Powertopes made easy!

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Powertopes made easy!

Postby wintersolstice » Sat Feb 18, 2012 12:59 pm

I've been looking for an alternate way of constructing powertopes. Combining and multiplying(by the lengths of orthotope in the power) co-ordinates, only gives you the co-ordinates of a powertope it doesn't give you the actually element counts and positions.

Here's a description for Powerchora (polygonal polygoltriates) of the convex type (I'm guessing the powertopes from self-intersecting powers are facetings of these?)

for 4D there are 3 types of "power" to the consider:

1) multiple of 4 side standing on an edge

2) non-multiple of 4

3) multiple of 4 side standing on a vertex

when the base has n-sides

1) this has two girdles of n n-prisms sepearated by layers of cuboids-like shapes (they've been stretched and squashed)
when the power has 4A sides the number of layers is A-1

2) this is made from the previous even number (standing on an edge) with one girdle "blend augmented" (I'll explain later)

3) this is made from the previous multiple of 4 (standing on an edge) with both girdles "blend augmented"

blend augmenting:

when the base is an n-gon consider the n n-prisms

take a n-gon pyramid prism (a prism of the n-gon pyramid) and place it on it's n-gon prism cell (there will be an two vertices not contained in this cell, they form an edge)

augment each prism cell on the girdle with one of these (the n-gon pyramid prism)

now take the "edge" from each pyramid prism and stretch them so they meet and make a complete polygon (with is not a face)

I'm currently working on how to adapt this method in higher dimensions :D

EDIT btw this method is only for the element count.

Edit "the [power standing on end] of a base" is the dual of "the [power standing on an edge] of the dual of the base" and the
wintersolstice
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