This glossary contains terms relating predominantly to polyhedra and stellation theory in three dimensions. I'm afraid you'll have to look elsewhere for anything relating to higher dimensions!
A cell may also refer to a polyhedron that forms part of the surface of a four-dimensional polytope. It is the 4D equivalent of a face in 3D.
For example, the cube and octahedron are duals, as are the dodecahedron and icosahedron. The tetrahedron however is its own dual. For simple models like these, the dual can be create by connecting the midpoints of faces around each vertex to each other, forming new faces.
More generally though, the exact operaton used to create the dual is called spherical reciprocation, which is done with respect to a sphere. The midsphere is usually used if one exists. Using different spheres will distort the resultant dual, although repeating the operation with the same sphere will always bring you back to the original model. Note that any faces in planes passing through the centre of the sphere will lead to infinite vertices in the dual, and exactly how to draw such a model becomes hard to define.
For a 4D polytope, the Schlegel diagram is a 3D structure, created along similar lines.
A polyhedron is made up of faces, and each face lies in some plane. If we consider the whole of each plane, rather than just the area bounded by each face, we get a bunch of planes which all intersect each other many times. You may think of these planes as carving up space. One plane divides space into two halves. Two planes divides this further into four parts. A third plane will generally divide space into eight parts, but so far all the parts are infinite, that is, no part is bounded yet. When we add a fourth plane, space is divided up again, and this time we might have a single bounded region of space. The tetrahedron has four faces, and here the four planes do indeed enclose a region of space: the tetrahedron itself. As we add more planes, many more bounded regions of space, called cells, are created.
A collection of cells which together follow the symmetry group of the model being stellated (and where no smaller collection does) is referred to as a cell type. Generally (and from here on) when I say cell I really mean cell type, since you don't often want to refer to just one single cell on its own.
So finally, a stellation is some combination of these cells put together to form a single polyhedron. Sometimes the stellation may have disconnected parts floating around separately in space, or it may have parts connected by vertex or edge only, or it could be one solid piece. Due to the huge number of possible combinations of cells, various people proposed various criteria for deciding whether a given combination should be considered valid or not. Other people came up with different criteria in order to help find "interesting" stellations, or at least ones that could be physically built! Here are some of the criteria used, ordered roughly from least restrictive (allowing the most valid stellations) to most restrictive (allowing the fewest stellations): Miller's rules, fully connected, fully supported, monoacral (single-peaked), primary, and main-line. See some examples of stellations here.
The term extends to 4D. Again, all vertices are the same and faces must be regular, but we only require that cells be uniform, not necessarily regular. If the cells themselves are not uniform, then the polytope is called scaliform.
For uniform polyhedra there is only one type of vertex figure, since all vertices are the same. The vertex figure can be used in this case to find the shape of the face of the dual polyhedron. This method is called the Dorman Luke construction. Note also that in this case all vertices in the vertex figure lie on a circle.
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Copyright © 2001-2013, Robert Webb.