If, instead of using circles, we used n-sided regular polygons, is it possible to fit these together perfectly edge-edge to make a ring, and if so, what is the relationship between the number of sides of each polygon, and the number of polygons that fit into a circle? Yes, it is possible, for example, here is a bracelet constructed using eighteen regular nonagons 9 sided shapes. Below are representations of the first ten regular polygons; starting with the triangle with three equal sides and angles and three vertices through to the Dodecagon with twelve equal sides and angles.
Which of these tessellates? We can place one on the plane, then add another sharing an edge and common vertex, then another sharing the same vertex, tiling as we go around. The last triangle placed fits in perfectly. If these were tiles on the floor they would fit together with no gaps, and be flat on the floor no bumps. Triangles tesselate on a flat plane.
Similarly, we can do the same with squares. In Figure 1, we can see why this is so. The angle sum of the interior angles of the regular polygons meeting at a point add up to degrees.
Looking at the other regular polygons as shown in Figure 2, we can see clearly why the polygons cannot tessellate. The sums of the interior angles are either greater than or less than degrees. In this post, we are going to show algebraically that there are only 3 regular tessellations.
Circles are a type of oval —a convex, curved shape with no corners. Circles can only tile the plane if the inward curves balance the outward curves, filling in all the gaps. While they can't tessellate on their own, they can be part of a tessellation There are three different types of tessellations source :. Tessellations figure prominently throughout art and architecture from various time periods throughout history, from the intricate mosaics of Ancient Rome, to the contemporary designs of M.
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. For instance in the top row we see on the left a semi-regular tessellation with at every vertex a 3,6,3,6 configuration. We see a 3-gon, a 6-gon, a 3-gon and a 6-gon.
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