The Icosahedron is not a superior choice compared to the
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

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)

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 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)