Stewart's 6J91(P4)

Stewart's J91(P4)
Stewarat's 6J91(P4)
Type	Polyhedron
Panels	24 × 6 × 24 ×
Rods	120 ×
Spheres	56 ×
Author	PolyClare 14:44, 14 January 2008 (UTC)

Stewart's 6J₉₁(P₄) -- Six bilunabirotundas (6J₉₁) surrounding a central cube (P₄). Although the Geomag model shown at the right retains the inner (yellow) cube inside (for extra stability), this is not part of 6J₉₁(P₄).

The bilunabirotunda is known as Johnson Solid J₉₁, and Johnson used the notation P₄ for a cube (P for prism, 4 for square). Thus Stewart (page 127, 2nd ed.) labels this complex made from multiple polyhedra 6J₉₁(P₄).

The J₉₁(P₄) combination is formed by matching a square face of a bilunabirotunda to a side of a cube. Each subsequent bilunabirotunda is attached to another face of the cube, with its long axis oriented orthogonally to the long axes of bilunabirotundas attached to adjacent cube faces. Each bilunarotunda loses a complex of one square surrounded by four triangles in this process, and the cube disappears entirely (though for Geomag model building it may be desirable to retain the cube for extra stability).

This J₉₁(P₄) may be used to form the central 6-way hole for the Lunar-holed Rhombicosidodecahedron, Stewart's E₅/6J₉₁(P₄).

These models were built using Geomag Pastelles. I'm finding that I like using the Pastelles panels for complicated constructions, because their relatively high transparency lets me see more of the entire structure.

Other views

1. A complete model of Stewart's 6J₉₁(P₄)

2. One bilunabirotunda separated off (with matching parts filled in)

3. One separated bilunabirotunda oriented to show how it fits in

4. A former 6J₉₁(P₄), after its tragic crash to the floor

5. Pieces of the 6J₉₁(P₄); all parts accounted for

6. The rebuilt Stewart's 6J₉₁(P₄), with different triangle colorings.

Related links & references

• Book: Adventures Among the Toroids: A Study of Quasi-Convex, Aplanar, Tunneled Orientable Polyhedra of Positive Genus Having Regular Faces With Disjoint Interiors by B. M. Stewart

• Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200.

• Wikipedia:Bilunabirotunda

• Mathworld:Bilunabirotunda