Icosidodecahedron as it appears from several considerations of

advantage.

In the latter case it is possible to build a dome using continuous

interwoven relatively flexible strips that can be rigidized by adding

similar strips to make curved I-beams. A continuous reinforcement is

more favourable for force distribution.Construction of higher frequency

triangular elements is possible here also, there is material economy

and stress distribution is improved without concentrations at so many

mechanical joint interfaces.

The more symmetric Platonic solid shell using icosahedron faces and

sub-divisions of higher frequency is not necessarily a better choice

than that of an Archimedean solid shell icosidodecahedron.

What continuous, unbroken great circle arrangements are there on a

sphere of associated solid polyhedra with full spherical symmetry?

To this question only three cases are seen as an answer from Wolfram

site for Archimedean and Platonic solids.

Among Platonic solids we have only one case corresponding to an

octahedron.Here three great circles in mutually perpendicular planes

(E.g.,on a globe containing 3 great circles: Equator

*/*

0 deg Greenwich /-90 deg US Central Time meridians). Among other

0 deg Greenwich /

cases using a cube or tetrahedron, 3 lines come together at a vertex,

so a continuous projected great circle on the sphere cannot be drawn

from them, they are not suitable choices.

Among Archimedean solids there are only two cases, those projected from

the icosidodecahedron and the cuboctahedron onto the sphere.

The former has particular appeal, more than any other

arrangement_ perhaps because of its association with the GoldenRatio

(phi ~ 1.618 ) in several ways. On the surface of sphere, tan (half

interior angle of pentagon) is phi and tan of (pentagon center to

corner angle seen at sphere center) equals 1/phi.The side of

Icosidodecahedron solid/Sphere radius ratio is anyhow golden, from

geometry of a plane decagon central section. There are only six

cutting planes/great circles producing 32 faces (20 spherical

triangles and 12 spherical pentagons). Cutting plane is at arctan(0.5)

to axis of symmetry through center of pentagons. It is a strange but

pretty combination of dodecahedron and icosahedron.A RealTime3D view

with great circles looks even better than a solid model. It is rather

easy to make a model, I made one of 500 mm dia from 6 plastic strips

each divided into 10 equal parts for great circle intersection joints.

Three layers are carefully interwoven/locked together. Practically

chose 11equal parts, one overlapped for assembly convenience.

Comments on any aspect are welcome.

Narasimham

(A part of this was posted earlier on de.sci.mathematik 23 Oct 2004.

Favourite Icosododecahedron based Great Circles)