Message #2260

From: schuma <mananself@gmail.com>
Subject: Re: Regular abstract polytopes based on {5,3,4} and {4,3,5}
Date: Sun, 10 Jun 2012 22:13:16 -0000

Hi all,

After learning a little bit of hyperbolic geometry, I made an applet to visualize the {5,3,4} honeycomb:

http://people.bu.edu/nanma/InsideH3/H3.html

This visualization is similar to Andrey’s MHT633. Imagine that you have a spaceship flying in a {5,3,4} honeycomb in a hyperbolic space. Then this is what you will see (suppose light travels along geodesics). Dragging is to rotate the spaceship. Shift+up/down dragging is to drive the spaceship forward and backward. Try navigating in this space!

Because it’s just for my own education, I haven’t implement the automatic extension of the tessellation when you move "outside" of the several initial cells. So we can drive the spaceship away from these cells and look back from outside.

In case anyone is curious, internally I’m using a hyperboloid model. Among the several models, I found this one intuitive for me.

Nan

— In 4D_Cubing@yahoogroups.com, Roice Nelson <roice3@…> wrote:
>
> Hi Nan,
>
> This is wonderful information, and puzzle versions definitely sound
> realizable. I do not have access to Jstor, but would love to see a copy of
> the paper.
>
> As far as the coordinates being in Minkowski space, that means they are in
> the Hyperboloid Model <http://en.wikipedia.org/wiki/Hyperboloid_model>. I
> have written code to go between this model and some other models
> (Poincare/Klein), and I’m happy to share if you think it could help in your
> adventures on this topic. It is code for 2D geometry, but should be
> adaptable to the 3D case.
>
> Roice
>
>
> On Fri, Jun 8, 2012 at 12:48 AM, schuma <mananself@…> wrote:
>
> > Hello,
> >
> > The regular abstract polytopes based on hyperbolic tessellations {5,3,4}
> > and {4,3,5} have been mentioned by Andrey several times here. Recently I
> > read more about them and found Gruenbaum talked about a polytope formed by
> > 32 hemidodecahedra, which was related to {5,3,4}. It should be this one:
> >
> > http://www.abstract-polytopes.com/atlas/1920/240995/5.html
> >
> > According to this page, it has 32 cells, each of which is a
> > hemidodecahedron. It has 96 faces, 120 edges and 40 vertices. The vertex
> > figure is an octahedron (note: not hemi-octahedron). Compared with the
> > 11-cell and the 57-cell, this 32-cell received little attention.
> >
> > It has a dual, which is based on {4,3,5}:
> >
> > http://www.abstract-polytopes.com/atlas/1920/240995/2.html
> >
> > The 40 faces are cubes (not hemi-cubes). The vertex figure is a
> > hemi-icosahedron.
> >
> > The vertex coordinates of {4,3,5} and {5,3,4} have been computed
> > analytically and can be found in this paper [Garner, Coordinates for
> > Vertices of Regular Honeycombs in Hyperbolic Space,
> > www.jstor.org/stable/2415373, Table 1]. This is of course a good news for
> > implementation. If any one wants to see the paper but has no access to
> > Jstor please email me. The coordinates are in a Minkowskian space. I need
> > to learn more hyperbolic geometry to understand the model.
> >
> > According to Colbourn and Weiss [A CENSUS OF REGULAR 3-POLYSTROMA ARISING
> > FROM HONEYCOMBS,
> > http://www.sciencedirect.com/science/article/pii/0012365X84900323], there
> > are more abstract polytopes based on {5,3,4} and {4,3,5}. But they cannot
> > be found in [http://www.abstract-polytopes.com/atlas/] because this atlas
> > contains information of "small" polytopes with up to 2000 symmetries.
> > Fortunately, the 32-hemidodecahedral-cell and its dual, 40-cubic-cell have
> > 1920 symmetries, which is just below the boundary. Something like 120-cell,
> > and 57-cell etc are not there because they are too large. But these two
> > things can keep me excited for a while.
> >
> > Nan
> >
> >
> >
>