MATHEMATICS AND ORIGAMI

Mathematics and Origami

18.12.2 STELLATE REGULAR POLYHEDRON nº 2

The auxiliary starting polyhedron is, in this case, a convex icosahedron with side AB =

L (fig. 1); ABH is one of its facial equilateral triangles; the pentagon of center O is the base of

its upper dome (Point 18.6.2). ABDH are some of its vertices.

OA = 0,8506508 L (r in Point 18.6.1)

OH = 0,525731 L (h in Point 18.6.2)

OC = 0,3249196 L (OD in Point 18.12.1)

HC = OC^2 +OH^2 = 0,618034 L (what shows that HC = CB = CA, Point 18.6.3).

Fig. 2 is the folding diagram for trihedron in C, ABH (closing lap join is shown).

Fig.3 is the folding diagram (with no lap joints at all) of dome in H (Fig. 1), plus the 5

triangles associated to it (Point 18.6.2). Of course, that Fig. 3 does not develop equilateral tri-

angles, but the trihedral angles to Fig. 2.

Two assemblies like that obtained with Fig.3, set in opposition, give what appears to be

an icosahedron. But it is worthwhile to analyse it closely to see what it is like, actually, the

stellate polyhedron we have got (Fig.4).

This stellate polyhedron nº 2 has 12 vertices, as the starting icosahedron; we have to ig-

nore the concave vertices of sank trihedrals because they do not lie on the sphere of reference.

Therefore we have 12 vertices composed by stellate pentahedral angles (V = 12); i.e. they have

species E = 2: we may recall (Point 11) that the species of a stellate pentagon is also 2.

H H

C

A B

2

C

A

H

B

3

H

H

H H

H