Envelopes into 3D of 6 regular polychora ?

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

Envelopes into 3D of 6 regular polychora ?

Postby begvend » Sat Dec 27, 2008 9:30 pm

Hi all,

Envelopes into 3D of 5 cell are:
- Cell-first projection is a tetrahedron.
- Face-first projection is a trigonal bipyramid.
- Edge-first projection is a trigonal bipyramid.
- Vertex-first projection is a tetrahedron.

Envelopes into 3D of 8 cell are:
- Cell-first projection is a cube.
- Face-first projection is a cuboid.
- Edge-first projection is a hexagonal prism.
- Vertex-first projection is a rhombic dodecahedron.

Envelopes into 3D of 16 cell are:
- Cell-first projection is a cube.
- Face-first projection is a hexagonal dipyramid.
- Edge-first projection is a ?
- Vertex-first projection is an octahedron.

Envelopes into 3D of 24 cell are:
- Cell-first projection is a cuboctahedron.
- Face-first projection is a ?
- Edge-first projection is a ?
- Vertex-first projection is a rhombic dodecahedron.

Envelopes into 3D of 120 cell are:
- Cell-first projection is a ?
- Face-first projection is a ?
- Edge-first projection is a ?
- Vertex-first projection is a ?


Envelopes into 3D of 600 cell are:
- Cell-first projection is a ?
- Face-first projection is a ?
- Edge-first projection is a ?
- Vertex-first projection is a ?

Many thanks in advance for your help.
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Re: Envelopes into 3D of 6 regular polychora ?

Postby wendy » Sun Dec 28, 2008 8:03 am

Please note that 'cell' as used by Coxeter is simply a count of faces (facets) in any dimension. A 7D simplex becomes an 8-cell. The correct terminology is eg 120-choron.
- Face actually means N-1 dimensions, Cells have no meaning here (it's actually a solid element of a tiling!), however from your usage, i suppose you mean hedron (margin) and choron (face), for your "face" and "cell".

16-ch
Edge-first hight 1:q tegumic-prism (a rhomb prism)

24choron (one looks at the model here to get these figures)
- Edge first = it looks like a hexagonal tegum
- Hedron first = hexagon barrel. (roughly a cuboctahedron + one rotated 60 degrees).

500choron = again, from the model
- vertex-first = icosadodecahedron, with the pentagonal faces apiculated = Conway-Hart notation: k5aD
- edge first - some kind of decagonal-symmetry thing, prolly a decagonal ball.
- hedron-first - not easy to explain, but the underlying symmetry is f&.
- vertex-first - presumably the middle section in Coxeter's Regular polytopes, with some faces apiculated.

100choron = again from the model
- face-first = rhombo-tricontahedron, with the pentagonal peaks truncated. = t5jD
- the rest are rather hard to explain, but get yourself a model and check this out.
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Re: Envelopes into 3D of 6 regular polychora ?

Postby Keiji » Sun Dec 28, 2008 1:11 pm

wendy wrote:Please note that 'cell' as used by Coxeter is simply a count of faces (facets) in any dimension. A 7D simplex becomes an 8-cell. The correct terminology is eg 120-choron.


No, it should be either "120-cell" or "hecatonicosachoron". The facet-term (edge, face, cell) is used with decimal numbers and the tope-term (gon, hedron, choron) is used with Greek prefixes. After 4D though, these terms overlap (teron, peton, exon, etc.) so will work for both writings.
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Re: Envelopes into 3D of 6 regular polychora ?

Postby wendy » Mon Dec 29, 2008 7:42 am

One must understand that numbers in latin and greek are not generally understood. Writing 44-gon for a polygon of 44-sides is perfectly legitimate.

The polytopes in four dimensions are called polychora, and it is quite legitimate to write 48choron as it is to write 48gon.

Coxeter used 'cell' to represent the N-1 element of an Nd polytope, so that a cube might be called a 6-cell. There are indeed references where he uses N-cell to refer to a six-dimensional polytope. The reference to a face of a 4d polytope comes from referring to N=4, since most folk are not interested in N>4. In the main, many of the interesting polytopes have lots of faces, and the numbers tend to run away too fast.

In any case, this is not the parent meaning of the word 'cell', such as used in the real world. Bearing this in mind, one is better to refer to these using the same suffix as appears in 'poly-xxx', ie 120-choron. Yes, i know, i converted too, but for the better. Calling it a cell would be totally wrong as calling the edges of a hexagon 'cells'.
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Re: Envelopes into 3D of 6 regular polychora ?

Postby quickfur » Tue Dec 30, 2008 8:53 pm

begvend wrote:[...]Envelopes into 3D of 16 cell are:
- Cell-first projection is a cube.
- Face-first projection is a hexagonal dipyramid.
- Edge-first projection is a ?

A flattened square dipyramid (flattened octahedron):

Image

The nearest edge is the vertical one in the center of the image.

- Vertex-first projection is an octahedron.

Envelopes into 3D of 24 cell are:
- Cell-first projection is a cuboctahedron.
- Face-first projection is a ?

I don't know what to call this shape, it's something like two irregular hexagonal antiprisms with enlarged bases, joined to each other at the enlarged bases.

Image

The nearest face is highlighted in red for ease of identification.

- Edge-first projection is a ?

Hexakis hexagonal prism (hexagonal prism with two hexagonal pyramids on its hexagonal faces).

Image

Nearest edge highlighted in red for easy spotting. Note that the hexagonal prism is not uniform; its tetragonal faces are rectangular, not square, so this shape is not the Johnson solid.

- Vertex-first projection is a rhombic dodecahedron.

Envelopes into 3D of 120 cell are:
- Cell-first projection is a ?

Truncated rhombic triacontahedron (almost like a buckyball/truncated icosahedron, except with a different number and arrangement of hexagons, in this case non-uniform elongated hexagons).

Image

- Face-first projection is a ?

It's an interesting shape, but I've no idea what to call it. It has a fascinating arrangement of pentagonal and decagonal faces. For clarity, I show only the faces/edges/vertices that lie on the envelope plus the closest face.

Image

- Edge-first projection is a ?

No idea what to call this shape either. All I can say is, there are dodecahedra, dodecahedra everywhere! :lol:

Image

- Vertex-first projection is a ?

Something with tetrahedral symmetry, of course. Still no idea what to call it.

Image

Like I said, dodecahedra, dodecahedra everywhere! :P

Envelopes into 3D of 600 cell are:
- Cell-first projection is a ?

An interesting shape, no doubt. I don't think there's a name for it:

Image

There are lots of triangles and kite-shaped faces. For clarity, I'm rendering only the envelope here.

- Face-first projection is a ?

Who knows what this is called? I don't. At any rate, it's something with triangular symmetry.

Image

- Edge-first projection is a ?

An interesting shape with two dodecagonal pyramids.

Image

Let's see. It consists of two identical halves, each of which looks like some kind of modified pentakis icosidodecahedron with dodecagonal pyramids at the top. Good luck naming this thing. I suggest "envelope of the edge-first projection of the 600-cell into 3D". :D

- Vertex-first projection is a ?

A pentakis icosidodecahedron (an icosidodecahedron with pentagonal pyramids on its pentagonal faces).

Image

Many thanks in advance for your help.

You're welcome. Sorry for replying so late; I've been quite busy recently.

P.S. I doubt you'll gain very much insight into these shapes just by looking at these projection envelopes. Much more insightful is the internal structure of these projections (which I've omitted in many of these images 'cos they're kinda complex to convey in a single 2D image), as described in:

<shameless plug>http://eusebeia.dyndns.org/4d/regular.html</shameless plug>
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Re: Envelopes into 3D of 6 regular polychora ?

Postby Keiji » Wed Dec 31, 2008 12:37 am

I don't know what to call this shape, it's something like two irregular hexagonal antiprisms with enlarged bases, joined to each other at the enlarged bases.


I'd call it the dodecagonal biprismic alternation.

I've tried and failed to come up with names for the remaining shapes, though all the envelopes of the 120-cell look like the exact same shape at least topologically.
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Re: Envelopes into 3D of 6 regular polychora ?

Postby begvend » Wed Dec 31, 2008 5:22 pm

Hi quickfur,

You're welcome. Sorry for replying so late;

I reply in French.

Merci beaucoup, vous n’étiez obligé de rien.

.
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Re: Envelopes into 3D of 6 regular polychora ?

Postby quickfur » Wed Dec 31, 2008 6:58 pm

Hayate wrote:[...]I've tried and failed to come up with names for the remaining shapes, though all the envelopes of the 120-cell look like the exact same shape at least topologically.

Depends on whether you mean 4D topologically, or 3D topologically, since the 3D projection envelope may have faces (like decagons, dodecagons, etc.) that result from the projection of several pentagons in 4D onto the same 3D plane.

On an unrelated note, I've recently come to really like the 120-cell. Even more than the 24-cell, actually. The 24-cell is really quite "ordinary" once you get to know it; it's just the 4D equivalent of the rhombic dodecahedron, which just happens to be regular by a pure freak coincidence. (I have this theory that the regularity of the 24-cell arises from the fact that 2^2 = 4 = 2*2. Look at its coordinates, if you know what I mean.) In fact, while rendering projections of the 24-cell family of uniform polychora, I realized that essentially it has exactly the same symmetry as the tesseract and 16-cell combined. That it happens to be regular allows for a few more uniform polychora among what would otherwise be non-uniform derivatives of the 4-cube/16-cell; but really, there is no novel symmetry here. I mean, even all the principal bounding hyperplanes simply correspond with the tetracube's and the 16-cell's cells and vertices.

The 120-cell, OTOH, is just incredibly beautiful... my favorite Platonic solid is the dodecahedron, and here we have 120 of them. So pretty! Moreover, if you look at the layers of cells in the cell-first projection (as I've put up on my 120-cell page), you can see that they are laid out in patterns with the symmetry of the dodecahedron's faces, edges, and vertices. First, you have a single dodecahedral cell, corresponding with the dodecahedron itself. Then there's a layer of 12 cells, corresponding with the faces of the dodecahedron. After that, a layer of 20 cells corresponding with the vertices of the dodecahedron, and then another layer of 12 cells corresponding to the faces of the dodecahedron. Finally, at the equator, there's a layer of 30 cells corresponding to the edges of the dodecahedron. From there, the layers repeat in reverse, until you reach the antipodal cell. It's just so beautiful.
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Re: Envelopes into 3D of 6 regular polychora ?

Postby Keiji » Thu Jan 01, 2009 2:52 am

quickfur wrote:
Hayate wrote:[...]I've tried and failed to come up with names for the remaining shapes, though all the envelopes of the 120-cell look like the exact same shape at least topologically.

Depends on whether you mean 4D topologically, or 3D topologically, since the 3D projection envelope may have faces (like decagons, dodecagons, etc.) that result from the projection of several pentagons in 4D onto the same 3D plane.


I mean that the shape formed from the envelope of the projection of the object is the same in all four cases.
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Re: Envelopes into 3D of 6 regular polychora ?

Postby quickfur » Thu Jan 01, 2009 5:24 am

Hayate wrote:
quickfur wrote:
Hayate wrote:[...]I've tried and failed to come up with names for the remaining shapes, though all the envelopes of the 120-cell look like the exact same shape at least topologically.

Depends on whether you mean 4D topologically, or 3D topologically, since the 3D projection envelope may have faces (like decagons, dodecagons, etc.) that result from the projection of several pentagons in 4D onto the same 3D plane.


I mean that the shape formed from the envelope of the projection of the object is the same in all four cases.

I dunno, they look distinct to me, unless you treat coplanar faces as distinct. Note that I don't cull coplanar edges/vertices when projecting, so from that point of view, they are topologically the same.
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Re: Envelopes into 3D of 6 regular polychora ?

Postby Keiji » Thu Jan 01, 2009 4:55 pm

Well yes, I was ignoring the extra coplanar facets. After all, you did for the third projection in your post. :D
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