REGULAR AND SEMI-REGULAR SOLIDS 295 
fully by Theaetetus; they were evidently only called Platonic 
because of the use made of them in the Timaeus, where the 
particles of the four elements are given the shapes of the first 
four of the solids, the pyramid or tetrahedron being appro 
priated to fire, the octahedron to air, the icosahedron to water, 
and the cube to earth, while the Creator used the fifth solid, 
the dodecahedron, for the universe itself. 1 
According to Heron, however, Archimedes, who discovered 
thirteen semi-regular solids inscribable in a sphere, said that 
‘ Plato also knew one of them, the figure with fourteen faces, 
of which there are two sorts, one made up of eight triangles 
and six squares, of earth and air, and already known to some 
of the ancients, the other again made up of eight squares and 
six triangles, which seems to be more difficult.’ 2 
The first of these is easily obtained; if we take each square 
face of a cube and make in it a smaller square by joining 
the middle points of each pair of consecutive sides, we get six 
squares (one in each face); taking the three out of the twenty- 
four sides of these squares which are about any one angular 
point of the cube, we have an equilateral triangle; there are 
eight of these equilateral triangles, and if we cut off from the 
corners of the cube the pyramids on these triangles as bases, 
we have a semi-regular polyhedron 
inscribable in a sphere and having 
as faces eight equilateral triangles 
and six squares. The description of 
the second semi-regular figure with 
fourteen faces is wrong: there are 
only two more such figures, (1) the 
figure obtained by cutting oft* from 
the corners of the cube smaller 
pyramids on equilateral triangular bases such that regular 
octagons, and not squares, are left in the six square faces, 
the figure, that is, contained by eight triangles and six 
octagons, and (2) the figure obtained by cutting oft* from the 
corners of an octahedron equal pyramids with square bases 
such as to leave eight regular hexagons in the eight faces, 
that is, the figure contained by six squares and eight hexagons. 
1 Timaeus, 55 n-56 B, 55 c. 
2 Heron, Definitions, 104, p. 66, Heib.