Tessellations in geometry
A couple of examples of tessellations in geometry are shown below. Basically, whenever you place a polygon together repeatedly without any gaps or overlaps, the resulting figure is a tessellation. A tessellation is also called tiling. Of course, tiles in your house is a reallife example of tessellation.
Tessellation with rectangles
Tessellation with equilateral triangles
Tessellations in geometry: How to quickly check if a geometric figure will tessellate.
What makes the above figures tessellations?
1) The rectangles or triangles are repeated to cover a flat surface.
2) No gaps, or overlaps between the rectangles or the triangles.
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. This is a very important concept since it will help determine when a figure can 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
A couple of examples showing how to use the formula above to determine which figures will tessellate.
Example #1
Determine whether a regular pentagon will tessellate
A pentagon has 5 sides, so n = 5.
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!
Example #2
Determine whether a regular 16gon will tessellate
An 16gon has 16 sides, so n = 16.
Interior angle of the 16gon = [180 × (162)] / 16
Interior angle of the 16gon = [180 × 14] / 16
Interior angle of the 16gon = 2520 / 16
Interior angle of the 16gon = 157.5 degrees
157.5 is not a factor of 360, so the 16gon will not tessellate.
Tessellations can happen with translations, rotations, and reflections and it can also happen with irregular figures.

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