triangles and quadrilaterals, and know that above six sides there is no hope, the classification of convex polygons which tessellate comes down to two questions:. Question 2 was completely answered in by K. Reinhardt also addressed Question 1 and gave five types of pentagon which tessellate.

In , R. Kershner [3] found three new types, and claimed a proof that the eight known types were the complete list. A article by Martin Gardner [4] in Scientific American popularized the topic, and led to a surprising turn of events.

In fact Kershner's "proof" was incorrect. After reading the Scientific American article, a computer scientist, Richard James III, found a ninth type of convex pentagon that tessellates. Not long after that, Marjorie Rice , a San Diego homemaker with only a high school mathematics background, discovered four more types, and then a German mathematics student, Rolf Stein, discovered a fourteenth type in As time passed and no new arrangements were discovered, many mathematicians again began to believe that the list was finally complete.

But in , math professor Casey Mann found a new 15th type. Recall that a regular polygon is a polygon whose sides are all the same length and whose angles all have the same measure.

We have already seen that the regular pentagon does not tessellate. We conclude:. A major goal of this book is to classify all possible regular tessellations. Apparently, the list of three regular tessellations of the plane is the complete answer. However, these three regular tessellations fit nicely into a much richer picture that only appears later when we study Non-Euclidean Geometry. Tessellations using different kinds of regular polygon tiles are fascinating, and lend themselves to puzzles, games, and certainly tile flooring.

A parallelogram is cut by either of its diagonals into two equal triangles. Conversely, two copies of the same triangle put next to each other after one of them has been rotated a half-turn form a parallelogram. Any triangle therefore also tessellates the plane. A honeycomb supplies a model of the hexagonal tessellation: a regular hexagon, too, tessellates the plane.

And this is almost it. Our list of tessellating shapes is almost complete. It can also be shortened. Squares, rectangles, parallelograms and trapezoids all are convex quadrilaterals with various degrees of regularity. However, no regularity is required of a quadrilateral to tessellate the plane: any simple , in particular a non convex, quadrilateral has this property.

The applet below allows you to experiment with arbitrary quadrilaterals. There is one present at the outset. Its shape can be modified by dragging its vertices. The buttons "Copy" and "Copy and rotate" help create copies if that basic shape. All so created polygons are draggable. This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not.