software3d.com Forum Index software3d.com
Software3D Forums
 
 FAQFAQ   SearchSearch   MemberlistMemberlist   UsergroupsUsergroups   RegisterRegister 
 ProfileProfile   Log in to check your private messagesLog in to check your private messages   Log inLog in 

Powertopes

 
Post new topic   Reply to topic    software3d.com Forum Index -> Stella Forum
View previous topic :: View next topic  
Author Message
Jabe



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

PostPosted: Wed Feb 04, 2009 11:35 pm    Post subject: Powertopes Reply with quote

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.
Back to top
View user's profile Send private message Visit poster's website
Display posts from previous:   
Post new topic   Reply to topic    software3d.com Forum Index -> Stella Forum All times are GMT
Page 1 of 1

 
Jump to:  
You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot vote in polls in this forum


Powered by phpBB © 2001, 2005 phpBB Group
Protected by Anti-Spam ACP