__Polyhedral "Twisters"__

An unusual deformation of polyhedra
can be constructed by taking a cuboctahedron
with the square faces removed leaving eight triangular
faces joined at the vertices (above left). If we allow these vertices to deform, we can
twist this figure to resemble an icosahedron (above
middle). A further
twist collapses the figure to an octahedron (above
right). This figure was first described by Buckminster Fuller, who
coined it a 'Jitterbug'. A formal mathematical description of the process,
and an enumeration of the Jitterbug-like set (which he terms 'dipolygonids') was
completed by Dr Hugo Verheyen in his paper: "**The complete set
of jitterbug transformers and the analysis of their motion**" a. Comp.
Math. Appl., 17, Pergamon, 203-250. Oxford, 1989. b. Symmetry 2:
Unifying Human Understanding, ed. I. Hargittai. Pergamon, Oxford, 1989.

I came across the simpler members of
this set in "**Platonic and Archimedean Solids**"
by Daud Sutton, published by Wooden Books Ltd in 1998 *(*ISBN
1-902418-34-4). The subject was also investigated independently in 1998 by
Dr Richard Klitzing (summarised here) who included consideration of
the regular tessellations.

My own, non-technical term for these objects is 'twisters', formal notation can be found in Verheyen.

If we consider the vertices
involved, then starting with a rhombic figure having vertices of the form (**a**,4,**b**,4)
we deform it into a snub figure with vertices of the form (**a**,3,**b**,3,3)
and finally merge two vertices together in the final semi-regular form (**a**,**b**,**a**,**b**).
In general terms I describe this process as forming an [**a**,**b**]-twister.

In addition to the three examples given in Sutton, four re-entrant twisters can be found amongst the Uniform Polyhedra. These seven are described in turn below, as is an additional case, that being a non-uniform twister which I do not believe has been previously described. (Note: this is not the complete set of dipolygonids described by Verheyen which includes more degenerate cases.) In each case, clicking on the snub image will display an animated VRML file:

Rhombic: cuboctahedron
or 'rhombic tetratetrahedron'

Snub: icosahedron or 'snub tetratetrahedron'

Semi-regular: octahedron or 'tetratetrahedron'.

Rhombic: rhombicuboctahedron

Snub: snub cuboctahedron

Semi-regular: cuboctahedron

Rhombic: rhombicosidodecahedron

Snub: snub icosidodecahedron

Semi-regular: icosidodecahedron

Rhombic: rhombidodecadodecahedron

Snub: snub dodecadodecahedron

Semi-regular: dodecadodecahedron

The rotation of the [5/2,5]-twister can be continued to form the following:

Rhombic: complex
rhombicosidodecahedron (degenerate)

Snub: vertisnub dodecadodecahedron

Semi-regular: dodecadodecahedron

Rhombic: small
complex rhombicosidodecahedron (degenerate)

Snub: great snub icosidodecahedron

Semi-regular: great icosidodecahedron

The rotation of the [5/2,3]-twister can be continued to form the following:

Rhombic: great
rhombicosidodecahedron

Snub: great
vertisnub icosidodecahedron

Semi-regular: great icosidodecahedron

As well as the figures given above,
an infinite family of dihedral twisters exists where the vertices are not uniform.
This family connects the antiprisms, snub
anti-prisms and gyrobicupolas. The latter of these sets a limit for
this family of ^{n}/_{d} < 6. The examples below show members of this
family for values of ^{n}/_{d} at 5, ^{5}/_{2}
and ^{7}/_{3}. For values of ^{n}/_{d} with d
even, the gyrobicupola is degenerate. Clicking on the images of the snubs
below leads to VRMLs of the twisters.

Rhombic: pentagonal
gyrobicupola Snub: pentagonal snub antiprism Semi-regular: pentagonal antiprism |
Rhombic: pentagrammic
gyrobicupola (degenerate) Snub: pentagrammic snub antiprism Semi-regular: pentagrammic antiprism |
Rhombic: heptagrammic
gyrobicupola Snub: heptagrammic snub antiprism Semi-regular: heptagrammic antiprism |

These twisters also exist for ^{3}/_{2}
< ^{n}/_{d} < 2. The crossed anti-prism setting the
lower bound. Cases have been generated for ^{n}/_{d
}= 5/3 and 7/4.