Isomorphs to the Rhombicuboctahedron

This 'space invader' is actually an isomorph of a uniform polyhedron, it shares the same net as the rhombicuboctahedron.  It's derivation is described below.

As previously mentioned in my cupola pages, the small rhombicuboctahedron above can be thought of as six overlapping square cupolae.

Were one of these cupolae to be pushed inwards (or 'inverted') an isomorph to the rhombicuboctahedron is formed which I name an inverted rhombicuboctahedron above.  Normally when such an operation is carried out on one face of such a polyhedron, a similar operation cannot be carried out on any neighbouring faces.  The rhombicuboctahedron however is an exception.  The relevant edges of the inverted cupola remain coplanar with the (now distorted) octagonal face of the neighbouring cupolae allowing these also to be inverted.

Two variants of the bi-inverted rhombicuboctahedron exist.  A meta-bi-inverted rhombicuboctahedron above left where two neighbouring cupolae are inverted, and a para-bi-inverted rhombicuboctahedron above right where two opposing cupolae are inverted.

Similarly two tri-inverted rhombicuboctahedra exist. I term these a meta-tri-inverted rhombicuboctahedron above left where three neighbouring cupolae are inverted,  and a para-tri-inverted rhombicuboctahedron above right where two of the three inverted cupolae are opposed.

For the tetra-inverted rhombicuboctahedron we one more have two variants. Again the terminology is similar, in this case the meta-tetra-inverted rhombicuboctahedron above left has two neighbouring cupolae left convex (or 'everted') and the para-tetra-inverted rhombicuboctahedron above right has two opposing cupolae left everted, these are now {4/3} cupolae due to the invertion of all the neighbouring cupolae.

Only one penta-inverted rhombicuboctahedron above exists with its one everted 4/3} cupola.

When we come to the hexa-inverted rhombicuboctahedron above we arrive at a familiar figure.  We have now generated a great rhombicuboctahedron.  This allows us to rename the penta-inverted rhombicuboctahedron as an everted great rhombicuboctahedron and the tetra-inverted rhombicuboctahedra as bi-everted great rhombicuboctahedra, this clarifies the use of the meta and para prefixes for these polyhedra.


The generation of the great rhombicuboctahedron from the small rhombicuboctahedron by invertion of its cupolae is the subject of the following animated VRML files.

1. All cupolae are inverted together

2. The cupolae are inverted serially (meta route). 

3. The cupolae are inverted serially (para route).

Equivalence to cube colourings

A feature of the above isomorphs to the rhombicuboctahedron is that they illustrate the ten distinct ways in which a cube can be coloured using two colours for the faces, as shown in the diagram below.   The plain faces relate to everted cupolae and the faces with green blobs to inverted cupolae.

ps. the 'space invader' could be either the meta-tri-inverted rhombicuboctahedron (if you want a back to his head) or a meta-tetra-inverted rhombicuboctahedron (if you prefer a two-faced version).

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