Incidence matrices for non-generalized graphs

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Incidence matrices for non-generalized graphs

Postby ubersketch » Tue Apr 30, 2019 12:26 am

By non-generalized graphs, I mean objects that are not made of collections of elements (like those in a polytope.)
Does anybody have any examples of some of these?
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Re: Incidence matrices for non-generalized graphs

Postby Klitzing » Tue Apr 30, 2019 8:31 am

not clear what you are after …
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Re: Incidence matrices for non-generalized graphs

Postby wendy » Tue Apr 30, 2019 11:49 am

The list I use is:

1. Solids. These have a discrete surface. The opposite here would be a cloud, where the gradient of volume is non-zero over an extended area. A mathematical cloud would be something like the gaussian curve \( De^{-r^2}\). A solid is taken to have a solid surface, and this iterates downward.

Examples of 'solid' include the cone and sphere. The cone has two faces, which share a common edge, but the vertex is incident on one face only, and not on the edge. The sphere has a face, but nothing less than that.

Convex polytopes are a subclass of solid, bounded by a number of planes. The intersection of m such planes of an n-solid leads to an (n-m)-solid. From this derives all sorts of derivitive figures.

A multitope is an accumulation of polytopes, which are joined such that the interior of an element shared in part, is shared in full. Thus to join two squares, either they can be connected to share only a common vertex, or a full edge (with all parts that are parts of the edge).

A polytope is then a multitope with closure. That is, the net of a dodecahedron is a multitope, but when it is closed up, it becomes a polytope. The meaning of 'closure' varies by application.

A peritope is a figure that encloses the space occupied by a polytope at single density. For example, in the pentagram, the five long chords are counted as edges, and the core pentagon is counted as twice-covered (since you need to cross two edges to get outside). Its peritope is the zigzag decagram such as you might cut out of cardboard. Peritopes is what modellers build.

An 'aperitope' is a figure with no perimeter, that is, it covers all of space. This is the general tiling, so a tiling of hexagons is a apeirohedron.

A polycell is a closure of solid cells, such as to fall in the space where the cells are solid. An example is the uniform hexagonal pyramid, where the height is zero, and the figure lies in the plane, with faces point up and faces pointing down.
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