by wendy » Sun Jan 04, 2009 8:02 am
Some terminology exists for frames, etc, but most of these are specific to particular problems.
One should note that the standard terminology is so far from common usage (by way of meaning shift), that it by itself inhibits understanding of the abstractions. Take for example "ridge". In polytopes, the faces are bounded by structures that look like ridges. But this is the nature of the particular classes of polytope studied, not the general nature of face-boundaries: the term remains valid because the area of study has never left the intersection of the meanings of mountions and of margin. However, in tilings, the connection of faces is no longer a mountion-range, but now is flat with the thing. In something like a bathroom tiling, the "ridge" is actually a valley :S.
It is also the fate of general terms to become the specific for unnamed. So while we have polytope = line, polygon, polyhedron, ...., and people are only looking at 4d, it is natural to slip polytope there. Coxeter uses =cell to refer to the facet-count of any polytope, not just the 4d ones. But with just 4d in mind, we have cell migrating from its usual meaning (think cellular automata, cells in games etc), via face of N-dimensions, to the unnamed 4d thing to the name of a 3d surtope. Norman Johnson even suggested "cellule" to bear its original meaning! (One sees the same with "face" / "facet"). Hyperspace for 4d, becomes the same sort of dimension that things turn around on in six dimensions!
Once you look at the present terminology from six dimensions, it does look totally stupid, haphazard etc. What I have done in the Polygloss is to restore the meaning of stems to something that still means what they do outside polytopes.
The Polygloss 'fabric' and 'patch' idiom bases on the "fabric of space", ie a manifold. Since fabric can be of any dimension (button, thread, cloth, clay), the fabric is formed as "hedrix" for 2d manifold. A patch is part of fabric, eg "hedron". You do things with patches of space, to make a "polyhedron", or an "apeirohedron". Thus, while "polyhedron" is "many 2d patches", /poly/ also implies a closure or completeness (cf multi), so it is taken with a content. One can hardly use 'surface' for hedrix in 4d, since a surface divides, and a hedrix (2d manifold), does not. This is why, for example, you can render a klein bottle or whatever, without crossing. You can knot 2d surfaces in 4d and 5d, but only weave them in 4d.
A frame is a unclosed polytope (multitope) with all of the surtopes up to the named dimension. So a "latroframe", is a frame with vertices + edges marked. A "hedroframe" has also the 2d elements marked. &c. The process of adding higher surtopes to close the polytope is 'inno-analysis".
George Olshevsky uses the terms based on an army, which is similar to the frame-concept, except that it requires a leader to make the frame. For example, an Icosahedron can be a "general" of an "army", consisting of the various polyhedra that are inscribed in the 12 vertices of the icosahedron (teeloframe), or the "colonel" of a regiment that have the 12 vertices + 30 edges of the icosahedron (ie latroframe). This is the divide and conquerer method used by Jonathon Bowers to derive the uniform starry polychora. However, ye see that it supposes the existance of rank-holders, whereas latroframe etc do not.
The notion of 'spheration', first appeared here, in order to describe what happens when the vertices and edges are replaced by spheres and pipes. One might want to spherate the vertices to make them larger than the points on the line, for example, is still covered under spheration, since this is done separately to classes of surtope (eg vertices, edges, ...).
An other method is the form used by da Vinci, where the latroframe is shown as a wood frame, ready for the application of faces. It is as if the faces were cut out of a solid box. At the moment, a name has not been allocated here.
In regard to spheres, etc.
The notation I use for spheres is to treat them as a polytope in O, eg circle, {O), sphere (O,O), glome (O,O,O), and so forth. Placing semicolons in the midst makes way for the assorted ellipsoids (by way of increasing diagonal), so (;O,O;0} corresponds to rss(x,x,y,y), where x<y. One notes that this is not the duocylinder (which is (;O&;O)).
The distinction between surface (fabric) and content (disk), is made by noting in the former, the surface is the thing (glomohedrix), while in the latter, the surface is a patch that bounds something (eg glomohedron).
Extracting a subset of surtopes is a separate function, or can be done by the products of surfaces and solids.
eg bi-glomolatral prism = disk(2) * disk(2); glomolatral glomolatric prism disk(2) * ring(2) ; bi-glomolatric prism = ring(2) * ring(2).
GLX *# GLX (circle=surface * surface sphere Prism).
I have, for example, described the space of great arrows as a bi-glomohedric prism = ring(3)*ring(3).
Note that as yet i have no distinction between glomohedrix (E2) as a space, vs glomohedrix (sphere-surface), as say, a ring in 4d.