### Powertopes

Posted:

**Wed Feb 04, 2009 11:35 pm**If we take the square of a polygon, we get a duoprism of that polygon - and this is the beginning of powertopes. Powertopes are the result of taking a polytope and taking it to some "power" where the power is usually a shape having "block" symmetries (i.e. rectangle, square, cuboid, cube, tesseract, etc). The square of a shape is the duoprism of that shape, the diamond (square standing on a corner) of a shape is the duotegum of the shape (duotegums are the duals of duoprisms).

So what happens when we take the octagon or the octagram of a shape - lets start with the octagon of the octagon (or ocavoc for short). When we look at the octagon, lets consider it to have square symmetry - the octagon has the four edges of a square, blown out a bit - and four diagonal edges (which connects a horizontal edge to a vertical edge by nearest points). Notice that if the edge length is 1, then the height of the octagon is sq2+1 (I call this length "vo"). The octagon contains the short edges of a 1 by vo rectangle and a vo by 1 rectangle along with the diagonals that connect nearest points. The ocavoc is similar, it contains the small sides of an octagon(size 1)-octagon(size vo) duoprism and an octagon(size vo)-octagon(size 1) duoprism, along with "diagonals" that connect nearest rectangles of one duoprism to the other. The diagonal looks like a 1 by vo rectangle atop a vo by 1 rectangle - but enough talk, lets look at some pics:

Here is the unfolded ocavoc:

Here is its projection:

Here is the dual "Duocavoc"

Ocavog is the octagon of the octagram - here is the projection and a cross section:

Ogavoc is the octagram of an octagon - here's the projection and a section.

Ogavog is the octagram of an octagram - here's the projection and a section.

More to come.

So what happens when we take the octagon or the octagram of a shape - lets start with the octagon of the octagon (or ocavoc for short). When we look at the octagon, lets consider it to have square symmetry - the octagon has the four edges of a square, blown out a bit - and four diagonal edges (which connects a horizontal edge to a vertical edge by nearest points). Notice that if the edge length is 1, then the height of the octagon is sq2+1 (I call this length "vo"). The octagon contains the short edges of a 1 by vo rectangle and a vo by 1 rectangle along with the diagonals that connect nearest points. The ocavoc is similar, it contains the small sides of an octagon(size 1)-octagon(size vo) duoprism and an octagon(size vo)-octagon(size 1) duoprism, along with "diagonals" that connect nearest rectangles of one duoprism to the other. The diagonal looks like a 1 by vo rectangle atop a vo by 1 rectangle - but enough talk, lets look at some pics:

Here is the unfolded ocavoc:

Here is its projection:

Here is the dual "Duocavoc"

Ocavog is the octagon of the octagram - here is the projection and a cross section:

Ogavoc is the octagram of an octagon - here's the projection and a section.

Ogavog is the octagram of an octagram - here's the projection and a section.

More to come.