Jesús de la Peña Hernández
If we assume that the dimension of a rhomb ́s small diagonal is 1, the great will measure
2 and the side0 , 8660254
22
21
2 2
=
+
On the other hand the rhomb ́s altitude is equal to its area divided by its base (a side)0 , 8164965
2 0 , 86602541 2
́ =
××
AJ=Hence the measure of angle A ́JE ́ will be:
: 0 , 8164965 120 º
22. ́ ́ 2 arcsen =
AngAJE=We must say that Fig. 5 is not the only way to get a “DIN A” rhomb. Another procedure
was already disclosed in Fig. 1, Point 18.2.5: one is as good as the other, but we have to bear in
mind that starting from rectangles of the same size, we end up with distinct, tough similar,
rhombs: 2 is the ratio of similarity.
Carrying on with Point 18.2.5, we saw there that 60º was the angle formed by the base
of the pyramid and one of its lateral faces. It means that if we join together two of those pyra-
mids by setting in common two lateral faces, the rhombic bases will form an angle of 120º in
the new figure.
In consequence, that new figure holds two rhombic faces of a rhombic-dodecahedron
since we got to know that 120º is also the angle formed by two faces of this polyhedron. Be-
sides, the common vertices of those two pyramids coincide with the center of the rhombic-
dodecahedron: of course, the radii of the rhombic-dodecahedron are different depending
whether we join its center with an acute or an obtuse vertex. It happened in the rhombic pyra-
mid as regards to its lateral sides.
Fig. 9 shows the 4 pyramids with center O (that of the rhombic-dodecahedron) associ-
ated to the acute vertex C of Fig. 1. At left we can see in shade the only seen rhomb.
To construct the polyhedron we ought to have 12 pyramids, which, in turn, are obtained
this way:
From Fig. 10 we get an acute vertex formed by 4 rhombic faces: it is shown in Fig. 9.
Folding Fig. 11 we get two more pyramids; it ́s a matter of redoing this folding four
times to complete the 12 pyramids we need.
Two comments: first, the two triangles forming the facial rhombs seem to be equilateral
but they are not; second, the rhombic-dodecahedron obtained in the last process is more con-
sistent than the others since its interior is reinforced by 12 pyramids.