How do regular polygons tessellate

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. The number of polygons meeting at a point is. The product is therefore. Factoring and simplifying, we have , which is equivalent to.

Observe that the only possible values for are squares , regular hexagons , or equilateral triangles. This means that these are the only regular tessellations possible which is what we want to prove.

The pattern at each vertex must be the same! To name a tessellation, go around a vertex and write down how many sides each polygon has, in order There are also "demiregular" tessellations, but mathematicians disagree on what they actually are! All these images were made using Tessellation Artist , with some color added using a paint program. Hide Ads About Ads. You can create irregular polygons that tessellate the plane easily, by cutting out and adding symmetrically.

I found some figures here though the language is Japanese. For example, you can divide a hexagon of 4 into two congruent pentagons. Second, let's see the case we can use more than two distinct polygons and its copies to tessellate the plane. You can find helpful comments in other's answer. Also, you'll find some figures in the same page as above. For example, 3,3,3,3,6 means there exist four equilateral triangles and one hexagon at every vertex. Edit 1 : This is a question which I asked at mathoverflow.

You may be interested in the question. Fedorov found that there are exactly five 3-dimensional parallelohedra. You can see beautiful figures here. You'll be interested in these figures. It's all about angle sums. The question you are asking is by no means trivial. But you can gain some intuiton using the Euler Characteristic.

A graph can be viewed as a polygon with face, edges, and vertices, which can be unfolded to form a possibly infinite set of polygons which tile either the sphere, the plane or the hyperbolic plane. If the Euler characteristic is positive then the graph has an elliptic spherical structure; if it is zero then it has a parabolic structure, i. When the full set of possible graphs is enumerated it is found that only 17 have Euler characteristic 0.

You can test the function Euler Characteristic with any polyhedron, for example. So, in some sense, it is a measure of the curvature of the space you are in. Proofs involving the Euler Characteristic can be extremely simple, but may be really complex too it is widely used in algebraic topology.

In any case, however, the function gives conditions on the polygons you are working with. I remember a very simple of proof of the fact that any polyhedron has at least a face that has 5 sides or less:.