Regular tessellations are constructed with triangles, squares or hexagons. The tessellations are named according to the number of sides in each intersecting shape. For example, a pattern of four lines intersecting at right angles is a 4. The eight semi-regular tessellations are composed of two or more regular polygons.
They use the following combinations of shapes:. Like regular tessellations, the pattern at each vertex is the same. However, because more than one shape is used, the pattern will contain more than one number.
You can find tessellations of all kinds in everyday things—your bathroom tile, wallpaper, clothing, upholstery Because tessellations repeat forever in all directions, the pattern can't have unique points or lines that occur only once, or look different from all other points or lines.
Additionally, a tessellation can't radiate outward from a unique point, nor can it extend outward from a special line. While any polygon a two-dimensional shape with any number of straight sides can be part of a tessellation, not every polygon can tessellate by themselves! Furthermore, just because two individual polygons have the same number of sides does not mean they can both tessellate. Only three regular polygons shapes with all sides and angles equal can form a tessellation by themselves— triangles , squares , and hexagons.
What about circles? 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. The other tessellations on the top row have a 3,4,6,4 , a 3,12,12 , and a 3,3,3,4,4 configuration. These configurations are unique up to cyclic reordering and possibly reversing the order. For example 3,12,12 can also be written as 12,12,3 or 12,3, In the bottom row we have 4,8,8 , 3,3,4,3,4 , 4,6,12 and 3,3,3,3,6 configurations. This means that 3 triangles and 2 squares will give us a vertex type. In this case we can arrange these polygons around the vertex in two different ways: 3,3,3,4,4 and 3,3,4,3,4.
Both of these will give rise to a semi-regular tessellation. There are only 21 combinations of regular polygons that will fit around a vertex.
And of these 21 there are there are only 11 that will actually extend to a tessellation. Below are the different vertex types. An asterisk indicates that this vertex type cannot be extended to a tessellation. For some more information see Archimedean Exploration.
Polygonal Tessellation Exercises. All parallelograms tessellate. All quadrilaterals tessellate. No convex polygon with seven or more sides can tessellate. There are three regular tessellations of the plane: by triangles, by squares, by hexagons. Theorem 1: There are 3 regular tessellations of the plane.
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