MATHEMATICS AND ORIGAMI

Mathematics and Origami

18.12 STELLATE REGULAR POLIHEDRA

To try to explain them we shall apply to the stellate polygons (Point11).

Then we said that the species was the indicator of the number of turns to give around

the circumference to end up at the starting vertex after all the sides were generated.

We considered as vertices those lying on the circumference, and not the interior inter-

sections of sides.

A convex regular polyhedron can be inscribed within a sphere in such a manner that if

the polyhedron is projected from the center over the sphere, we get on this sphere a network of

spherical polygons covering, without any overlapping at all, the whole spherical surface.

One stellate regular polyhedron can also be inscribed in one sphere, but if we try to

project its faces on it, we observe that the spherical polygons obtained do overlap each other

with the consequence that the sphere is covered by them more than once.

The amount of times ε that the sphere is covered upon after that projection, is the spe-

cies of the stellate regular polyhedron; ε is a function of the number of its faces, vertices and

sides, and also of the species e of its faces (whether they are convex or stellate polygons), and

of the species E of the polyhedral vertices (likewise they may be convex or stellate).

This fact led Euler to enunciate his generalised theorem that is expressed:

2

Ce+EV−A

ε=

We can note that this theorem is an extension of that disclosed in Point 18 for the con-

vex polyhedra.

We may recall that regarding stellate polygons, neither triangle, square or hexagon

could produce this kind of polygons. Likewise in the case of the convex regular stellate polyhe-

dra, only the convex regular dodecahedron and icosahedron can generate them.

Even so, and because of internal restrains, we end up with only four types of stellate

regular polyhedra. From here on, we shall study them in detail just assigning an identification

number to each of them. The reason is not to lead the reader to confusion because of the fact

that initial and final configurations may be equal or different with regard to the number of

faces, in several cases. Therefore, out of the five platonic polyhedra, we only get four stellate

regular polyhedra.

18.12.1 STELLATE REGULAR POLYHEDRON nº 1

• We start with a dodecahedron of side BC = L (Fig. 1)


• Having a pentagonal face as base, we build one stellate pyramid with H as vertex.


• H is the center of homothety represented by A ́ in Fig. 4, Point 18.6.3.


• Let ́s figure out the faces of the stellate polyhedral angle determined by H and the
  stellate pentagon of one face (Fig. 1).

From Points 18.6.3 and 18.6.1 we infer:

BH=1,618034L ; L

L

L

L

D

d

l L 0 , 381966

1 , 618034

0 , 618034

= = = (side of the small pentagon within the

stellate one)

OD= 0 , 8506508 × 0 , 381966 L= 0 , 3249196 L (radius of former pentagon)

OB= 0 , 8506508 L (radius of pentagon with side L)

OH= BH^2 −OB^2 = 1 , 3763819 L ; OE = 0,6881909 L (apothem of pentagon with side L)