Regular polygons are polygons whose side lengths are equal. The regular polygons with three, four, five, and six sides are shown below.
Regular polygons are special because they have congruent sides, congruent angles, and have at least one line of symmetry. Aside from those shown above, we can create more regular polygons, and it is not hard to see that we can create as many as we please.
In three dimensions, we have also regular solids which we call Platonic solids. They are solids whose faces are regular polygons and with the same number of faces meeting at each vertex. Unlike regular polygons, there are only five Platonic solids, and it can be shown mathematically that no other Platonic solids exist.
The Platonic solids were studied extensively in the ancient times, particularly by the Greeks. In the play Timeaus, Plato associated these solids with the four classical elements — the octahedron for air, the tetrahedron for fire, the icosahedron for water, and the hexahedron (or cube) for earth. Plato also spoke of the dodecahedron: “There still remained a fifth construction, which the god used for embroidering the constellations on the whole heaven”
Despite its antiquity, the beauty of the Platonic solids continue to fascinate many mathematicians and artists of the modern times. M.C. Escher, for example, celebrated them in his woodcut The Four Regular Solids. The woodcut is the intersection of four Platonic solids with their symmetries aligned. Escher made it translucent so all solids can bee seen.
Can you see which solid is missing?