Ad Quadratum Construction and Study of the Regular Polyhedra

53
These dipyramids in turn can be divided into sets of oppositely congruent tetrahedron
similar to Schläfli’s orthoschemes 22 . The case of the SSD (fig. 35C) is straightforward:
pentagonal pyramids base to base divided by planes passing through the pyramids apexes
and the vertices of the common pentagonal base.
For the GD (fig. 35D), it may be easier to consider the dipyramids as made up
of a positive pyramid, part of the internal structure of the enveloping convex
icosahedron and a negative pyramid representing a dimple.
The GSD (fig. 35E) will have the same internal pyramids as the GD, and an
external pyramid with triangular basis built on an enveloping convex
icosahedron.
The GI (fig. 35F) is somewhat more complicated. The dipyramids of its
structure as made up of 12 pentagrammal external pyramids, base to base with
the 12 pentagonal internal pyramids of a convex dodecahedron with, for each
dipyramid, a set of five negative dimples to subtract from the internal
pyramid.
All these forms are easily divided into orthoschemes. The geometry of the
pyramids and dimples being known, formulas similar to Schläfli’s can be
derived for the stellated forms.
3. The Four Regular Stellated Polyhedra
We now proceed to the study of the regular stellated polyhedra. We start therefore from
the regular convex dodecahedron and the icosahedron. Lengthening the edges of the
dodecahedron till they meet gives rise to the small stellated dodecahedron (fig. 35C).
This is the first step in the stellation process previously described (fig. 35A(a)). This
figure can be visualized as a regular dodecahedron on the faces of which pentagonal
pyramids would be attached. Generally attributed to Kepler, its image can however be
seen in a marble marquetry design in the Basilica San Marco, Venice (fig. 36). This
design, dated 1425-27, is ascribed there to Paolo Uccello, monk and geometer friend of
Leonardo da Vinci. Having 12 faces (the intersecting pentagrammal stars), it justifies its
dodecahedral name 23 , in spite of its 20 vertices relating it with the icosahedron.
If the 12 vertices (apexes of the pyramid) are joined together, we obtain the figure of the
regular convex icosahedron enveloping, so to speak, the small stellated dodecahedron. If
the process of expansion of the edges (or faces) is continued, enveloping dodecahedra
and icosahedra will keep alternating in an ever-expanding pulsation of growth. This
process of growth obeys a geometric progression, as will be shown in section 4.
22
Kappraff, op. cit. p. 287ft.
23
These names of the stellated polyhedra were introduced by the British mathematician Arthur Cayley in
1859.