[Cos-top] "vertical" faces

[Jeff Weeks]

Boud,

>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.

No, the hexagons on the surface of layer 4 really are flat (not concave) !
Indeed they must be flat, because the 30 dodecahedra in layer 5
are completely flat (zero volume!) in this projection.

The situation is analogous to placing a cube on a tabletop
and then projecting it orthogonally onto that tabletop.
The cube's four side faces project to line segments.
Those projections have zero area, even though the side faces
themselves have positive area.

The analogous thing happens with the orthogonal projection
of the dodecahedron from 4D to 3D.  The 30 dodecahedra near
the equator have positive volume, but because they are perpendicular
to the 3D space we're projecting onto, their shadows (projections)
have zero volume.

I hope that's clear!

Best wishes,
Jeff

P.S.  I'm not sure whether my message which bounced yesterday
has made it to the list yet or not, so for sake of completeness
I'll reattach those images to the present message for the sake
of the cc readers.  Please forgive the duplication.
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