Totally Tessellated: Semiregular Tessellations, page 1/4

Semiregular Tessellations (1/4)

1. Introduction
2. Filling 360 degrees
3. Determining which
Arrangements Tessellate
4. The Semiregular
Tessellations
5. Hands-On Activities

Introduction
We have investigated the regular polygons that tessellate by themselves. The next logical step is to allow more than one regular polygon in a tessellation. Semiregular tessellations are tessellations of more than one type of regular polygon such that the polygon arrangement at each vertex is the same (more about this stipulation as you read on).

Filling 360 degrees
In order to find out which combinations of regular polygons will tessellate, we should first find out which combinations of regular polygons will fill the 360 degrees around a single vertex, similar to we did in the previous section. Then, if we find that some of these arrangements can be extended to cover a plane, we will have found some new tessellations.

Let us refresh our memory about the interior angles of regular polygons:

number of sides interior angle (degrees)

3 60

4 90

5 108

6 120

7 128 4/7

8 135

9 140

10 144

11 150

... ...

n 180(n-2)/n

We need to determine which interior angles will sum to 360°. Since there are an infinite number of different interior angles, this may seem like a difficult task. But let us proceed with a gradual approach.

Suppose exactly two polygons met at a vertex. Then at least one of the interior angles will have 180 degrees or more.

Two Angles Meeting at a Vertex (animation)

If two angles share their vertices, then at least one will be 180° or greater

But interior angles of 180 degrees or more is impossible. Thus, we must have at least three polygons meeting at a vertex.

Real examples of semiregular tessellations:

To browse full-page templates of the semiregular tessellations that are ready to be printed, proceed to the templates page:

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