MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

18.13 PSEUDORREGULAR POLIHEDRA

All their faces are equal (as in the regular polyhedra), or symmetrical; they are not

regular, though: they are irregular polygons. We are going to consider two examples with a

common octahedric base.

Another common feature is that they represent the crystalline structure (regular system)

of some nesosilicates type Me 2 Me 3 (SiO 4 ) 3 : Me may be Al, Fe, Mg, Cr.

18.13.1 RHOMBIC-DODECAHEDRON

In the Cosmo Caixa Museum at Alcobendas, Madrid, one can see a perfect example of a

green garnet (there exist garnets of different colours) crystallised as a rhombic-dodecahedron.

It is shown in Fig.1 and has the following characteristics:

V = 14; there are two groups of vertices: 6 acute polyhedral angles (4 faces) corre-

sponding to the acute angles of the rhombic faces (and in direct relation to the 6 vertices of the

basic octahedron). Besides, 8 obtuse polyhedral angles (3 faces) corresponding to the obtuse

angles of the rhombic faces and closely related to the 8 faces of the basic octahedron.

C = 12; the 12 faces are equal rhombs with diagonals in the ratio of 2. All the dihedral

angles formed by the faces are also equal. This leads to what could be named a regular rhombic

dodecahedron. On the other hand, if large to small rhombic diagonals ́ ratio is different from

2 but smaller than 3 , we get the so-called irregular rhombic-dodecahedron: 10 equal basic

faces plus 2, also equal to each other, but consisting in rhombs different from the other 10. This

will hold true as long as the dihedral angles of those 10 faces will have the value of













RHOMBsAltitude

RHOMBsGreaterDiagonal

2 ́

́

2.arcsen

A = 24; all the sides are equal to the rhomb sides. Therefore a side has one extremity on

an acute vertex and the other on an obtuse one.

C + V = A + 2 ; 12 + 14 = 24 + 2

In Fig. 1, ABCDE are the viewed acute vertices of the rhombic-dodecahedron; F and G

are obtuse vertices. Those 5 acute vertices have being segregated into Fig. 2 to set on the start-

ing octahedron. Fig. 3 is the clue to understand the relation between the rhombic-dodecahedron

and 2 , and hence, with the DIN A rectangle. As we can see, Fig. 3 is Fig. 2 after adding to it

the sides in F and G of Fig. 1. By means of that we have constructed the triangular pyramids F

ABE and G ABD. The former is cut by the plane EFH (normal to diagonal AB): in it is repre-

sented the altitude FO of said pyramid. Fig. 4 is a partial enlargement of Fig. 3 to allow calcu-

lation of angle in H.