Euclid's Elements ends (Book XIII) with the
construction of the five Platonic solids:
- Tetrahedron (pyramid, 4 triangles)
- Cube (6 squares)
- Octahedron (two pyramids on a square base, 8 triangles)
- Icosahedron (20 triangles)
- Dodecahedron (12 pentagons)

and they proof there are no others.
See them
The first three are generally regarded as
Pythagorean and known to the ancient Egyptians.
The latter two are usually thought to be discoveries
at Plato's Academy.

However, it seems they were
known to the megalithic people of Scotland,
from whom the **carved stone balls**
have come down to us. Some 387 are catalogued
by the National Museum of Antiquities of Scotland.

Cf. Dorothy N. Marshall, Carved stone balls, Proc. Soc. Antiquaries
Scotland, v. 108 (1976-77) pp. 40-72.

#### Some examples

About the use of these balls: nobody knows. Among the speculations:
weights and measures, money or trade objects, ball games,
models for the celestial sphere.

Ralph H. Abraham, 25 April 1996.