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 degrees.

Figure 1 – Tessellating regular polygons.

Looking at the other regular polygons as shown in Figure 2, we can see clearly why the polygons cannot tessellate. The sums of the interior angles are either greater than or less than 360 degrees. » Read more

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.

The Platonic Solids

Regular polyhedra are also known as Platonic solids — named after the Greek philosopher and mathematician Plato. The Greeks studied Platonic solids extensively, and they even associated them with the four classic elements: cube for earth, octahedron for air, icosahedron for water, and tetrahedron for fire. » Read more

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