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2.3 Tessellations

A tessellation is a pattern of polygons fitted together to cover the entire plane without overlapping. A regular tessellation is a tessellation is when a regular polygon, a figure whose sides all have the same length and whose angles are all equal, tiles the plane.

A tiling or a tessellation is a covering of the entire infinite plane by non-overlapping figures. A monohedral tiling who tile is a regular polygon is called a regular tiling.

The three regular polygons that can tile the plane are an equilateral triangle, a square, and a regular hexagon. This is possible because the interior angles of these three regular figures can be evenly go into 360 (the number of degrees in a circle).

For instance: The interior angle of an equilateral triangle is 60 degrees, which can evenly go into 360 degrees. This makes an equilateral triangle one of the three regular polygons that can tile the plane. Let's take an pentagon. A pentagon has an interior angle of 108 degrees. This does not go into 360 evenly, meaning it cannot tile the plane without overlapping or gaps being left.

Theorem: Any triangle and quadrilateral can tile the plane. Even convex quadrilaterals can tile the plane. This is possible because a side of a triangle or quadrilateral can be modified and translated or rotated. Then that tile is used to tile the plane.