MATHEMATICS AND ORIGAMI

Mathematics and Origami

18.14 MACLES

In crystallography are so named the twinned crystals oriented in such a way that one can

overlap the other if properly moved, rotated or subjected to a symmetry. The last two opera-

tions have as reference what are called macles ́ planes or axles. As a matter of fact, macles are

steps in a crystal growing process.

18.14.1 TETRAHEDRIC MACLE

It is shown in Fig. 1. In it, two equal tetrahedra inter-penetrate each other: one is the

ABCD; of the other we can see its vertices XYZ.

It seems to be a stellate polyhedron, but is not (see Point 18.12). One of the tetrahedra

may become the other under this process: to get its symmetric with respect to one of its faces,

to move 1 / 2 of its altitude and rotate it 60º.

The macle ́s folding diagram is not shown for we consider it made up of a big tetrahe-

dron and four small ones centred on the faces of the big; the small tetrahedra have a side half of

the big ́s. See Point18.2.1 to construct a tetrahedron: it is obvious that the paper strip width to

produce the big tetrahedron is double than the one needed for the small ones.

The macle has 4 × 2 = 8 vertices belonging, in turn, to a cube (Fig. 2). If L is the side of

the big tetrahedron, the cube ́s side is the distance between two opposite sides of that big tetra-

hedron: its value is

2

L 2

(see Point 18.2.1.1).

The blende, zinc sulphide (SZn), crystallises in the cubic system, tetrahedric mode,

polysynthetic macles.

18.14.2 MADE OF CUBES

Figs. 1 and 2 show the aspect of a multiple macle of these characteristics:

• Big cube of side AB = L.


• Small cubes with sides
  2

L

.

• Their relative position is such that section ABC is an equilateral triangle.
  The conditions that follow define the small emergent cube whose folding diagram is Fig. 3: