Drilled truncated icosidodecahedron

Drilled Truncated Icosidodecahedron
Stewart's K₅/12Q₅S₅(E₅)

Type	Stewart Toroid

Rods	720 ×
Spheres	260 ×
Author	PolyClare

Geomag realization of a large Stewart Toroid, named in the spirit of Stewart's Drilled Truncated Dodecahedron, which inspired this model. The original Drilled Truncated Dodecahedron has pairs of triangular faces (where the edges of the pentagonal cupolae meet) that have a too-small-for-geomag angle between them. This model inserts a square between those steeply-angled trianges so that the geomags can handle it. Additionally the original Drilled Truncated Dodecahedron has too-small-for-geomag angles between triangles where the internal edges of the Pentagonal Antiprisms meet. This is solved for this model by the insertion of a square between these two triangles as well. The widening squares are added in parallel corresponding positions to both the inner and outer shells, so that the distance between the inner and outer shells is maintained, meaning they can still be connected by Pentagonal Cupolae with attached Pentagonal Antiprisms. Thus the polyhedra in the inner and outer shells are modified from those in the original design to those in the list below.

Stewart referred to the drilled truncated dodecahedron as the jewel in the crown of the toroids, and this model has the same beautiful property that it is composed of squares and triangles, and yet pentagons are in its structure.

Using Stewart's notation for his Toroids, I'd say that he'd describe this model as K₅/12Q₅S₅(E₅) (See page 188 of Stewart's book, 2nd ed.).

Robert Webb calls this model the "Cupola-Drilled Truncated Icosidodecahedron" in the library accompanying his (very excellent) Great Stella program.

The white triple triangles (plus the three surrounding/connecting metallic blue rods) are supposed to be hexagons, but I don't like to use the rhombic geomag panels to make hexagons -- they end up too floppy and fragile. I ended up leaving them unpanelled, leaving places that are easy to open up to view the true interior of this Toroid.

This Stewart Toroid is composed from the following parts:

The outer shell of this Toroid is a Truncated Icosidodecahedron (white rods plus decagons of metallic blue rods) with Pentagonal Cupolae Excavations (metallic blue rods and blue panels), with central Pentagonal Holes (red rods).

The inner shell of this Toroid is a Small Rhombicosidodecahedron with Pentagonal Holes (panelled in red, red and silver rods, very hard to see in photo).

The twelve holes connecting the inner shell and outer shell of this Toroid are Pentagonal Antiprisms with missing Pentagons (red rods, green triangular panels).

Rod & Ball Calculations:

White Triple Triangles:

180 = 20 units × 9 rods

Blue Pentagonal Cupolae (minus central pentagon):

240 = 12 units × 20 rods

Red Antiprisms:

240 = 12 units × 20 rods

Silver Rhombicosidodecahedron (minus pentagons):

060 = 1 unit × (120 - 5×12) rods

TOTAL: 720 Rods

White Triple Triangles (just the central ball):

020 = 20 units × 1 ball

Blue Pentagonal Cupolae (including central pentagon):

180 = 12 units × 15 balls

Red Antiprisms (counted elsewhere)

000 = 12 units × 0 balls

Silver Rhombicosidodecahedron (including pentagons):

060 = 1 unit × 60 balls

TOTAL: 260 Balls

For comparison, the original Stewart Toroid that inspired me to build this one is the Drilled Truncated Dodecahedron, which has a Truncated Dodecahedron (with Pentagonal Cupolae Excavations (joined at the triangles) with central Pentagon Holes) for an outer shell; a Dodecahedron inner shell (missing all pentagons); connected by the same Pentagonal Antiprisms (missing the pentagons).

Thanks to Karl Horton for the inspiration to explore geomag constructions of (or inspired by) the Drilled Truncated Dodecahedron.