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Tessellations of Triangles (1/2)
 

Introduction
The first logical step after investigating tessellations of regular polygons is to investigate tessellations of non-regular polygons. The simplest non-regular polygons are triangles, since they have the least possible number of sides.

 

1. Introduction
2. Technique: Rotations
Around Midpoints
3. Technique: Variation of
Rotations Around Midpoints
Technique: Rotations Around Midpoints
A useful method for investigating tessellations is to consider the arrangement of shapes around a single vertex. This technique is critical in our exploration of tessellations of regular polygons. Is there a way to arrange a triangle around a vertex to fill 360 degrees?

Recall that the sum of the interior angles of any triangle is 180 degrees. Then if we took two copies of each angle and fit them all together, the sum of the angles would be 360 degrees:

Some Arrangements of Triangles that will not Tessellate

There are many different ways to arrange the angles of a triangle around a point to fill 360 degrees. Note that around each vertex, there are two copies of angle 1, two copies of angle 2, and two copies of angle 3


As you can see from the examples, most of the arrangements will not tessellate, mainly because the sides of the triangles do not line up. Let us try to keep the sides lined up.

How to Make a Triangle Tessellate

Rotating the triangle around the midpoints of its sides produces an arrangement that will tessellate.


Notice that this technique ensures that there are two copies of each angle around the vertex. This technique of multiple rotations can be used to make a tessellation out of any triangle.

 

Additional examples:

Example of a Single Triangle Tessellation

An application of the technique. The shaded area represents the rotated area
Example of a Single Triangle Tessellation

An application of the technique. The shaded area represents the rotated area

 

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Now, experiment yourself! (No template required.) Draw any triangle and use one of the two techniques to create a tessellation.

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