SharkRetriver wrote:I also don't know what non-orientable means...chiral?
Edit: http://www.polytope.net/hedrondude/icoes.htm There's a whole group of hemis here! Now only if I knew what the acronyms meant. Are CRFs listed on that site?
wendy wrote:The class exists in all dimensions, and the class is very common where AB polytopes exist.
This means, for example, the AB polytopes /3.3.3/ and /3.3.4 , as does /3,3,3,3/ and /3,3,3,4 in 5D and so forth. Generally, there are not all that many figures that girth on polyhedra.
wendy wrote:Bower's site is devoted to finding the uniform star-polyhedra. In other words, these correspond to the 75 uniform starry polytopes in 3d. In four dimensions, there are something like 8200 of them. He brought in some pretty fancy terms to do it: armies and regiments and colonels and so forth. [...]
SharkRetriver wrote:wendy wrote:The class exists in all dimensions, and the class is very common where AB polytopes exist.
This means, for example, the AB polytopes /3.3.3/ and /3.3.4 , as does /3,3,3,3/ and /3,3,3,4 in 5D and so forth. Generally, there are not all that many figures that girth on polyhedra.
What does 3.3.3 stand for here? (I would read it as the vertex figure of a tetrahedron, although that doesn't really make sense here.)
Klitzing wrote:Hehe, this is one of Wendy's cryptix!
On the first run somthing like p.q.r.s etc. in her notation just means the symmetry group o-p-o-q-o-r-o-s-o. I.e. the dot (or comma) represents an unringed node (those at the ends are to be understood in addition). The slash is a representant of a ringed node. So /3.3.3/ just means the Dynkin symbol x3o3o3x. (Beware: sometimes she uses semicolons instead of slashes too.)
But this is only the begin. The next step would be to understand her content. What has a x3o3o3x and a x3o3o4o in common?
This is the true mystery here, which has to be uncovered. :-)
It happens to be an analysis of the rectified cross-polytopes o3x3o3o...o4o : those can be given as a bistratic lace tower o3x3o...o3o || x3o3o...o3x || o3o3o...x3o . In fact, within an other direction they likewise can be given as a similar tower x3o3o...o4o || o3x3o...o4o || x3o3o...o4o . Therefore there can be a hemipolytope be constructed therefrom by using those hemifacets x3o3o...o3x together with those x3o3o...o4o of the other direction. (The hull clearly would be the o3x3o3o...o4o we started with.)
Esp. in 3d we have o3x4o (co) which can be given as x3o || x3x || o3x or likewise as x4o || o4q || x4o. Accordingly there is a polyhedron using the diametral x3x for hemifacets and additionally the facets x4o of the former. That one is cho (cubihemioctahedron).
In 4d we get o3x3o4o (a different description of ico, the 24-cell). It can be displayed as tower as o3x3o || x3o3x || o3x3o or alternatively as x3o4o || o3x4o || x3o4o. (In fact those alternate descriptions happen to be identical stacks of oct || co || oct. This is due to the additional symmetry of the 24-cell.) Accordingly we have a hemipolytope using as the hemifacets those x3o3x and as outer facets x3o4o.
In 5d we use o3x3o3o4o as o3x3o3o || x3o3o3x | o3o3x3o resp. as x3o3o4o || o3x3o4o || x3o3o4o. Therefore there is a hemipolytope using those hemifacets x3o3o3x and those outer x3o3o4o ones.
Etc.
--- rk
SharkRetriver wrote:On a side note, what's a simpler definition for a powertope? triangle^octagon led me to a {3}x{8} duoprism.
SharkRetriver wrote:[...]
I have yet to learn even the simplest of coxeter-dynkin diagrams (e.g. ones with more than one x or one that's non-linear).
Also, I've never heard of lace towers, although I've heard of || being used for segmentochora. (Same thing?)
above
AB /AA\ AB /AA\
\----/----\----/----\
\BB/ BA \BB/ BA \
below
quickfur wrote:Aha! Found your polychoron in Bowers' list: http://www.polytope.net/hedrondude/regulars.htm (look near the bottom for "Tho"). Apparently it's non-orientable. Fun, fun. The cells are exactly as we predicted. Interestingly enough, the vertex figure of Tho is Thah (i.e., tetrahemihexahedron). So it's a direct 4D analogue.
Users browsing this forum: No registered users and 21 guests