Miscellaneous Polyhedra
Here is a set of miscellaneous polyhedra which don't fit into the previous
categories. Only the Johnson solids, their duals, and the Stewart toroids can
be made using
Small Stella.
All models except for the four-dimensional ones can be made using
Great Stella.
And of course all models can be made using
Stella4D.
Click on the images below to see a bigger picture and get more information
about how they were built.
Polychora
Polychora are four-dimensional polytopes. They can't be truly realised in our
three-dimensional world of course, but there are a few different ways to
visualise them in 3D. Below are a few models I've made giving 3D
representations of 4D polytopes.
Johnson solids and their duals
The non-uniform convex regular-faced polyhedra. There are 92 in total. Here
are just two, and their duals.
Zonohedra
Great Stella can create a zonohedron based on
any polyhedron (look up zonohedron in
the glossary).
Stewart toroids
B. M. Stewart's Adventures Among the Toroids was an investigation of
regular-faced non-self-intersecting polyhedra of genus greater than zero (ie
with holes). Excavation was the term used for subtracting one smaller
polyhedron from a larger one, to create a hole. The term Drilling was
used when an excavation cut right through the model. See my paper
Stella: Polyhedron Navigator for more
details.
Various Stewart toroids are built into
Small Stella and
Great Stella. The latter also supports
augmentation/excavation/drilling so that you can create your own new toroids.
Sub-symmetric stellations
Great Stella lets you see the current
symmetry group of any model, and a list of all possible sub-symmetry groups.
You may select to use a sub-symmetry group instead of full symmetry, to produce
sub-symmetric stellations (also applies to faceting and augmentation).
Faceted polyhedra
Great Stella can create faceted polyhedra.
A faceted polyhedron has the same vertices as the model being faceted, but they
are connected together with different faces. In a special mode for creating
facets, you click on each vertex of a new face in turn. When all new faces
have been made Great Stella can repeat each
one over the symmetry group to generate all the faces.
Compounds
Great Stella can create various compounds of
polyhedra. Depending on the compound, these can be made via stellation,
faceting, augmentation, or a combination of methods. A great many compounds
are included in the polyhedron library included with
Great Stella.
Augmented polyhedra
Great Stella can augment any polyhedron with
any other polyhedron, which just means to connect them at some common face.
The augmentation may be done symmetrically, in which case the second model is
augmented to all faces of a type rather than just one. See my section about
Augmented Uniform Polyhedra for more information.
Cupolae
Great Stella can create any regular-faced
pyramid, cupola, cuploid or cupolaic-blend. These are all models where the
vertices lie in two parallel planes. (See the definitions of
cupola,
cuploid and
cupolaic-blend in the glossary).
Topological models
These models are listed in other categories, but are worth another mention
here. They are stylized versions of polyhedra, the idea being that only true
edges of the polyhedron should connect. The false edges that normally
appear in a model with intersecting faces have been removed by hiding parts of
each face, allowing the faces to pass through each other without collision.
This lets you better see the internal structure of the polyhedron. I call
these topological models.
Copyright © 2001-2008,
Robert Webb.