A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

ON SEMI-REGULAR POLYHEDRA 99
angular, but not similar, polygons ; those discovered by
Archimedes were 13 in number. If we for convenience
designate a polyhedron contained by m regular polygons
of oc sides, n regular polygons of /? sides, &c, by (m a
, %...),
the thirteen Archimedean polyhedra, which we will denote by
jF}, P 2
...P IZ
,
are as follows:
Figure with 8 faces: P x
= (4,, 4 G
).
Figures with 14 faces: P 2
= (8 3
, 6 4 ), P 3
= (6 4
, 8 6 ),
P 4
= (83J 6 8 ).
Figures with 26 faces : P 5
= (8 3
, 18 4 ), P 6
= (12 4
,
8 6
, 6 8 ).
Figures with 32 faces: P 7
= (20 3
, 12 5 ), P 8
= (12 5
, 20 6 ),
P 9
= (20lf 12 10 ).
Figure with 38 faces: P 10
= (32 3
, 6 4 ).
Figures with 62 faces: P n = (20 3
, 30 4
,
12 5 ),
P ]2
EE(30 4
,20 G
,12 10 ).
Figure with 92 faces: P 13
= (80 3
, 12 5 ).
Kepler 1 showed how these figures can be obtained. A
method of obtaining some of them is indicated in a fragment
of a scholium to the Vatican MS. of Pappus. If a solid
angle of one of the regular solids be cut off symmetrically by
a plane, i.e. in such a way that the plane cuts off the same
length from each of the edges meeting at the angle, the
section is a regular polygon which is a triangle, square or
pentagon according as the solid angle is formed of three, four,
or five plane angles. If certain equal portions be so cut off
from all the solid angles respectively, they will leave regular
polygons inscribed in the faces of the solid ; this happens
(A) when the cutting planes bisect the sides of the faces and
so leave in each face a polygon of the same kind, and (B) when
the cutting planes cut off a smaller portion from each angle in
such a way that a regular polygon is left in each face which
has double the number of sides (as when we make, say, an
octagon out of a square by cutting off the necessary portions,
1
Kepler, Harmonice mundi in Opera (1864), v, pp. 123-6.
H 2
100 ARCHIMEDES
symmetrically, from the corners). We have seen that, according
to Heron, two of the semi-regular solids had already been
discovered by Plato, and this would doubtless be his method.
The methods (A) and (B) applied to the five regular solids
give the following out of the 13 semi-regular solids. We
obtain (1) from the tetrahedron, P 1
by cutting off angles
so as to leave hexagons in the faces ; (2) from the cube, P 2
by
leaving squares, and P 4
by leaving octagons, in the faces
(3) from the octahedron, P 2
by leaving triangles, and P 3
by
leaving hexagons, in the faces ; (4) from the icosahedron,
Bj by leaving triangles, and P g
by leaving hexagons, in the
faces; (5) from the dodecahedron, P 7
by leaving pentagons,
and P 9
by leaving decagons in the faces.
Of the remaining six, four are obtained by cutting off all
the edges symmetrically and equally by planes parallel to the
edges, and then cutting off angles. Take first the cube.
(1) Cut off from each four parallel edges portions which leave
an octagon as the section of the figure perpendicular to the
edges ; then cut off equilateral triangles from the corners
(see Fig. 1)
; this gives P 5
containing 8 equilateral triangles
and 18 squares. (P 5
is also obtained by bisecting all the
edges of P 2
and cutting off corners.) (2) Cut off from the
edges of the cube a smaller portion so as to leave in each
face a square such that the octagon described in it has its
side equal to the breadth of the section in which each edge is
cut; then cut off hexagons from each angle (see Fig. 2); this
""r "jr
Fig. 1. Fig. 2.
gives 6 octagons in the faces, 12
""
\*Y~
•
! l—
,
squares under the edges and
exactly
8 hexagons at the corners; that is, we have P 6
. An