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## Convex Geometry

Csaba Vincze (2013)

University of Debrecen

9.3 Regular polyhedra

## 9.3 Regular polyhedra

The Platonic solids (regular convex polyhedra) have been known since antiquity. The ancient Greeks studied them extensively. Some sources (such as Proclus) credit Pythagoras with their discovery. Other evidence suggests he may have only been familiar with the tetrahedron, cube and dodecahedron and the discovery of the octahedron and icosahedron belong to Theaetetus, a contemporary of Plato. Theaetetus gave a mathematical description of all five Platonic solids. He may have been responsible for the first known proof that there are no other convex regular polyhedra. Euclid also gave a complete mathematical description of the Platonic solids in the Elements. Propositions 13 - 17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron and dodecahedron. For each solid Euclid found the ratio of the diameter of the circumscribed sphere to the edge length. InProposition 18 he argues that there are no further convex regular polyhedra.

Figure 58: The tetrahedron.

Definition A convex polygon in the plane is called regular if it is equiangular and equilateral, i.e. all internal angles are equal in measure and all sides have the same length. The facets of a regular polyhedron are congruent regular polygons with the same dihedral angle along each edge and the same number of edges concur at each vertex: the pair (m,n) is the symbol of the regular polyhedra if the facets are regular m-gons and n is the common number of edges meeting at each vertex).

Figure 59: The cube.

Theorem 9.3.1 The possible symbols of a regular convex polyhedron are

Proof Since each vertex has the same defect and their sum is 4π (Descartes' theorem) we have that each vertex has the same positive defect:

 $2\pi -n\alpha \left(m\right)>0$ (9.14)

where

 $\alpha \left(m\right)=\frac{m-2}{m}\pi$

is the common measure of the internal angles in a regular m-gon. From here

 $2>n\frac{m-2}{m}=n\left(1-\frac{2}{m}\right)>\frac{n}{3}$

because the value of m is at least three. It follows that n is less than six and, consequently, its possible values are 3, 4 or 5. If n=3 then 9.14 says that m < 6 and, consequently, its possible values are 3, 4 or 5. In case of n=4 we have by 9.14 that the only possible value of m is 3. Finally, if we substitute n=5 into equation 9.14 we have that m=3 (because it must be less than 10/3) ▮

Figure 60: The octahedron.

Remark The regular tetrahedron is of type (3,3). The cube is of type (4,3). The convex hull of the centers of the facets of a cube is the so-called octahedron of type (3,4). The cube and the octahedron are "dual" (see also polar sets in subsection 7.3.2. The edges of a regular octahedron can be subdivided in the golden ratio so that the resulting vertices define a regular icosahedron of type (3,5). It can be done by first placing vectors along the octahedron's edges such that each facet is bounded by a circle, then similarly subdividing each edge into the golden mean along the direction of its vector. The convex hull of the centers of the facets of an icosahedron is a dodecahedron of type (5,3). They are also "dual". The convex hull of the centers of the facets of a tetrahedron is a tetrahedron. It is "self-dual" (see also polar sets in subsection 7.3.2).

Figure 61: The icosahedron.

Figure 62: The dodecahedron.

In what follows we summarize the basic data of regular polyhedra.