Regular Tessellations: Why only three of them?

In Tessellations: The Mathematics of Tiling post, we have learned that there are only three regular polygons that can tessellate the plane: squares, equilateral triangles, and regular hexagons. In Figure 1, we can see why this is so. The angle sum of the interior angles  of the regular polygons meeting at a point add up to 360 […]
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Platonic Solids: Why only five of them?

Regular polygons are polygons with congruent sides and congruent interior angles.  In three dimensions, the equivalent of regular polygons are regular polyhedra — solids whose faces are congruent regular polygons.  The most common regular polyhedron is the cube whose faces are congruent squares.  The other regular polyhedra are shown below. Regular polyhedra are also known […]
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