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