Download 71. The Five Regular Solids. m = =Ÿ 2

71. The Five Regular Solids.
Divide the surface of a sphere into congruent regular sherical polygons.
As posed and solved, this problem involves spherical geometry and spherical
trigonometry. These topics might not be as well known today as they were in Dörrie’s time,
and I feel that a literal translation would be of little benefit to most readers. However, to give
a flavor of what’s going on here, note
Theorem. If dABC is a spherical triangle on a sphere or radius r, then its area is r 2 /,
where / is the spherical excess, i.e., the sum of the angles in dABC (in radian
measure) minus =. (A spherical triangle is a triangle on the sphere, the sides of
which are arcs of great circles of the sphere.)
A proof can be found in many geometry books.
One of the main results of this section is
Theorem. There are only five possible regular divisions of a spherical surface,namely by
1.
2.
3.
4.
5.
four regular triangles,
six regular tetragons (squares),
eight regular triangles,
twenty regular triangles,
twelve regular pentagons.
These solids are called the tetrahedron, cube, octahedron, icosahedron
and dodecahedron, respectively.
Proof. A regular polygon is one with equal sides and equal angles (even on a sphere).
We assume that the sphere is covered completely and without gaps by z regular
n-gons and that at every vertex of such an n-gon, m sides come together. We divide
each n-gon into n isosceles (spherical) triangles, each with central angle 2n= and
(two) base angles m= (since at each vertex 2m such base angles come together).
The spherical excess of each such triangle is / 2n= 2m= " = =Ÿ 2n m2 " 1 .
There are z of them covering the sphere, so znr 2 / 4=r 2 or znŸ 2n m2 " 1 4 or
2
2
4
n m 1 zn . Since n, m 2, there are only a limited number of possibilities,
namely
1
n
m
z
3
3
4
3
4
8
3
5
20
4
3
6
5
3
12
R
Note 1. Dörrie continues to show (generally by spherical trigonometry) that all of these
possibilities can in fact occur.
Note 2. The regular solids were already known to the Pythagoreans and thus go back to
the sixth century B.C.
Note 3. The proof that there are only five regular solids probably comes from Euclid (ca.
330-275 B.C.). See his Elements Book XIII.
Note 4. Even modern proofs of existence (and uniqueness) of the regular solids are
pretty involved. See for example Geometry: Euclid and Beyond by Robin
Hartshorne, Springer, 2000, section 44. As he says "defining something is no
guarantee of its existence." His proof is based on that of Legendre (1823).
Note 5. Models can be easily constructed by cutting out "nets" of regular polygons,
folding them up, and taping them together. This is a common way of presenting
them:
Tetrahedron.
Cube.
Octahedron.
2
Dodecahedron.
Icosahedron.
3