Stewart's polyhedra

the "G3"

We can search for polyhedra with a specific vertex; a pqr-vertex is common to three faces: a p-gon (regular polygon with p sides), a q-gon and a r-gon. The list of all types of vertices is long, and we find several examples among the Johnson polyhedra and the uniform polyhedra. Nevertheless none of these polyhedra has a 345-vertex!
Here is a quiet simple one: Bonnie Stewart's "G3".


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With its 13 faces and its 13 vertices, this polyhedron has six 345-vertices and a 3-fold dihedral symmetry, but it is not minimal, neither for the number of faces nor for the number of vertices.

"G3" has also a 555-vertex as has the regular dodecahedron; the first has a minimum of vertices and the second a minimum of faces.

Searching - and finding! - such polyhedra needs imagination!.

some examples of toroids

B.M. Stewart described many toroids (polyhedra with at least one "tunnel"), and in particular those which are assemblings of (semi)regular polyhedra. Here are four examples:


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assembling of twenty cubes
 

ring of eight regular dodecahedra



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ring of eight regular octahedra
(minimal deltahedron: 24 vertices and 48 faces)
 

to get a ring of eight regular icosahedra
(deltahedron with 144 faces)
we have just to inscribe an icosahedron in each octahedron



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But a polyhedron is a "Stewart's toroid" only if:
•  its faces are regular, non intersecting, and two faces which share an edge are not coplanar,
•  it is "quasi-convex" (its convex envelop has no new edges) and has at least one "tunnel".
Here are two simple examples:


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Johnson 18 drilled
with a triangular cupola (Johnson 03)
augmented with a triangular prism.
 

truncated octahedron drilled
with four triangular cupolas
and a regular octahedron (at the center)



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references: •  Adventures Among the Toroids  by B.M. Stewart, 1970.
•  http://www.orchidpalms.com/polyhedra/ (pages "acrohedra" and "toroids") by Jim McNeill


January 2004
updated 30-01-2008