## [Updated] Envelopes of polychora

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

### [Updated] Envelopes of polychora

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
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Mercurial, the Spectre
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### Re: [Updated] Envelopes of polychora

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.
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wendy
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### Re: [Updated] Envelopes of polychora

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...
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Mercurial, the Spectre
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### Re: [Updated] Envelopes of polychora

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.
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the dream we dream together is reality.

wendy
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### Re: [Updated] Envelopes of polychora

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.
Hadean society is is a social democracy in which people, irrespective of their background, are entitled equal rights in which they all cooperate according to each others' needs.

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Mercurial, the Spectre
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### Re: [Updated] Envelopes of polychora

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
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### Re: [Updated] Envelopes of polychora

The ugly old dragon of the meaning of "face" raises its head again...

Nowadays I try to avoid using the unqualified word "face", because it is just too ambiguous. In dimensions higher than 3, there is an ambiguity between a 2D element of the lattice vs. an (n-1)-D element of the lattice. Etymologically speaking, Wendy's usage is "more accurate" (in some sense), because an (n-1)-D element is a subset of an (n-1)-dimensional subspace of n-space, which can be described as an oriented surface with an unambiguous normal, so the element can be said to be "facing" the direction of the normal (assuming the normal points away from the center of the polytope). However, due to the historical bias of geometry towards 3D, the word "face" has come to acquire the more specific meaning of 2D element, because it just so happens that in 3D, a 2D element is the same as an (n-1)-dimensional element, and therefore also has the property of "facing a direction". In the context of higher dimensional geometry, however, such a terminology seems ill-fitting, because a 2D element in 4D, for example, does not "face" anything at all, being nothing more than the joint between two 3D elements, the latter of which are what has the "facing" property. So in 4D, a 2D "face" plays a role closer to an edge than a "face"; and if one goes to 5D or beyond, it becomes more and more distant from its original role in 3D of "facing" something.

And in the grand scheme of things, given an arbitrarily high dimension n, there seems to be no good reason to single out the role of a 2D element above any of the other elements, since the most prominent features of an n-dimensional polytope involve its vertices (0D elements) and (n-1)-D elements, and to a lesser extent, its edges (1D elements) and ridges ((n-2)-D elements). So it would seem wiser to reserve a general purpose word like "face" for one of these more significant roles than the essentially-arbitrary choice of 2D element.

However, due to the precedent set by the common usage of "face" referring to 2D element, changing its meaning now to something else only engenders confusion and miscommunication. This is why I tend to avoid using this word in its unqualified form nowadays.

Thus far, I have been using "facet" to refer to (n-1)-dimensional element, but from what Klitzing said, this causes another lexical conflict with the process of "faceting", which has a different meaning altogether. So now I don't even know if there is any word that can be used unambiguously to refer to an (n-1)-dimensional element anymore. (Perhaps Wendy can coin a new word here? )

For elements of other dimensions, I generally write i-face where i is some integer from 0 to n. Though this use of the word "face", in spite of being unambiguous, nevertheless still carries the wrong connotation of "facing" something where such may not be applicable at all. The only alternative I know of is Wendy's terminology, surtope (i.e., surface polytope), either with the radix-derived form (surhedrix for 2D element, surchoron for 3D element, etc.), or perhaps a numerically-prefixed form like 2-surtope, 3-surtope, etc..
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