Ad Quadratum Construction and Study of the Regular Polyhedra

Introduction:
1
Ad- Quadratum Construction and Study of
Regular Polyhedra
As a geometer who fully concentrates
In squaring the circle and succeeds not
Pondering principles that he would need
Such was I at that new sight:
Wishing to see how image to sphere conformed
And how one within the other found its place.
- Dante, Paradiso XXX111 (133-138)
The Ad Quadratum method is a geometric construction for the platonic polyhedra based
on the simple trigonometry of the internal (or Maraldi) angle, i.e., the angle between two
consecutive diagonals in a regular polyhedron 1 . This trigonometry is based on the ratio
of the first three integers 1, 2, and 3. For example, in a cube, the internal angle C ˆ i is such
that cosC
ˆ i 1
(fig. 1A.) For the tetrahedron, cosT ˆ i
3 1
(fig. 1B); For the octahedron,
3
sin ; For the dodecahedron, sin
ˆ O i 1
D ˆ 2
i ; and for the icosohedron, tanI ˆ i 2 .
3
Through properties of duality, it is easy to show that the internal angle of one solid are
related to the dihedral angle 2 (fig. 1C) of its dual through simple relationships.
Thus the dihedral angle of the cube C ˆ D O ˆ i , the octahedron internal angle; conversely,
the octahedron dihedral angle O ˆ D C ˆ i where C ˆ i is the cube internal angle.
Analogous relationships obtain for all pairs of duals as we shall see later.
The Ad-Quadratum method allows for easy compass and ruler construction of all regular
polyhedra. It yields much more as we shall establish: from the construction of the
Pythagorean triples, the gnomic golden rectangle series, the golden spiral, the exponential
spiral, the tuning of the monochord, and much more. All this based on “the little matter
of distinguishing one, two and three” to use the words of Socrates in The Republic 3 .
1 i.e., those pairs of symmetry axes passing through the center of the circumsphere and two adjacent
vertices belonging to the polyhedron under consideration. Mr. James Armstrong of London first drew my
attention to the interest of the internal angles viewpoint.
2 The dihedral angle is the angle between two intersecting planes (here the adjoining faces) measured
between two lines contained in the planes and mutually perpendicular at a common point to the line of
intersection. On the other hand, the dual or reciprocal of a polyhedron is another polyhedron, the vertices
of which are the centers of the faces of its dual and conversely. Thus cube and octahedron, dodecahedron
and icosahedron are duals of one another while the tetrahedron is its own dual, being self-reciprocating.
3 Plato: Republic vii 522