Gyrated antiprisms

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

Gyrated antiprisms

Postby wintersolstice » Thu Jul 12, 2012 1:58 pm

OR Gyrated duprisms since there made by gyrating duprisms

these were inspired by the "antiprismatic rings" and play an important role in "gyromulticupolic screw guages" (see other thread for them)


btw here I've used CP to mean a prism or a duprism and cp to mean the operation of Cartesian product:D


If you take a CP all the elements will be made from a cp* now consider the to shapes A and B that are combined together,

pick one of these shapes (it doesn't matter which) lets say A

this shape will appear as an element in the CP. they appear as the cp of the "body" of A and the "vertices of B" now if you were to "dual" some (if you did it to all of them it would be "Dual A X B") of them then the B girdle (where the facet are arranged the same way as the facets in B) will now become "Gyrated CP" of lesser dimensions

If you did to

a prism you would get an antiprism (so it could be regarded as a type of antiprism hence the name "Gryated Antiprism" **)

3,n duoprism you would get an "n-gonal antiprismatic ring"

Now I haven't yet worked what facets fill in the void when one girdle is gyrated, (this is an idea I had only recently:D)

*stricly speaking some of the cell are not CP's there just the cp of the vertices in one of the shapes to the elements in the other:D

**now if the prism case only creates a "typical antiprim" (two bases one the dual of the other) then that's already named so it could be called a "Gyrated Duoprism"
wintersolstice
Trionian
 
Posts: 91
Joined: Sun Aug 16, 2009 11:59 am

Re: Gyrated antiprisms

Postby quickfur » Thu Jul 12, 2012 4:36 pm

Would this construction be related to alternating a 2m,2n-duoprism? AFAIK, the only uniform case is m=n=2, which is also regular, and gives you the 16-cell. All other cases I believe will have various tetrahedra of sorts filling the gaps, but none of them will be uniform.

I was actually wondering if there are some CRFs in there that was missed by the people searching for uniforms, but OTOH if we're not worried about uniformity or CRF then you can get lots of polychora with antiprism cells in two rings.

(Oddly enough, the grand antiprism cannot be constructed as an alternated duoprism; I think you need a cantellated duoprism or something, I'm not 100% sure.)
quickfur
Pentonian
 
Posts: 2173
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Gyrated antiprisms

Postby wintersolstice » Thu Jul 12, 2012 10:43 pm

quickfur wrote:Would this construction be related to alternating a 2m,2n-duoprism?


yes it would :D but it is slightly different:

Gyrated Antiprism

If you had a 6,n duoprism and you gyrated the hexagons, there would be n hexagonal antiprisms.

Duoantiprism

if you took 12,2n duoprism and alternatre it, the 2n dodecagon prisms will become 2n hexagonal prisms

Of course you could have used a 6,2n duprism in the first case, but this demostrates that alternation halfs the polygon in the duoprism where as the gyration doesn't

two other differences are in the following two examples

if you gyrate a 3,n duoprism (just one face) you create an antiprimatic ring with two antiprisms* and square pyramids and triangular prismd seperating them but I don't think they're joined end to end anymore??**

if you take a 4,n duprism and gryate it on two faces you create an "antiduoprism" (not to be confused with dupantiprism) which is an antiprism prism, with 2 prisms and antiprisms* and 2n triangualr prisms

*refers to the difference that there could be a mixture of prisms and antiprisms instead of just antiprisms

** referes to what seperates the girdles: not tetrahedra, and the fact that the other girdle probably disappears???

like I say I'm still need to invesigate :D


quickfur wrote:I was actually wondering if there are some CRFs in there that was missed by the people searching for uniforms


I did wonder that myself actually, when I first started thinking about duoprismatoids with antiprisms in the rings :D

quickfur wrote:(Oddly enough, the grand antiprism cannot be constructed as an alternated duoprism; I think you need a cantellated duoprism or something, I'm not 100% sure.)

yes sort of a runcination but with a difference: when the prisms are shrunk they're then stretched so there's still the same number in each girdle (10,10 duprism is used for the grand antiprism and there are still ten when it's expanded) you put rectanglular trapezaprisms between the girdles (it's equiviant to an octagol triate, I think I mentioned that elseare on the forum :D
wintersolstice
Trionian
 
Posts: 91
Joined: Sun Aug 16, 2009 11:59 am

Re: Gyrated antiprisms

Postby quickfur » Fri Jul 13, 2012 4:49 pm

wintersolstice wrote:[...]
quickfur wrote:(Oddly enough, the grand antiprism cannot be constructed as an alternated duoprism; I think you need a cantellated duoprism or something, I'm not 100% sure.)

yes sort of a runcination but with a difference: when the prisms are shrunk they're then stretched so there's still the same number in each girdle (10,10 duprism is used for the grand antiprism and there are still ten when it's expanded) you put rectanglular trapezaprisms between the girdles (it's equiviant to an octagol triate, I think I mentioned that elseare on the forum :D

In that case, would it work if you runcinate a 5,5-duoprism?

EDIT: oh wait, nevermind, the prisms wouldn't be decagonal prisms, so it wouldn't work. My bad.
quickfur
Pentonian
 
Posts: 2173
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Gyrated antiprisms

Postby wintersolstice » Tue Jul 17, 2012 7:31 pm

these shapes were actaully also based on an idea I had for more complex duoprismatoids (such as duoprisms and duo antiprisms)

I've even wondered about duocupola (but haven't figured how these might work, but I'm getting some ideas together)

these could (assuming they exist) be extended to multicupolic screw gauges
wintersolstice
Trionian
 
Posts: 91
Joined: Sun Aug 16, 2009 11:59 am

Re: Gyrated antiprisms

Postby quickfur » Wed Jul 18, 2012 12:11 am

wintersolstice wrote:these shapes were actaully also based on an idea I had for more complex duoprismatoids (such as duoprisms and duo antiprisms)
[...]

I've often wondered about generalizations of the grand antiprism, i.e., objects with two rings of antiprisms instead of prisms. Obviously only the grand antiprism will be uniform, but I wonder if there are CRFs variants of it.

Also, after my recent realization that the bi-icositetradiminished 600-cell exhibits a kind of swirlprism-like symmetry (8 rings of 6 cells each, swirling around each other according to the fibres in the Hopf fibration of the 3-sphere), I'm starting to wonder if we will find any CRFs with similar kinds of symmetries. These will have drastically different structures than our usual uniform polychora. (Well, actually most uniform polychora also exhibit this kind of symmetry, but it's not obvious because the more "usual" kinds of symmetry are more prominent.)

(Yeah I've been thinking a lot about CRFs recently... haven't gotten very far, though I do have a couple o' ideas that might be worth posting about. Seems that the CRF thread has been silent for a while though, except me posting renders and talking to myself. :P)
quickfur
Pentonian
 
Posts: 2173
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North


Return to Other Polytopes

Who is online

Users browsing this forum: Google [Bot] and 1 guest