Tessellations in geometry
A couple of examples of tessellations in geometry are shown below:
Tessellation with rectangles
Tessellation with equilateral 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
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 × (n2)] / n
Let's say n = 5. This is a pentagon
Interior angle of the pentagon = [180 × (52)] / 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:
The gap is shown with a red arrow!
Tessellations can happen with translations, rotations, and reflections.

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