The Octahedron, having eight equilateral triangles as faces.

The Dodecahedron, having twelve regular pentagons
(or five-sided figures) as faces.

The Icosahedron, having twenty equilateral triangles as faces.[1]


[1] If the reader will copy figs. 4 to 8 on cardboard or stiff paper,
bend each along the dotted lines so as to form a solid, fastening together
the free edges with gummed paper, he will be in possession of models
of the five solids in question.


Now, the Greeks believed the world to be composed of four
elements--earth, air, fire, water,--and to the Greek mind the
conclusion was inevitable[2a] that the shapes of the particles of
the elements were those of the regular solids. Earth-particles were
cubical, the cube being the regular solid possessed of greatest
stability; fire-particles were tetrahedral, the tetrahedron being the
simplest and, hence, lightest solid. Water-particles were icosahedral
for exactly the reverse reason, whilst air-particles, as intermediate
between the two latter, were octahedral. The dodecahedron was, to these
ancient mathematicians, the most mysterious of the solids: it was by
far the most difficult to construct, the accurate drawing of the
regular pentagon necessitating a rather elaborate application of
PYTHAGORAS' great theorem.[1] Hence the conclusion, as PLATO put it,
that "this [the regular dodecahedron] the Deity employed in tracing
the plan of the Universe."[2b] Hence also the high esteem in which
the pentagon was held by the Pythagoreans. By producing each side of