[Cos-top] 120 cell

[Jeff Weeks]

Dear Boud et al.,

You intuition is correct -- there are some additional in-between layers.
The attached images show layers 1 through 5.  Layers 6 through 9 are of course
symmetrical as you work your way towards the south pole.
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).

Anyhow, here's the accounting:

    layer 1   1
    layer 2  12
    layer 3  20
    layer 4  12
    layer 5  30
    layer 6  12
    layer 7  20
    layer 8  12
    layer 9   1
    -----------
    total   120

Note that all numbers are "dodecahedal numbers",
respecting the dodecahedral symmetry of the whole construction.

The 120-cell is quite beautiful, isn't it?

Best wishes to all of you,
Jeff
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