Question

Use complete sentences to differentiate between a regular tessellation and a pure tessellation. Be sure to include specific types of polygons in your explanation.


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Answer 1

A regular tessellation is a tiling of the plane by congruent regular polygons in which the same number and arrangement of those polygons meet at every vertex. Because the interior angle of a regular p-gon is 180(p − 2)/p degrees, only three regular polygons can meet in a way that fills 360 degrees around a point: equilateral triangles (six at a vertex), squares (four at a vertex), and regular hexagons (three at a vertex). These three cases are the only regular tessellations of the plane.

A pure tessellation (often called a monohedral tessellation) is a tiling made entirely from congruent copies of a single polygonal shape, but that shape need not be regular and the vertex arrangements need not all be identical. For example, there are many monohedral tilings by congruent convex pentagons or by congruent irregular hexagons; these are pure tessellations because only one tile shape is used, but they are not regular because the tiles are not regular polygons or the vertex arrangements vary. Note that a tiling by congruent regular pentagons is impossible, so a pure tessellation can use a single polygon type that is not regular, whereas a regular tessellation requires a congruent regular polygon and identical vertex configuration.