Ad Quadratum Construction and Study of the Regular Polyhedra

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as previously found. Note however that the thinking remains palpably “solid” rather than
“structural,” i.e., in terms of surfaces and volumes rather than radially from the center.
Polyhedra & Regularity
Intuitively, a polyhedron can be naively defined as a three-dimensional region of space
totally enclosed by a set of plane figures. If these plane figures are regular polygons (i.e.,
having all sides and all angles equal), the polyhedron is regular. Only triangles, squares
and pentagons can give rise to regular polyhedra. For stellated polyhedra, we can add the
pentagram or five-pointed star (fig. 30).
However, both the concepts of polyhedron and regularity have evolved with time. Some
modern definitions of polyhedra are such that stellated polyhedra for instance are
excluded 15 . The ancients seem to have thought of polyhedra as solids bounded by
polygons. This appears also to be the case for Descartes (1596-1650) and other
mathematicians such as Legendre (1752-1834). Judging from his drawings (fig. 31A),
Kepler (1571-1630) seems to have thought of them as hollow surfaces. Cauchy (1789-
1857) regarded them as potentially flexible deformable surfaces.
A more modern approach, often better suited to the interests of designers and structural
engineers, takes a more topological view. Having perhaps its roots in the Renaissance
drawings of Leonardo (1452-1519) (fig. 31B) and Wentzel Jamnitzer (1508-1585) (fig.
31C and 31D), it looks upon polyhedra as skeletons, considering as fundamental the
relationships between edges rather than faces.
For our purpose, limited to the study of regular convex and stellated forms of potential
utility in our design, an intuitive approach that selectively uses surfaces, volumes and
topological approach seems best as long as it remains consistent. Our goal is more to
look for the structurally useful rather than the mathematically all-encompassing for its
own sake.
We alluded to a definition of regularity at the beginning. Again, as the definition of
polyhedron, the concept of regularity has evolved over time. From Euclid down,
regularity has involved equality of faces and faces being polygons with equal sides and
angles. But this conception of regularity is incomplete. A case in point is that of the five
so-called deltahedra, made up of congruent equilateral triangles as shown in fig. 31E.
15 Peter Cromwell: Polyhedra. Cambridge U.P., 1999, p. 209.