## Platonic Solids Revisited I will not talk about Sacred Geometry. This is a tutorial on geometry. I guarantee that if you read this tutorial you will learn at least one fun fact that you did not know before.

How many 3-dimensional geometric objects (convex polyhedra) are there that satisfy the following properties?

1. All vertices touch the circumsphere
2. The insphere touches all the faces.
3. All faces consist of identical polygons.
4. All the vertices are surrounded by the same number of faces.
5. All the dihedral angles are equal.
6. All the solid angles are equivalent

There are only 5 such objects. They are called Platonic Solids. They are named after Plato even though he did not invent these solids. Theaetetus who was one of Plato’s contemporaries proved that only 5 such solids exist.  Later, Euclid gave complete descriptions of these solids in his book “The Elements.”

Tetrahedron 6 edges of equal length
4 faces (equilateral triangles)
4 vertices (3 triangles meet at each vertex)

Octahedron 12 edges of equal length
8 faces (equilateral triangles)
6 vertices (4 triangles meet at each vertex)

Cube 12 edges of equal length
6 faces (squares)
8 vertices (3 squares meet at each vertex)

Icosahedron 30 edges of equal length
20 faces (equilateral triangles)
12 vertices (5 triangles meet at each vertex)

Dodecahedron 30 edges of equal length
12 faces faces (pentagons)
20 vertices (3 pentagons meet at each vertex)

An interesting property (Euler’s formula)

Number of vertices + number of faces = number of edges + 2

Circumsphere of a Platonic Solid

Circumsphere is the smallest sphere that holds a 3-dimensional object. The Platonic Solids are very special because all vertices of a Platonic Solid touch the circumsphere. Image credit: http://mathworld.wolfram.com/Circumsphere.html

Insphere of a Platonic Solid

Insphere is the largest sphere that can be inscribed to a 3-dimensional object. The Platonic Solids are very special because the insphere touches all the faces of the Platonic Solid. Image credit: http://mathworld.wolfram.com/Insphere.html

Dual of a Platonic Solid is another Platonic Solid

The dual of a Platonic Solid can be constructed by connecting the midpoints of the sides surrounding each vertex, and constructing the corresponding tangential polygon (tangent to the circumcircle).

• Dual of a cube is an octahedron
• Dual of an octahedron is a cube
• Dual of a dodecahedron is an icosahedron
• Dual of an icosahedron is a dodecahedron

Image credit: http://mathworld.wolfram.com/DualPolyhedron.html The top row shows the original Platonic Solids. The middle row shows the corresponding duals. The third row is the combined forms of the Platonic Solids and their duals.

Tetrahedron is self-dual

Tetrahedron is the only Platonic Solid which is self-dual meaning that it’s dual is also a tetrahedron.

Midsphere of a Platonic Solid

The midsphere of a Platonic Solid touches all edges, as well as the edges of the dual of that solid.

There is an interesting relationship 

R r = ρ2

If we assume edge length=1 References

 Shing-Tung Yau, “The Shape of Inner Space” (2010), Basic Books 