Favorite Icosidodecahedron based Great Circles.

From: Narasimham G.L. (mathma18_at_hotmail.com)
Date: 10/23/04 

Date: 23 Oct 2004 14:37:17 -0700

What continuous great circle arrangements are there on a sphere of
associated solid polyhedra with full spherical symmetry? From Wolfram
site for Archimedean and Platonic solids only three case are seen:

Among Platonic solids we have only one case, three great circles in
mutually perpendicular planes (containing 3 great circles: Equator +
meridians of Greenwich and US/Canada Central Time),corresponding to an
octahedron.

Using a cube or tetrahedron, 3 lines come together at a vertex, so a
continuous projected great circle on the sphere cannot be drawn.

Among Archimedean solids only two cases, those projected from the
icosidodecahedron and the cuboctahedron onto the sphere.

The former has particular appeal to me, 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.
Care is needed in placing of rings as a two torus interlocked mode.
Practically chose 11 equal parts, one overlapped for assembly
convenience.

I wonder if there could be easy, simple symmetric position/ direction
descriptions of all the points of intersections of icosidodecahedron
using quaternions or in any other suitable way. Please comment on any
aspect you like.

Narasimham

Relevant Pages

  • Re: The sphere and hyperbolic geometry
    ... My geometry - the surface of the sphere, all points and all circles ... axiom, ...
    (sci.math)
  • Re: What would Sam have done?
    ... winter, which meant that either the earth stood still, or the ... that same sphere of fixed stars. ... As was the replacement of all those perfect circles (the ...
    (rec.arts.books.tolkien)
  • Re: The sphere and hyperbolic geometry
    ... My geometry - the surface of the sphere, all points and all circles ... axiom, ...
    (sci.math)
  • The sphere and hyperbolic geometry
    ... The sphere with great circles as lines is not a hyperbolic geometry. ... north and south poles) missing and lines of longitude ...
    (sci.math)
  • Re: 2 equations
    ... circles (basically any old shape defined over one quadrant and then ... f^an other symmetry. ...
    (sci.math)
Loading