MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

Fig. 1 shows an example made out of one octahedron of side L to Fig.1 (Point 18.9.1)

plus 8 tri-right-angled vertex pyramids (Point 18.2.1.2), centered on the corresponding faces of

the octahedron. Fig. 3 evidences those pyramids associated to the reference cube (the interior of

the cube appears in shade to set off those pyramids on its vertices). From the figures we can

deduce the measure of cube ́s side:

2

L

.

The galena, lead sulphide (PbS), among a great variety of combinations (cubes, octahe-

dra, rhombic-dodecahedra), occurs in macles of tabular aspect frequently with the shapes of

octahedra and hexahedra.

18.14.5 PYRITOHEDRON nº 1

The pyrite crystallises in the regular system mainly in shape of cubes, pentagonal-

dodecahedra or in combinations of both of them. At least 25 different shapes are known.

The first we are going to deal with is shown in Fig. 1 and is the result of chamfering the

sides of a pentagonal-dodecahedron like the one in Fig. 2. Discarded wedge shaped chamfers

are represented in Fig. 3. One can see that the chamfer may be performed in many ways, but in

this occasion we have chosen that which produces in the wedge an oblique section like the tri-

angle ABC of Fig. 1 (Point18.6.1).

If we look at present Fig. 1 we ́ll see that it consists in 18 faces: 12 equal irregular

hexagons coincident with the faces of the polyhedron of Fig. 2, and 6 equal rectangles, which

are the bases of the respective wedges.