A history of Greek mathematics - Wilbourhall.org

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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
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Fig. 1. Fig. 2.
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,
gives 6 octagons in the faces, 12
squares under the edges and
exactly
8 hexagons at the corners; that is, we have P 6
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