[Updated] Envelopes of polychora

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

[Updated] Envelopes of polychora

Postby Mercurial, the Spectre » Sat Jul 22, 2017 6:08 pm

This was uplifted from a 2008 post about the same topic.
So I have considered the envelopes (the outermost convex covering of a polytope assuming orthogonal projection) of the six regular polychora. Here is the list:

5-cell
Cell/Vertex: Tetrahedron
Edge/Face: Equilateral-triangular bipyramid (the convex hull of a unit triangle and a perpendicular edge of length 1)

Tesseract
Cell: Cube
Face: Square prism (height is sqrt(2) times the square edge length)
Edge: Nonuniform regular-hexagonal prism (height is sqrt(3/2) times the hexagon edge length)
Vertex: Rhombic dodecahedron

16-cell
Cell: Cube
Face: Regular-hexagonal bipyramid (the convex hull of a unit hexagon and a perpendicular edge of length sqrt(6))
Edge: Square bipyramid (the convex hull of a unit square and a perpendicular edge of length 1)
Vertex: Octahedron

24-cell
Cell: Cuboctahedron
Face: Regular-hexagonal biantiprism (two hexagonal antiprisms joined at each other by a large hexagonal face)
Edge: Hexakis regular-hexagonal prism
Vertex: Rhombic dodecahedron

120-cell
Cell: Chamfered dodecahedron (or order-5 truncated rhombic triacontahedron)
Face: something with D10h symmetry (two decagons, which represent dodecahedra, straddle on the axis of symmetry; the rest are pentagons)
Edge: something with D6h symmetry (it has two dodecagons (again representing dodecahedra) with all other sides being pentagons)
Vertex: something with Oh symmetry (it is composed of 12 decagons in the planes of the cuboctahedron and 72 pentagons)

600-cell
Cell: Hexakis chamfered cube (or hexakis order-4 truncated rhombic dodecahedron)
Face: something with D6h symmetry (this object has 12 trapezoids, the rest are triangles)
Edge: Decakis order-10 truncated order-10 decakis regular-decagonal biantiprism (a biantiprism is composed of two antiprisms with different-sized faces joined at a larger face)
Vertex: Pentakis icosidodecahedron

I will do some research on the envelopes of the uniform polychora. Stay tuned!
Mercurial
User avatar
Mercurial, the Spectre
Dionian
 
Posts: 33
Joined: Mon Jun 19, 2017 9:50 am

Re: [Updated] Envelopes of polychora

Postby wendy » Tue Sep 05, 2017 12:16 pm

The vertex envelope of the 600ch is what Conway writes as k5aD This is a icosadodecahedron, with small pyramids raised over the pentagons.

The vertex-envelope of a face-first 120ch is the t5jD, a tricontahedron, with the vertices with five faces truncated to pentagons.
The dream you dream alone is only a dream
the dream we dream together is reality.
User avatar
wendy
Pentonian
 
Posts: 1783
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Re: [Updated] Envelopes of polychora

Postby Mercurial, the Spectre » Tue Sep 05, 2017 12:38 pm

wendy wrote:The vertex envelope of the 600ch is what Conway writes as k5aD This is a icosadodecahedron, with small pyramids raised over the pentagons.

The vertex-envelope of a face-first 120ch is the t5jD, a tricontahedron, with the vertices with five faces truncated to pentagons.

I think you meant the cell-first projection of the 120-cell...
User avatar
Mercurial, the Spectre
Dionian
 
Posts: 33
Joined: Mon Jun 19, 2017 9:50 am

Re: [Updated] Envelopes of polychora

Postby wendy » Tue Sep 05, 2017 12:49 pm

Actually it's face or chora-first. You are right.

Cell refers to a bubble in a foam. In RL it is used for both tiles of the plane (cell phone), and tiles of space (cells of the body). This is why i use cell/wall to refer to elements of tiling, rather than a specific dimension. Chora refers to 3d polytopes as part of larger structures.
The dream you dream alone is only a dream
the dream we dream together is reality.
User avatar
wendy
Pentonian
 
Posts: 1783
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Re: [Updated] Envelopes of polychora

Postby Mercurial, the Spectre » Tue Sep 05, 2017 1:58 pm

wendy wrote:Actually it's face or chora-first. You are right.

Cell refers to a bubble in a foam. In RL it is used for both tiles of the plane (cell phone), and tiles of space (cells of the body). This is why i use cell/wall to refer to elements of tiling, rather than a specific dimension. Chora refers to 3d polytopes as part of larger structures.

Don't be confused. In geometry, face refers to a 2D subcomponent of a higher-dimensional polytope or shape, while cell refers to a 3D subcomponent. Terons represent the 4D subcomponents. Or one could simply call faces as 2-faces or cells as 3-faces.
User avatar
Mercurial, the Spectre
Dionian
 
Posts: 33
Joined: Mon Jun 19, 2017 9:50 am

Re: [Updated] Envelopes of polychora

Postby Klitzing » Sat Sep 09, 2017 12:48 pm

Mercurial, the Spectre wrote:
wendy wrote:Actually it's face or chora-first. You are right.

Cell refers to a bubble in a foam. In RL it is used for both tiles of the plane (cell phone), and tiles of space (cells of the body). This is why i use cell/wall to refer to elements of tiling, rather than a specific dimension. Chora refers to 3d polytopes as part of larger structures.

Don't be confused. In geometry, face refers to a 2D subcomponent of a higher-dimensional polytope or shape, while cell refers to a 3D subcomponent. Terons represent the 4D subcomponents. Or one could simply call faces as 2-faces or cells as 3-faces.

This one too is a mere matter of terminology.

Wendy here was knowingly using and re-originating "face" in the original sense of "one equation", as we usually apply that term in 3D. Even in non-mathematical live, a face is describing the directly visible part.

Other mathematicians similarily would like to keep "face" here, but then also use the very term as the 2D-within-nD intension. Thus instead they try to use "facet" in a nD-within-(n+1)D usage. But that derived term then conflicts in turn to the usage of "facet" as in facetings, which refers to internal vertex spanned parts of according hyperplanes.

Thus, in fact, we here likewise are in the realm, where extrapolation of commonly known terms, refering to some specific geometric idea does lift up through dimensions much too easily. But sadly this does apply for more than just a single such geometric idea, which all happen to coincide within the original low dimension. Therefore the extended usage of such terms no longer can be seen unique. It might be viewed such and is used as being uniquely defined within one community, and thus every one of their members directly associates the right intended geometric meening. But accross communities such ambiguous terms generally should be looked at very, very carefully, as on what is its intended meaning.

Well, I know, this is kind of a nowadays re-occurrance of the Babylonian confusion. But again, there cannot be any right or wrong! This is hard to understand by newcommers, I know. Moreover any of such communities of term usage does have good theoretical reasons, and most often even historical references, for their specific association of ideas and terms. These can be - and even should be detailed in such conversations. But be assured, there is no repelling harm intended when others would not willingly follow one owns usage of terms! Instead any ones own mental flexibility is asked for, to provide according "translations" as well, to give members of other communities a chance to grasp the intended topic. Or, the other way round, if not already provided, anyone should feel free to ask! (Esp. as this is a Forum, and any Forum does live through discussions!)

Btw. such debates are not a new thing at all. Within all decades, centuries and even millenia of history of mathematics examples of this can be found! A very common such Babylonian confusion of mathematical terms even can be found outlined within Wikipedia: just look up https://en.wikipedia.org/wiki/Trapezium, which is understood quite different inside and outside of America!

--- rk
Klitzing
Pentonian
 
Posts: 1311
Joined: Sun Aug 19, 2012 11:16 am
Location: Heidenheim, Germany


Return to Other Polytopes

Who is online

Users browsing this forum: No registered users and 1 guest

cron