Ad Quadratum Construction and Study of the Regular Polyhedra

47
We shall now determine the diameter of the circumsphere of the new figure and the side
of the cube circumscribing it.
A top view of the stella octangula (fig. 33) shows that the edge of one of the typical
pyramids (OC) is equal in length to the edge of the original octahedron(DE). Therefore
the construction of the pyramids is straightforward. From the Ad Quadratum diagram
measure a, the edge of the octahedron. Draw a circle (fig. 34) of radius a and measure a
with a compass along the circumference three times. Draw chords and radii to obtain a
half hexagon. This is the development of the pyramid. Cut out and fold. Repeat the
operation 8 times (one for each pyramid). They can now be assembled to form the stella
octangula.
Referring to fig. 33, if DE=a is the edge of octahedron, the edge of pyramid OC also
equals a, then the edge of circumscribing cube is
AB AO 2 OB 2 2a 2
AB a 2
Therefore, the circumsphere radius which is half the cube diagonal will be:
AB
2
3
a 2
2
3 a 3
2
The relationship between dihedral angles of the stella octangula and the original
octahedron can readily be written.
We have:
O ˆ
D ˆ C i
and ˆ
T D ˆ
T i ˆ
T D ˆ
O D
or ˆ
O D ˆ
T D
ˆ
T i
Of course, the stella octangula can also be constructed from two intersecting internal
tetrahedra.
2. The Process of Stellation