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?

- All vertices touch the circumsphere
- The insphere touches all the faces.
- All faces consist of identical polygons.
- All the vertices are surrounded by the same number of faces.
- All the dihedral angles are equal.
- 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. [1] 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.

**Inradius, circumradius, midradius**

r : inradius : radius of the insphere

R : circumradius : radius of the circumsphere

ρ : midradius : radius of the midsphere

There is an interesting relationship [2]

R r = ρ^{2}

If we assume **edge length=1**

**References**

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

[2] http://mathworld.wolfram.com/PlatonicSolid.html