Laminate Polytope

Higher-dimensional geometry (previously "Polyshapes").

Laminate Polytope

Postby wendy » Sun Jan 05, 2014 8:42 am

This is a picture that Roice Nelson did for Tom Ruen for the Wikipedia, and i ended up with a copy.

Anyway, ye take a x4x3o8o, and remove all the truncated cubes. You get a vast space completely bounded by flat "x3o8o"! It's a hyperbolic version of what you get with a layer of prisms between two x3o6o (ie from the x2x3o6o.), only it's hyperbolic.

All those funny circle things you see are COMPLETE PLANES. Really. And the space "between" them is completely convex. I wonder if the lads want to add this to their CRF project!
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Re: Laminate Polytope

Postby quickfur » Mon Jan 06, 2014 3:39 pm

wendy wrote:This is a picture that Roice Nelson did for Tom Ruen for the Wikipedia, and i ended up with a copy.

Anyway, ye take a x4x3o8o, and remove all the truncated cubes. You get a vast space completely bounded by flat "x3o8o"! It's a hyperbolic version of what you get with a layer of prisms between two x3o6o (ie from the x2x3o6o.), only it's hyperbolic.

All those funny circle things you see are COMPLETE PLANES. Really. And the space "between" them is completely convex. I wonder if the lads want to add this to their CRF project!

So if I understand this correctly, each of those circular regions contain a complete x3o8o tiling? So this is basically a mosaic of x3o8o tilings that connects the "infinite" edges of the tiling to make ... some kind of "hypertiling"? I'm not sure if I got this right. It seems to somehow make sense, yet not quite.
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Re: Laminate Polytope

Postby wendy » Tue Jan 07, 2014 9:03 am

It's a laminatope: it's bounded by unbounded planes.

Look at the pink bit of the attached picture. That's a cross-section along any of the circles you see.

In essence, you are sitting in the green bit to the left, looking right. The green things you see on the right are the x3o8o. The line between the green bit and the purple bits, are supposed to be straight, but Tyler plots poincare points, and makes straight lines between them. But you can only see the boundary between them, these are the etchings on the x3o8o.

If you follow a circle on the first picture, you will see that some are projected 'vertex-first' (ie they have a vertex in the middle). These points centrally invert to one of the crossings outside the picture, this is represented by the line from the big green bit to the little green bits on the right.

If you follow the line along, you will see some projections are 'edge first'. They have an edge in the middle. On the second figure, these are the ones on the opposite side of the octagons to the big green area. You see they're edge first too.

The next set you see are pairs of circles between the vertex- and edge- first x3o8o. You see that there are octagons not directly attached to the green bit on the left, and they have three little pink legs. Between these little pink legs comes a little green dot, not resolved, this leads to a pair of x3o8o, each edge-first, but too far for one to see that detail.

Each of those little legs are actually more octagons, ad infinitum.

It's not a mosaic. It's a convex thing, like a 2d version of the pink thing in the second diagram. And you wonder why i invent new words for all this stuff. Layer is a bit simplistic, although it is exactly what you get in x2x3o6o (two layers of triangular tilings). Just a larger kind of 2.
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Re: Laminate Polytope

Postby Klitzing » Tue Jan 07, 2014 3:48 pm

If I get her correctly, Wendy here describes 2 hyperbolic honeycombs.

One of them is the mere Wythoffian x4x3o8o.
Its incidence matrix can be given as:
Code: Select all
x4x3o8o   (N,M → ∞)

. . . . | 6NM |   1    8 |   8    8 |   8  1
--------+-----+----------+----------+-------
x . . . |   2 | 3NM    * |   8    0 |   8  0
. x . . |   2 |   * 24NM |   1    2 |   2  1
--------+-----+----------+----------+-------
x4x . . |   8 |   4    4 | 6NM    * |   2  0
. x3o . |   3 |   0    3 |   * 16NM |   1  1
--------+-----+----------+----------+-------
x4x3o . |  24 |  12   24 |   6    8 | 2NM  *  truncated cubes
. x3o8o |  3M |   0  12M |   0   8M |   * 2N  order 8 triangle tiling (used as bolohedra)


But it occurs in this structure, that the bolohedra have exactly the same surface- (i.e. tiling-) -curvature as the honeycomb itself. That is, those become flat in this geometry. Therefore these tilings could serve for mirror planes and thus themselves get blended out, reconnecting the truncated cubes of either side directly in the dyadic manner.

What then arises is the laminate honeycomb, which Wendy calls the lamina-truncate derived from x4x3o8o.
The incidence matrix of that one thus would be:
Code: Select all
lamina-truncate( x4x3o8o )   (N → ∞)

. . . . | 3N |  2   8 | 16  8 | 16
--------+----+--------+-------+---
x . . . |  2 | 3N   * |  8  0 |  8
. x . . |  2 |  * 12N |  2  2 |  4
--------+----+--------+-------+---
x4x . . |  8 |  4   4 | 6N  * |  2
. x3o . |  3 |  0   3 |  * 8N |  2
--------+----+--------+-------+---
x4x3o . | 24 | 12  24 |  6  8 | 2N  truncated cubes


--- rk
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Re: Laminate Polytope

Postby wendy » Tue Jan 07, 2014 11:31 pm

I thought the closest description is x3o8oW6.828o. That makes it regular, i suppose.

W6.828 is a polygon, whose shortchord square is 6.828. This gives 2.613126 .

It does not seem right to me that last comment. The pink area is a cell of x4oW6.828o, come to think of it.
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