SV: [Cos-top] MG11 - Topology of the Universe (fwd) (and PDS - how many tiles?)

[Boud Roukema]

Hi Jeff, Jesper, cos-top,

On Tue, 7 Feb 2006, Gundermann, Jesper wrote:

> You were right about the intermediate layers, in fact, there are 9
> layers
>
> Note that the central dodecahedron has 12 faces, 30 edges and 20
> vertices. The distance of the centers of
> The 120 dodecahedrons are
>
>
> Distance from origin:     direction   number
>
> 0                                        1
> pi/5                  midpoint faces    12
> Pi/3                      vertices      20
> 2*pi/5                midpoint faces    12
> Pi/2                  midpoint edges    30
> 3*pi/5                midpoint faces    12
> 2*pi/3                    vertices      20
> 4*pi/5                midpoint faces    12
> Pi                                       1
> Sum                                    120


Thanks for the explanation - it's the equatorial layer - "layer 5" - with
30 dodecahedrons which solves my intuitive problem of needing
(2 * (layer with an odd number of dodecahedrons)).


Jeff wrote:

> Note that the cells in layer 5 sit "vertically" with respect
> to the equatorial hyperplane (i.e. they're orthogonal to the equatorial
> hyperplane) which is why they appear flat in the attached image
> (each dark blue hexagon is the 2D shadow of a 3D cell when you project
> from 4D space to 3D space).

i guess another slightly confusing thing is in the picture of layer 4,
where the hexagons look "flat" whereas if i understand correctly, these
should be concave in order that layer 5 cells can be stuck on here.

It might be nice to have a picture of the equatorial S^2 surface,
showing where layer 4 cells touch layer 6 cells (in whole faces), and
the cross-sections through the layer 5 cells.


> Note that all numbers are "dodecahedal numbers",
> respecting the dodecahedral symmetry of the whole construction.
>
> The 120-cell is quite beautiful, isn't it?


Definitely :)

cheers
boud