MATHEMATICS AND ORIGAMI

Mathematics and Origami

Finally we get polihedron nº 4 as shown in Fig. 4. To draw it we have used a very sim-
ple contrivance: each pyramid is symmetric to other adjacent to it, with respect to the plane
formed by the base ́s side they have in common and the center of the icosahedron.
In the polyhedron nº 4 we can see 20 vertices (V = 20) made of trihedral angles, i.e. of
species E = 1 (obviously a trihedral angle cannot be stellate).
Out of its vertices start 3 faces that are stellate pentagons (e = 2): 3 × 20 = 60 planes.
But as each of those planes are common to 5 vertices, we ́ll have: 60 / 5 = 12 faces (C = 12).
As far as the sides is concerned, since 3 of them start out of each vertex, and being each

of them common to 2 vertices, the result is 30
2

3 20

=

×

A= ; (A = 30).

Summary: The regular stellate polyhedron nº 4 is a dodecahedron (C = 12) with stellate
pentagonal faces (e = 2); 20 vertices (V = 20) which are convex polyhedral (E = 1) having a
trihedral configuration; the resultant polyhedron has species ε = 7:

7

2

12 2 1 20 30

2

=

× + × −

=

+ −

=

Ce EV A

ε

Interlude