MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

Let ́s compute the faces. From each vertex start 5 equilateral triangles common, in turn,

to 3 vertices; therefore 20

3

5 12

=

×

faces: C = 20. These 20 faces are equilateral triangles, con-

sequently convex with species e = 1.

Summarising, the obtained polyhedron has:

C = 20 ; e = 1 ; V = 12 ; E = 2 ; A = 30

Euler ́s theorem for stellate polyhedra gives its species ε:

7

2

20 1 2 12 30

2

=

× + × −

=

+ −

=

Ce EV A

ε

Therefore, polyhedron nº 1 is an icosahedron (C = 20) with triangular faces, of 7th spe-

cies (ε = 7), and 12 pentahedral angles (V = 12) of 2d species (E = 2).

Interlude