Tessellations in geometry

A couple of examples of tessellations in geometry are shown below:

Tessellation with rectangles

Tessellation made with rectangles
Tessellation with equilateral triangles

Tessellation made with triangles
what makes the above tessellations?

1) The rectangles or triangles are repeated to cover a flat surface

2) No gaps, or overlaps between the rectangles or the triangles

A tessellation is also called tiling.Of course tiles in your house form a tessellation

Not all figures will form tessellations in geometry

When a figure can form a tessellation, the figure is said to tessellate

Every triangle tessellates

Every quadrilateral tessellates

Why can we say with confidence that the above 2 statements are true?

Well, since there are no gaps and no overlaps, the sum of the measures of the angles around any vertex must be equal to 360 degrees as seen below with red circles

360 degrees angle in a  tessellation
360 degrees angle in a tesselation
Therefore if the measure of an angle of a figure is not a factor of 360, it will not tessellate

We can use then the formula to find the interior angle of a regular polygon to check if a figure will tessellate

Interior angle of a regular polygon = [180 × (n-2)] / n

Let's say n = 5. This is a pentagon

Interior angle of the pentagon = [180 × (5-2)] / 5

Interior angle of the pentagon = [180 × 3] / 5

Interior angle of the pentagon = 540 / 5

Interior angle of the pentagon = 108 degrees

There is no way to make 360 with 108 since 108 + 108 + 108 = 324 and 108 + 108 + 108 + 108 = 432

Therefore the pentagon will not tessellate as you can see below:

Example of an image that is not a tessellation
The gap is shown with a red arrow!

Tessellations can happen with translations, rotations, and reflections.

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