spherical polyhedra

the "Campanus' sphere"

Very popular during the Renaissance, this polyhedron is composed with n rings of 2n faces.
Inscribe a regular 2n-gon in an "equatorial circle", then inscribe 2n-gons in n "meridian circles" using as vertices the two poles and two of the preceding points (opposite on the "equator-circle"). To complete the polyhedron connect the vertices of these n polygons with lines parallel to the equator.
For n=2 you get the regular octahedron.

Just for the fun, here is an approximation of an oblate spheroid (ellipsoid with two axis of same length, and the "polar axis" smaller).

Our earth looks like this polyhedron, but it is much nicer!

the "geodesic domes"

These polyhedra are constructed starting from Plato's polyhedra; a tessellation is drawn on each face and the central projection on the circumsphere defines the new vertices. Fuller widely popularized the structures based on this concept. Here are three examples constructed from the regular octahedron (triangulation in 9), dodecahedron (triangulation in 5) and icosahedron (triangulation in 4).

8x9=72 faces

12x5=60 faces

20x4=80 faces

distributing points on a sphere

There are different ways to distribute uniformly points on a sphere; Martin Trump used a model of n electrical particles linked on a sphere and stabilized the system; so he got a convex polyhedron S_n with n vertices.
We obviously expect to get "very symmetrical" configurations. For the small sets of points we actually find well known configurations with "rich" symmetries, but quickly the symmetry groups become impoverished while we get several stable configurations for a same number of points. For greater sets of points the polyhedra are reminiscent of geodesic domes, with sometimes a few quadrilaterals (not always squares).

S₄, S₆ and S₁₂ are the regular tetrahedron, octahedron and icosahedron.
S₅ and S₇ are diamonds with identical isosceles faces.
S₈ looks like the square antiprism (with isosceles triangles).
S₉ is related to the D14 deltahedron but with only two opposite equilateral faces (the others are isosceles).
S₁₀ is the assemblage of a square antiprism and two square pyramids (two kinds of isosceles triangles).
S₂₄ (which looks like the snub cube), one S₄₈, one S₇₂ have six square faces; one S₁₆, one S₃₂, one S₄₈ have only two.
S₈, S₁₆, S₃₂, S₄₈ ... is a sequence where new rings of triangles appear between two squares;
the same happens with two pentagonal pyramids in the sequence S₇, S₁₂, S₁₇, S₂₇ ...

Here are three examples (data provided by Bob Allanson's applet, see references) where edges of same color have same length and symmetry axes are dashed:

S₁₅ : 26 faces, symmetry group C₃
(one 3-fold symmetry axis, no symmetry plane)

S₁₆ : 24+2 faces, symmetry group C_4v
(one 4-fold symmetry axis and four planes)

an other S₁₆ : 28 faces, symmetry group T
(four 3-fold symmetry axes, no plane)

two other spherical polyhedra

If we erect pyramids on the six square faces of a snub cube we get a "spherical polyhedron" with 32+6×4=46 triangular faces; if we do so with the twelve pentagonal faces of a snub dodecahedron we get 80+12×5=140 triangular faces.

references:

• The Penguin dictionary of curious and interesting geometry by David Wells (Penguin Books, 1991).
• Polyhedra (pages 106-107) by Peter R. Cromwell (Cambridge University Press, 1997).
• Martin's Pretty Polyhedra and the Repulsion Polyhedra Generator by Bob Allanson.
• Distributing Points on a Sphere by Paul Bourke.
• a polyhedral celestial globe to be realize by yourself
• géodes by J.B. Roux (web page in French)

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March 2004
updated 08-07-2005