Jabe
Joined: 12 Jan 2008 Posts: 46 Location: Somewhere between Texas and the Fourth Dimension

Posted: Wed Feb 04, 2009 11:35 pm Post subject: Powertopes 


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. _________________ May the Fourth (dimension) be with you. 
