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.
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 example, in an equilateral triangle, two sides come together to form a 60 degree angle.
In a tessellation, a vertex refers to the point where three or more shapes come together to equal degrees. For example, three hexagons, whose interior angles equal degrees, come together to form a vertex of degrees, while a pentagon, whose interior angles measure degrees cannot equal a vertex of degrees. Polygons used in a tessellation must have at least one line of symmetry.
Symmetry can be defined as equal parts facing each other around an axis, sometimes referred to as a mirror image. Because regular tessellations are created by repeated polygons, a tessellated figure can be divided evenly down the middle, from various angles, to create two symmetrical shapes on either side of the dividing line. Regular tessellations should have multiple lines of symmetry.
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Facts About Parallelograms. How to Solve a Hexagon. How to Calculate Volumes of Pentagonal Prisms. How to Make a 3D Hexagon. Get creative and try making tessellated masterpieces of your own using this handy tessellation creator courtesy of the National Council of Teachers of Mathematics!
There are three different types of tessellations source : Regular tessellations are composed of identically sized and shaped regular polygons. Semi-regular tessellations are made from multiple regular polygons.
Only eight combinations of regular polygons create semi-regular tessellations. Meanwhile, irregular tessellations consist of figures that aren't composed of regular polygons that interlock without gaps or overlaps. As you can probably guess, there are an infinite number of figures that form irregular tessellations! All rights reserved.
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