16 years ago
Icosidodecahedron as it appears from several considerations of
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.
(A part of this was posted earlier on de.sci.mathematik 23 Oct 2004.
Favourite Icosododecahedron based Great Circles)