MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

18 POLYHEDRA

They are solids bounded by plane faces (polygons, obviously in number of 4 or more).

• Their faces C are those polygons.


• Their vertices V are the vertices of the polyhedral angles.


• Their sides A are the intersection of two faces, forming a dihedral angle.
  All polyhedra are governed by the Euler theorem that relates C, V, A. The generalised Euler
  theorem should be applied to stellate regular polyhedra; we ́ll see to it later on. For the rest of polyhe-
  dra, here we have the Euler theorem:
  C + V = A + 2
  We shall not prove it now. The reader should recall a similar approach given to plane figures in
  Point 1.3.1.
  According to various criteria, which in turn may be related to each others, polyhedra are classi-
  fied in:
  Regular, irregular, pseudorregular, concave, convex, stellate, platonic, archimedean, conjugate,
  etc, etc.
  Regular polyhedra are the five platonic ones: all their faces are equal regular polygons. Let ́s
  see why there are but five.
  To begin with, their faces can only be equilateral triangles, squares, or regular pentagons, hexa-
  gons, etc.
  It is obvious that the sum of the plane angles forming a polyhedral angle must add up to less
  than 360º.
  With the 60º of the angle of an equilateral triangle we can construct a polyhedral angle with a
  maximum of 5 faces (6 would not form a polyhedral angle, but a perigon: 60 x 6 = 360º). Since the
  trihedral is the smallest possible polyhedral angle, it follows that with equilateral triangles we ́ll be
  able to form polyhedral angles of 3 faces (3 60 = 180 < 360); of 4 faces (4 60 = 240 < 360); of 5
  faces (560 = 300 < 360); and no more faces (6 60 = 360).
  For the square: 3 90 = 270 < 360; 4 90 = 360. With the square we can construct only trihe-
  dral angles.
  Let ́s see what happens with the regular pentagon (the value of its interior angle is
  ()
  108 º
  5

1805 2

=

−

). We can construct a pentagonal trihedral angle, for 108 * 3 = 324 < 360; four pen-

tagonal faces are too many: 108 4 = 432 > 360.
It is not feasible to construct a trihedral angle with hexagonal faces (120 3 = 360).
Therefore, with the different regular polygons we can construct:

FACE NUMBER OF FACES IN POLYHEDRON ́S

POLYGON POLYHEDRAL ANGLE NAME

EQUILATERAL ∆ 3 TETRAHEDRON

4 OCTAHEDRON

5 ICOSAHEDRON

SQUARE 3 HEXAHEDRON

PENTAGON 3 PENTAGON-DODECAHEDRON

The latter 5 regular polyhedra, besides the faces, have also respectively equal, sides and angles:
dihedral as well as polyhedral.
The archimedean polyhedra are obtained by truncating the polyhedral angles of the platonic. It
is well known that Archimedes dealt with 13 of them. They are a special source of inspiration for
imaginative unit folders.