Which regular polygons tessellate the plane

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:. This completes the proof. As for the tessellations in itself, it's not exactly the shape of the polygon that matters, but its simmetry group.

Imagine you want to invent a pattern to make a tessellation. A symmetry of a pattern is, loosely speaking, a way of transforming the pattern so that the pattern looks exactly the same after the transformation. For example, translational symmetry is present when the pattern can be translated shifted some finite distance and appear unchanged. Think of shifting a set of vertical stripes horizontally by one stripe. The pattern is unchanged. Sometimes two categorizations are meaningful, one based on shapes alone and one also including colors.

When colors are ignored there may be more symmetry. The types of transformations that are relevant here are called Euclidean plane isometries translations, rotations, reflections and glide reflections. So, in detecting a pattern of a tessellation, sometimes it is easier to detect the isometries than the pattern itself. And is using these isometries that patterns can be created. There are exactly 17 distinct groups, which means that there are 17 different ways to make a tessellation all of which, by the way, can be found at the Alhambra.

It is the two dimensional case of a more general problem: the 3D case, for example, can be interpreted as the number of different crystaline structures. If you want to see how to create all 17 different patterns, look at here. There are som animated gifs that I made some time ago. For more information on the wallpaper groups, read this.

Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. 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. Try the Pattern Block Exploration. An Archimedean tessellation also known as a semi-regular tessellation is a tessellation made from more that one type of regular polygon so that the same polygons surround each vertex.

We can use some notation to clarify the requirement that the vertex configuration be the same at every vertex. We can list the types of polygons as they come together at the vertex. Subtracting 2 n from both sides and factoring the left hand side, we have. Notice that the only possible ordered pair n , a for this to be true are 4,4 , 6,3 and 3,6. These are the representation of square, regular hexagon, and equilateral triangle respectively as we have stated above.

Your email address will not be published. Can you think of other shapes that can tile the plane individually? Leave a Reply Cancel reply Your email address will not be published. In this post, we are going to show algebraically that there are only 3 regular tessellations. We will use the notation , similar to what we have used in the proof that there are only five platonic solids , to represent the polygons meeting at a point where is the number of sides and is the number of vertices.

Using this notation, the triangular tessellation can be represented as since a triangle has 3 sides and 6 vertices meet at a point. In the proof, as shown in Figure 1, we are going to show that the product of the measure of the interior angle of a regular polygon multiplied by the number of vertices meeting at a point is equal to degrees. Theorem: There are only three regular tessellations: equilateral triangles, squares, and regular hexagons.