Mathematics and OrigamiTo figure out that angle, let ́s set up first the hypothesis that a rhombic face has L as thegreat diagonal and L / 2 as the small one. In that case, we can write (Figs. 3 and 4):
35 , 26439 º
63 2 2
arccos
2 2:
31
23. arccos arccos =
×
=
×=
= L L
FHOH
AngHIf the hypothesis holds true, the half-rhombs AFB and AGB (Fig. 3) will be coplanar,
i.e., Ang. FHG = 180º.
Recalling (Point 18.9.5), that the dihedral angle between faces of an octahedron (in this
case that of side AB in Fig. 2) measures 109,47123º, mentioned angle Ang. FHG will measure:
109,47123 + 2 × 35,26439 = 180
what proves that the hypothesis is true.
Finally, let ́s construct a paper rhombic-dodecahedron by three different methods. In all
the cases we shall use rhombs obtained from DIN A rectangles. Fig. 5 shows how to fold one of
those rectangles to get a rhomb whose diagonals keep the ratio 2. Fig. 6 is the disposition to
be given to the 12 rhombs that will form the polyhedron. Fig.7 suggests a modular solution
with interlocked rhombs.
Recall Point 18.9.4 to perform that interlocking. Fig. 5 is a module and we need 4
groups of 3 interlocked modules like the one in Fig. 7. Then interlock the 4 groups with each
other.
Before proposing the third method, we shall figure out the value of the dihedral formed
by two rhombic-dodecahedron ́s faces. Fig. 8 is the same Fig. 1 after its side FB has been cut
by a plane normal to it through G (Fig. 1). So, in Fig. 8 A ́E ́ is equal to the diagonal AE (par-
allels between parallels) and A ́J = E ́J (altitudes of two adjacent rhombs); therefore, Ang.
A ́JE ́ measures the wanted dihedral.
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