# 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".
 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:
 assembling of twenty cubes  ring of eight regular dodecahedra 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

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:
 Johnson 18 drilledwith a triangular cupola (Johnson 03)augmented with a triangular prism.  truncated octahedron drilledwith four triangular cupolasand a regular octahedron (at the center)

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

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