User:EricABQ/Glossary

Translation table between Wendy/abstract polytopes/standard terminology
"We should set up a Violeta translation table.  Sections are taken perpendicular to a line through the polytope."

-- Wendy Krieger on the usage of the term "section" in abstract polytopes

Coordinate terms across dimensions
Many terms used in the polytope community are parts of patterns that cross dimensions.

These are the terms I use, but most of them are standard in the community, at least to the degree that the ideas are discussed at all. The terms that aren't common words originated from a variety of sources, including Jonathan Bowers' and George Olshevsky's multidimensional glossaries, and the hi.gher.space glossary.

Polytopes

 * General: polytope
 * 0D: point
 * 1D: polytelon
 * 2D: polygon
 * 3D: polyhedron
 * 4D: polychoron
 * 5D: polyteron
 * 6D: polypeton
 * 7D: polyexon/polyecton
 * 8D: polyzetton
 * 9D: polyyotton
 * 10D: polyxennon/polyronnon (?)
 * 11D: polydakon/polyquetton (?)
 * This system has been extended up to 10^45 dimensions.

Elements of polytopes

 * General: element
 * 0D: vertex
 * 1D: edge
 * 2D: face
 * 3D: cell
 * 4D: teron
 * 5D: peton
 * 6D: ecton/exon
 * 7D: zetton
 * 8D: yotton
 * 9D: xennon/ronnon (?)
 * 10D: dakon/quetton (?)
 * This system has been extended up to 10^45 - 1 dimensions.

Hyperspheres

 * General: hypersphere
 * 1D: circle
 * 2D: sphere
 * 3D: glome
 * 4D: pentorb/pentasphere
 * 5D: hexorb/hexasphere

Hyperspheres

 * General: hyperball
 * 1D: dyad/line segment
 * 2D: disk
 * 3D: ball
 * 4D: gongyl/gongol
 * 5D: pentaball

Hypersurfaces

 * General: hypersurface/surtope (but note that this is different from Wendy's use of "surtope" because it can apply to curved objects)
 * 1D: boundary (I haven't seen this in a multidimensional glossary but it sounds like the obvious term)
 * 2D: surface
 * 3D: surcell
 * 4D: surteron

Flexible n-D objects in (n+1)-D space
Note: the dimension refers to the dimension of the object.

These terms are from hi.gher.space. They correspond to Wendy's hedrix, chorix, etc.


 * 1D: string
 * 2D: sheet
 * 3D: swock

Cardinal directions in nD
These are defined in terms of planet rotation. The terms "marp" and "garp" were invented by Jonathan Bowers.


 * 2D: east and west
 * 3D: north and south
 * 4D: marp and garp

Content measures
The term "bulk" was invented by George Olshevsky.


 * General: Hypervolume/content
 * 1D: Length/distance (length is the measure of a 1D object, distance is between two points)
 * 2D: Area
 * 3D: Volume
 * 4D: Bulk

Divisions of content

 * 1D: Hole
 * 2D: Chasm
 * 3D: Gully

Affine (flat) spaces

 * General: Flat/affine space
 * (n - 1)-D: Hyperplane
 * 1D: Line
 * 2D: Plane
 * 3D: Realm
 * 4D: Flune

Terms invented by me
These are polytope terms that I invented, usually several years ago.


 * Linear Wythoffian operation: A flag-Wythoffian operation: the result of applying the expansions caused by shading/unshading nodes on the diagram of a regular polytope, to any polytope.
 * Hypersnub: A hypothetical operation that maps pentachoron to hexacosichoron, just as "snub" (or what is now called omnisnub) maps tetrahedron to icosahedron. This can probably be defined as an operation that decorates each flag with a certain multi-vertex design.
 * Skewary: An alternative to "skewed" in the Bowers names of uniform polytopes containing "skew" not followed by "-verted", by analogy with "sphenary".
 * Merge: A family of linear Wythoffian operations across dimensions defined by ringing the middle node (in odd dimensions) or the middle two nodes (in even dimensions) of the Coxeter-Dynkin diagram. So called because it "merges" the base and the dual by containing facets on the hyperplanes of both.
 * Merging operation: Any polytope operation (especially a linear Wythoffian one) producing the same result when applied to the base and the dual.
 * Dual (of an operation): If A is an operation applied to n-D polytopes, the dual of A is the operation dual∘A∘dual.
 * Kaleidoscopic: Like Wythoffian, but where the shape of the fundamental region can be any polytope, not just a simplex.
 * Flag-decoration operation: An operation on a polytope that generalizes flag truncates, their duals, and compositions of such. Intuitively, it works by projecting the polytope onto a sphere and drawing the same "design" in each flag, which may consist of multiple vertices, vertices on the edges or vertices of the spherical simplex, and edges crossing mirrors. I'm not sure how to define it formally.
 * Retrogradization: An operation on a Wythoffian polytope that replaces all n/d around a ringed node of the CD with n/(n - d). This has the effect of contracting an edge until it goes backwards with the same length as the original. Retrogradization can also be used to describe the corresponding transformation of the unringed CD, which is guaranteed to produce a CD with the same symmetry.