Ad Quadratum Construction and Study of the Regular Polyhedra

49
The stellation process is one whereby stellated forms arise from convex forms. The
process can originate from either face or edge extension. As we have seen, it only applies
to the dodecahedron and the icosahedron since tetrahedron and cube do not enclose new
spaces through extension and that the octahedron only gives rise to a compound (the
stella-octangula).
Due to the principle of duality, the outcome of the extension of the dodecahedron and
the icosahedron will alternate so that a particular figure may be said to have as kernel,
or seed, either of these polyhedra depending on the starting point.
We want to restrict our study to that of regular stellated forms. The stellation process
gives rise to many other forms besides the regular ones. There are, for example, 59
varieties of icosahedral stellations. 20 . The four regular stellated polyhedra, the only
possible ones (Cauchy), allow however a number of different viewpoints, some easier
to visualize, some easier in model making, some easier for purposes of structural
analysis.
The process of stellation can also be visualized directly in three-Dimension as one of
addition of volumic cells such as pyramids onto the surfaces of the convex polyhedra.
Of course, for the resulting figure to be regular, the only cells that can be added are
those whose edges or faces are themselves extensions of the kernel polyhedron (edges
or faces). Though it is not therefore really another method of stellation, it constitutes
a convenient visualization and conceptualization tool.
We shall briefly consider the surface and edge extension process and their afferent 3-
D approach to start with. However, most of our attention will be given to the process
resulting from the dodecahedron and icosahedron common internal structure of three
mutually perpendicular golden rectangles, since it is directly related to the concept of
internal angle and our adquadratum method.
(a) Face Stellation 21
Except for the tetrahedron, the face-planes of the regular convex polyhedra come in
parallel pairs. But since only the dodecahedron and icosahedron generate regular
stellated forms, we consider these two only. If one of the faces in one of the parallel
pairs is chosen as base and the other as top, a regular star polyhedron will be the
result of other faces extension forming a regular polygon in the plane of the base or
top. For this, these other faces must be arranged symmetrically around an axis
through the top and base center.
20
The Fifty-nine Icosahedra, by J.F. Petric, H.T. Flather, H.S.M. Coxeler, and P. Du Val, U. of Toronto
Press, 1951.
21
We follow here P. Cromwell op. cit. pp. 260-280.