**Toroids formed from augmented duals of uniform
polyhedra**

An interesting set of figures, a number of which are toroids, can be constructed from the duals of polyhedra with vertex figures of (m,n,m,n). The simplest case is probably that derived from the cuboctahedron, vertex figure (4,3,4,3). Take the dual of the cuboctahedron, the rhombic dodecahedron (model). Augment all of its faces with prisms (model). The 'V' shaped valleys in between the prisms can be filled with triangular prisms (model), these meet at the vertices of the original dual where they can be joined using tetrahedra and octahedra. Remove the original dual, and the rhombic prisms to leave a toroidal figure with rhombic holes (model).

Dual

Augmented Dual

Compound of 3-gonal prisms

Final Figure

Replacement of caps with pyramids

The above figure has been known for some time. I believe however, that the three figures shown below are new.

If we refer to the above figure as **[4,3],3**, where
**
[4,3]** relates to a cuboctahedron and the last ** 3** refers to the triangular prisms,
then we can refer to the general case as **[m,n],p**. The original polyhedron
has vertex figure of (m,n,m,n), p-gonal prisms are added, and the 'capping'
polyhedra are p^{m} and p^{n}.

Remarkably, if we take the set of figures (3, 5, ^{5}/_{2})
these can be set into ** [m,n],p** in any order. Three distinct figures result:

The dual of the icosidodecahedron (the rhombic triacontahedron) is augmented, valleys filled with pentagrammic prisms, caps are small stellated dodecahedra and great stellated dodecahedra.

Dual

Augmented Dual

Compound of pentagrammic prisms

Final Figure

Final Figure with dodecahedra
highlighted

The dual of the dodecadodecahedron is augmented, valleys filled with triangular prisms, caps are icosahedra and great icosahedra. (The image above shows the construction before the addition of the icosahedral caps).

Dual

Augmented Dual

Compound of triangular prisms

Addition of great icosahedra only

Final Figure

Replacement of caps with pyramids

The dual of the great icosidodecahedron is augmented, valleys filled with pentagonal prisms, caps are dodecahedra and great dodecahedra. (The image above shows the construction before the addition of the icosahedral caps). Note that this example is not toroidal, at least not in the sense of having visible holes..

Dual

Augmented Dual

Compound of pentagonal prisms

Addition of great dodecahedra only

Final Figure

**[3,3],4**

One further, simple figure is possible.
This can be thought of as the tetrahedral case, **[3,3],4**. For this
we have to regard the octahedron as a 'tetratetrahedron',
it's dual is a cube. This can be
augmented with cubes, the valleys are filled with cubes and the caps are also
cubes. The resulting toroid is shown above

**Credits**

The polyhedra on this page were initially generated using
Robert Webb's Great Stella
program and post-processed using **HEDRON**.

For users of Great Stella: Stella files of all these polyhedra are included in this archive: dual-toroid.zip