proposed definition for "Gyrated"

Discussion of known convex regular-faced polytopes, including the Johnson solids in 3D, and higher dimensions; and the discovery of new ones.

proposed definition for "Gyrated"

Postby wintersolstice » Thu Jun 28, 2012 2:04 pm

in 3D "gyrated" is easy to define: take off part of the shape (usually a cupola), rotate it and put it back.

however in 4d+ it's gets harder, take a look at these gyrations:

1)take a runcinated tesseract and remove a cube cupola off the shape and replace it with a octahedron cupola

2) take a cube cupola and take a "square orthobicupolic ring" of it and replace it with a "square gyrobicupolic ring"

3) take an octahedral prism and cut it in two (I haven't figured out what the two halves are yet) and rotate one and put it back, (this is listed on Klizting's paper as "reflected orthogonal trigon||trigon"

4) take a runcinated 5-cell and remove a tetrahedron cupola off the shape and rotate it and put it back

in 3d the term can be defined as either "replacing a facet (face in this case) with its dual" or "rotating a face"

1) demonstrates the former while 3) represents the latter and 4) represents both. 2) represents "dualing/rotation*" face (which is a lower element


so maybe the definition should be

Take an element, not necessary a facet, and rotate** or dual it

*for a polygon these two mean the same thing.

**assuming that in "2)" there will be a triangular prism that gets rotated! (I'll need to check that:D)
wintersolstice
Trionian
 
Posts: 91
Joined: Sun Aug 16, 2009 11:59 am

Re: proposed definition for "Gyrated"

Postby quickfur » Thu Jun 28, 2012 7:24 pm

Gyration, as we know it, is peculiar to 3D. In 2D, such an operation is impossible (you can cut off the top part of a polygon, but there's no way to "gyrate" it before putting it back on; the most you can do is to rotate it vertically and glue it back on another edge, which usually makes the result non-convex). 2D, however, has the peculiar property that the dual of a regular polygon equals the same regular polygon rotated by some amount.

In 3D, that peculiarity of 2D comes into play: cutting off the top of a rhombicuboctahedron and rotating it 45° just so happens to be equal to taking the dual of the top face and making a cupola of it. This simultaneous rotation/top face dualization is called "gyration". However, 3D also has the property that the dual operation is no longer equal to rotating the polyhedron (except for the simplex, which is interesting -- see below).

This divergence of rotation vs. dual in 3D causes 4D to have two distinct analogs of "gyration": (1) you cut off the top of a polychoron, do some rotation to it, and glue it back; (2) you cut off the top of the polychoron, replace it with an analogous shape made from the dual of its top face, and then glue it back. So there are really two different kinds of gyration going on here, and there may be some polychora for which both are applicable and result in different shapes.

Interestingly enough, the self-duality of the n-simplex causes the 3D concept of gyration to continue to exist, albeit in a more limited form. When the top face of the cut-off part of the polytope is a simplex, then taking the dual of this top face (and rebuilding the structure below analogously) is equal to rotating it, because the dual of the simplex is equal to itself except in orientation. This means that you can cut the runcinated 5-cell in half, and glue it back the "wrong" way -- i.e., rotate the tetrahedral top face to its dual orientation. This works not just in 4D, but in all higher dimensions with the facet-expanded n-simplex. So, in such cases, we have a direct analogue of 3D gyration, and we can properly call these rotated variants "gyrated" forms of their base polytopes.

The self-duality of the 24-cell also causes the same phenomenon, albeit restricted to 5D only. You can (probably) do this with the 24-cell bicupola (or its elongated version thereof), for example.
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: proposed definition for "Gyrated"

Postby quickfur » Thu Jun 28, 2012 7:31 pm

P.S. I was about to say, the concept of gyration in 4D applies to 4D analogues of Rubik's cube puzzles. Take a look at magic cube 4d, for example. The 4-cube's facets can undergo any 3D rotation, as long as its bottom cross section (which is a 3-cube) lines up with the rest of the puzzle again. So there's not just clockwise/counterclockwise rotations in this puzzle, but rotation around 3 axes (or as I prefer to call it, rotation in 3 orthogonal planes).

Now obviously you cannot take the dual of the top face of the rotating layer, and construct a prism below it, since you'd end up with an octahedral prism (and violating the (extrapolated) laws of physics and conservation of number/color of puzzle pieces etc.). So here's a clear example of where gyration splits into two distinct, incompatible operations.
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: proposed definition for "Gyrated"

Postby wintersolstice » Thu Jun 28, 2012 10:42 pm

quickfur wrote:Gyration, as we know it, is peculiar to 3D. In 2D, such an operation is impossible (you can cut off the top part of a polygon, but there's no way to "gyrate" it before putting it back on; the most you can do is to rotate it vertically and glue it back on another edge, which usually makes the result non-convex). 2D, however, has the peculiar property that the dual of a regular polygon equals the same regular polygon rotated by some amount.

In 3D, that peculiarity of 2D comes into play: cutting off the top of a rhombicuboctahedron and rotating it 45° just so happens to be equal to taking the dual of the top face and making a cupola of it. This simultaneous rotation/top face dualization is called "gyration". However, 3D also has the property that the dual operation is no longer equal to rotating the polyhedron (except for the simplex, which is interesting -- see below).

This divergence of rotation vs. dual in 3D causes 4D to have two distinct analogs of "gyration": (1) you cut off the top of a polychoron, do some rotation to it, and glue it back; (2) you cut off the top of the polychoron, replace it with an analogous shape made from the dual of its top face, and then glue it back. So there are really two different kinds of gyration going on here, and there may be some polychora for which both are applicable and result in different shapes.

Interestingly enough, the self-duality of the n-simplex causes the 3D concept of gyration to continue to exist, albeit in a more limited form. When the top face of the cut-off part of the polytope is a simplex, then taking the dual of this top face (and rebuilding the structure below analogously) is equal to rotating it, because the dual of the simplex is equal to itself except in orientation. This means that you can cut the runcinated 5-cell in half, and glue it back the "wrong" way -- i.e., rotate the tetrahedral top face to its dual orientation. This works not just in 4D, but in all higher dimensions with the facet-expanded n-simplex. So, in such cases, we have a direct analogue of 3D gyration, and we can properly call these rotated variants "gyrated" forms of their base polytopes.

The self-duality of the 24-cell also causes the same phenomenon, albeit restricted to 5D only. You can (probably) do this with the 24-cell bicupola (or its elongated version thereof), for example.

well that is more or less what I said in my above post: that there are two analogues of gyration and sometimes they can be both! (just not in so many words:D) LOL
wintersolstice
Trionian
 
Posts: 91
Joined: Sun Aug 16, 2009 11:59 am

Re: proposed definition for "Gyrated"

Postby quickfur » Fri Jun 29, 2012 12:22 am

wintersolstice wrote:[...]well that is more or less what I said in my above post: that there are two analogues of gyration and sometimes they can be both! (just not in so many words:D) LOL

Heh... I must be getting old, I'm so verbose. :\

But anyway, I was thinking about Klitzing's segmentotope made by cutting an octahedral prism in half and gluing it back together the "wrong" way. The cutting plane, obviously, must be a hyperplane, in this case, orthogonal to the "axis" of the prism, and oriented such that it cuts the octahedral facets in half. So the result is two square pyramid prisms. Each prism consists of two square pyramids, 4 triangular prisms, and 1 cube (square prism). The cube is produced by the cut, so there are 3 ways to glue the two pieces back, the "right" way, which gives an octahedral prism again, and two other ways where the original "axis" of the prisms in each piece is perpendicular. Because of the rotational symmetry of the square pyramids, though, these two other ways produce the same shape modulo a 90° rotation. So you end up with a polytope that has 4 square pyramids and 8 triangular prisms (the cube cells where the gluing happened become interior to the polytope).

A similar 3D shape is the gyrobifastigium (basically two triangular prisms glued together at a square face, with a 90° twist in orientation between the two pieces).
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: proposed definition for "Gyrated"

Postby quickfur » Fri Jun 29, 2012 3:59 am

wintersolstice wrote:[...]2) take a cube cupola and take a "square orthobicupolic ring" of it and replace it with a "square gyrobicupolic ring"
[...]
**assuming that in "2)" there will be a triangular prism that gets rotated! (I'll need to check that:D)

Hmm, I'm having trouble seeing where you can cut out a square orthobicupolic ring from a cube cupola. Are you sure a cube cupola has a square orthobicupolic ring as a sub-part? I can't see it, if there is one.

EDIT: Unless I'm mistaken, the square orthobicupolic ring has an octagonal prism base, which I assume is where it is cut off from the cube cupola, since joining it back on any other face would require the remaining shape to be non-convex. But I can't see where you would get an octagonal prism cross section in the cube cupola. (Am I just missing something obvious?)

Gah. I just realized that I confused the orthobicupolic ring with the magnabicupolic ring. Never mind. I'm still not sure if the cutting is possible, but at least it seems more likely now. I'll investigate this a bit more. Sorry for the false alarm!
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: proposed definition for "Gyrated"

Postby wendy » Fri Jun 29, 2012 8:58 am

Gyrated is used in the Conway notation for the dual of snub. It equates to the semis-stellated vaniate.
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Re: proposed definition for "Gyrated"

Postby wintersolstice » Fri Jun 29, 2012 10:17 am

quickfur wrote:Hmm, I'm having trouble seeing where you can cut out a square orthobicupolic ring from a cube cupola. Are you sure a cube cupola has a square orthobicupolic ring as a sub-part? I can't see it, if there is one.



I'll explain:D

a cube cupola has the following cells:

1 cube (top)
6 cubes (joined to the above cubes faces)
12 3-prisms (joined to the edges)
8 tetrahedra (joined to the vertices)
1 rhombicuboctahedron

if you remove a 4-cupola of the rhombicuboctahedron

EDIT: and take the convex hull

1 cube (joined to the top of the 4-cupola)
4 tetrahedra (joined to the 4 triangles)
4 triangular prisms (joined by their square faces to to squares in the cupola)

are all removed.

the cross section is a 4-cupola! because the removal of a cupola from the rhombicuboctahedron has exposed a 8-gon which needs to be joined to the square (once joined to the square in the cube on the top of the cube cupola) the triangles and squares from the cupola crosssection are from the 3-prisms and tetrahedra

so the fragment has two square cupola joined at the bases: one from the rhombicuboctahedron and one from the cross section, and a cube, which is joined to both. which a 4-bicupolic ring.

btw the magnabi forms come from what I call the "Partially-base truncated cupola" (truncate||cantelate) and the gyrobiforms from "Partially-base rectated cupola" (rectate||cantelate)
wintersolstice
Trionian
 
Posts: 91
Joined: Sun Aug 16, 2009 11:59 am

Re: proposed definition for "Gyrated"

Postby quickfur » Fri Jun 29, 2012 4:41 pm

Yeah I realized the validity of the construction after I posted my comment. :) I was going to post again, but didn't because I've been busy calculating coordinates for the result, which looks like a new CRF that isn't in Klitzing's list! (Although, one could argue that this is merely a composition of more basic CRFs: a square orthobicupolic ring composed with a square cupola prism composed with a square gyrobicupolic ring. But it's still an interesting example of a non-trivial CRF that can be derived by modifying a uniform polychoron -- in this case, cutting up a runcinated tesseract and modifying one end of it.)

Anyway, I'll post later when I got the coordinates figured out. With pictures. :)
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: proposed definition for "Gyrated"

Postby quickfur » Fri Jun 29, 2012 5:21 pm

P.S. I've decided to post the renders in the CRF thread, since this polychoron is CRF.
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: proposed definition for "Gyrated"

Postby wintersolstice » Sat Jun 30, 2012 12:56 am

quickfur wrote:Yeah I realized the validity of the construction after I posted my comment. :) I was going to post again, but didn't because I've been busy calculating coordinates for the result, which looks like a new CRF that isn't in Klitzing's list! (Although, one could argue that this is merely a composition of more basic CRFs: a square orthobicupolic ring composed with a square cupola prism composed with a square gyrobicupolic ring. But it's still an interesting example of a non-trivial CRF that can be derived by modifying a uniform polychoron -- in this case, cutting up a runcinated tesseract and modifying one end of it.)

Anyway, I'll post later when I got the coordinates figured out. With pictures. :)

actually I've just realized, the cube cupola CAN'T be gyrated but it CAN be diminished!
it's the Partially-base truncated cupola that gets gyrated:D(sorry about that)

what it is, I've lost all my paper work listing the gyrations and diminishes of what I call "the 29" (29 segmentotopes whose bases are within the Archimedian and Platonic solids excluding prisms and vertex-transitive cases)
wintersolstice
Trionian
 
Posts: 91
Joined: Sun Aug 16, 2009 11:59 am

Re: proposed definition for "Gyrated"

Postby quickfur » Sat Jun 30, 2012 5:47 am

wintersolstice wrote:
quickfur wrote:Yeah I realized the validity of the construction after I posted my comment. :) I was going to post again, but didn't because I've been busy calculating coordinates for the result, which looks like a new CRF that isn't in Klitzing's list! (Although, one could argue that this is merely a composition of more basic CRFs: a square orthobicupolic ring composed with a square cupola prism composed with a square gyrobicupolic ring. But it's still an interesting example of a non-trivial CRF that can be derived by modifying a uniform polychoron -- in this case, cutting up a runcinated tesseract and modifying one end of it.)

Anyway, I'll post later when I got the coordinates figured out. With pictures. :)

actually I've just realized, the cube cupola CAN'T be gyrated but it CAN be diminished!
it's the Partially-base truncated cupola that gets gyrated:D(sorry about that)

Did you see my posts in the other thread? I calculated the coordinates for two CRFs, both derived from the cube cupola by cutting off one (respectively, two) square orthobicupolic rings and replacing it (them) with square gyrobicupolic ring(s).

But you're right, though, it's not really a gyrate in a "true" sense, only a sub-part of it is gyrated. The base gets cut off because the square antiprism is shorter than the cube, so you don't get a pseudo-rhombicuboctahedron, but instead an elongated square cupola (rhombicuboctahedron with 1/3 cut off) and a square cupola cell (in rotated orientation) at a slight angle from it.

The diminishing of the cube cupola is simply because it can be decomposed into the sequence square orthobicupolic ring + square cupola prism + square orthobicupolic ring.

Or do you have another diminishing in mind?

what it is, I've lost all my paper work listing the gyrations and diminishes of what I call "the 29" (29 segmentotopes whose bases are within the Archimedian and Platonic solids excluding prisms and vertex-transitive cases)

Oh. :( Well, let us know when you find it; I'm interested to know what kind of new CRFs we can make.
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: proposed definition for "Gyrated"

Postby wintersolstice » Sat Jun 30, 2012 7:45 pm

quickfur wrote:Did you see my posts in the other thread? I calculated the coordinates for two CRFs, both derived from the cube cupola by cutting off one (respectively, two) square orthobicupolic rings and replacing it (them) with square gyrobicupolic ring(s).



actually no I haven't looked at the thread but shortly after my post I started to think maybe it was a valid CRF polychoron but thought I'd leave that to you to work out:D

quickfur wrote:The diminishing of the cube cupola is simply because it can be decomposed into the sequence square orthobicupolic ring + square cupola prism + square orthobicupolic ring.

Or do you have another diminishing in mind?


no that's right

I did actually make a recent thread to explain how that works in other dimensions if your interested:D

quickfur wrote:Oh. :( Well, let us know when you find it; I'm interested to know what kind of new CRFs we can make.


well I think I might just rennumerate them, but I can post "the 29" though.
wintersolstice
Trionian
 
Posts: 91
Joined: Sun Aug 16, 2009 11:59 am

Re: proposed definition for "Gyrated"

Postby wintersolstice » Fri Nov 16, 2012 7:36 pm

I've thought of a way to deal the two different analolgies of "3d gyrating", while maintaining the use of "gyrated" to mean rotated.

use a new term:

"dualated" this referes to #1 of the the list in the first post,

this means that "dualating" a polygon = "gyrating" a polygon

it would therefore be appriate to rename "gyrated antiprism" to "dualated" antiprism :D
wintersolstice
Trionian
 
Posts: 91
Joined: Sun Aug 16, 2009 11:59 am


Return to CRF Polytopes

Who is online

Users browsing this forum: No registered users and 12 guests

cron