Main Concept

What is a mathematically perfect 3D shape? The ancient Greek philosopher Plato argued that they are convex polyhedra whose faces are congruent polygons where each vertex touches the same number of faces. For example, a cube has 6 faces that are congruent squares and there are 3 faces touching each corner (vertex) of the cube.

In honor of Plato, polyhedra satisfying the above condition are known as Platonic Solids. There are only 5 Platonic solids and Plato even proposed that the ancient elements (earth, air  fire, water) were made of these mathematically perfect solids. He suggested that fire was made of tetrahedrons (4 faces), Earth was made of cubes (6 faces), air was made of octahedrons (8 faces), an lastly water was made of icosahedrons (20 faces). He postulated that the remaining solid, the 12 faced dodecahedron, was representative of the Universe as a whole.

One of the interesting things about such solids is that they satisfy Euler's formula for polyhedra, that the number of vertices minus the number of edges plus the number of faces of a convex solid is 2:

Euler's formula for polyhedra: $V- E \+ F \= 2$

The number on the right hand side of the equation is called Euler's characteristic $\mathbf{\χ}$, and relates to the genus, g, of the solid (a topological invariant) via the formula $\mathrm{\χ}\=2 -2 g$. In fact, any solid which is topologically equivalent to a sphere, where g=0, will have Euler characteristic $\mathrm{\χ}\=2$.

Many interesting and beautiful shapes are not convex polyhedra, one way to make such shapes is by stellation. Imagine extending the edges of a shape until they intersect. Now draw a polyhedra with vertices at the intersection of the lines. This process is called stellation, which comes from the Latin word for star. Some shapes have multiple stellations, while other have none. Try experimenting with the Platonic solids and their first stellations below. 

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