quickfur wrote:in Keiji's terminology, pyroperihedron
quickfur wrote:I noticed Keiji has started a project to classify Klitzing's segmentochora. I'd like to start this thread for discussing this.
First of all, a large number of segmentochora can be generated by choosing two uniform polyhedra with the same symmetry group and placing them in parallel hyperplanes. If the difference in circumradii is less than the edge length, then this will generate a segmentochoron. If the two polyhedra are equal, then the result is a prism; if one is a point, the result is a pyramid. Otherwise, the result is a kind of general cupola-like shape, which is a superset of what we traditionally considered cupolae in the past.
The segmentochoron cube||icosahedron is a notable exception, and is quite unique as far as segmentochoron constructions go.
Besides these, there are other sporadics such as line||orthogonal_3-prism (4.8.2), which occurs as a maximal diminishing of the rectified 5-cell, and a few other oddities. But the generalized cupola pretty fills up the bulk of Klitzing's segmentochora.
Note that due to the tetrahedron being the alternated cube, there is some overlap between the tetrahedron and cube family of uniform polyhedra, so there is somewhat a distinction between a cuboctahedron as the rectified cube (o4x3o), and a cuboctahedron as a "rhombitetrahedron" (x3o3x), for example. For the purposes of generating cupolae-like segmentochora with tetrahedral symmetry, we may regard the octahedron as a rectified tetrahedron (o3x3o), the cuboctahedron as a "rhombitetrahedron" (x3o3x) or, in Keiji's terminology, pyroperihedron, and the truncated octahedron as the omnitruncated tetrahedron (x3x3x).
One notable subclass of segmentochora worth classifying separately is the segmentochora in which the top cell's projection onto the bottom cell falls strictly within the bottom cell. These segmentochora can be used as augments of other polychora which have cells that match the shape of the bottom cell, and will produce a CRF if the dichoral angles add up to less than 180°. Segmentochora where the projection of the top cell protrudes outside the bottom cell cannot be used as augments, so it's a good distinction to draw when classifying them -- so that when we start considering stacks of monostratic CRFs or augmentations of uniform polychora, we will easily know which are possible augments and which aren't.
wintersolstice wrote:I've already classified these:D I did it over a couple of months I think I posted it in the CRF thread I can't remember. I'll try and find the post when I get chance:D
We all really should be posting results to the wiki [...] so that everyone knows what's the up-to-date information.
We should be spending time on new discoveries instead of redoing what has already been done (except for verification of previous results, of course, but that shouldn't be happening by accident!).
Keiji, is it possible to make the forum and the wiki share the same users/logins? That way we won't have the unnecessary additional barrier of making people sign up for another account on the wiki. The less barriers the more likely people will actually contribute.
wintersolstice wrote:
here's what I call "the 29"
segmentotopes whose bases are either Platonic/Archimedean (excluding vertex transitive cases)
Key:
here are the catagories, underneath which are the Platonic solid/Platonic solid pairs that are Johnson cases, underneath that is why the others are invalid. Next to the name is a bracket showing the pair of bases for the segmentotope. the "X" isn't part of the name it just means you change it for the name of the Platonic Solids for it to name the shapes in that catagory.
meanings:
snubdis means truncate then alternate (based on snubdis 24-cell)
Birectate = Dual in 3D
Bitruncate = Truncate of the dual in 3D
Platonic = original Platonic solid
other notes:
I know the cupolas and antiprisms are already known (they're here just for completeness )
Cupoliprism was not my invention (based on the non-uniform scaliform polychoron name :
"Truncated tetrahedral cupoliprism"
the forward slashes mean that either platonic solid (or hypertruncate) can be used in the pair and it would be the same
the other names I did invent
the list:
X Cupola (Platonic||Cantellate)
1) Tetrahedron
2) Cube
3) Octahedron
4) Dodecahedron
Note: the icosahedron cupola is not CRF
X Antiprisms (Platonic||Birectate)
5) Octahedron/cube
6) Dodecahedron/icosahedron
Note: the tetrahedron antiprism is vertex transitive( 16-cell)
X Anticupola (Platonic||Rectate)
7) Cube
8 ) Octahedron
9) Dodecahedron
10) Icosahedron
Note: the tetrahedron anticupola is vertex transitive (rectified 5-cell)
Partially-base truncated X cupola (Truncate||Cantellate)
11) Tetrahedron
12) Cube
13) Octahedron
14) Dodecahedron
15) Icosahedron
Partially-base rectified X cupola ((Rectate||Cantellate)
16) Octahedron/cube
17) Dodecahedron/icosahedron
Note: the “Partially-base rectified tetrahedron cupola” = “octahedron anticupola”
Partially-base truncated X anticupola (Truncate|| Rectate)
18) Tetrahedron
19) Cube
20) Octahedron
21) Icosahedron
Note: the “Partially-base truncated dodecahedron anti cupola” is not CRF
Completely-base truncated X anticupola (Truncate||Cantitruncate)
22) Tetrahedron
23) Cube
24) Octahedron
25) Dodecahedron
Note: the “Completely-base truncated icosahedron anticupola” is not CRF
Truncated X cupoliprisms (Truncate||Bitruncate)
26) Octahedron/cube
27) Dodecahedron/icosahedrons
Note: “Truncated tetrahedron cupoliprism” is vertex transitive
Partially-base snubdis X antiprism (Snubdis||Birectate)
28) Octahedron
Note: only the octahedron can be “snubdis”ed
Semi cupola (Platonic|| Truncate)
29) Tetrahedron
Note: this shape is unique, the others in this group are not CRF
wintersolsticeTrionian Posts: 74Joined: Sun Aug 16, 2009 11:59 am
Klitzing wrote:That crown-jewel, cube || ike, as Wendy lately pointed out, in fact is not so especial after all. It is a diminished element from ex: cube is a faceted dodecahedron, and the next vertex layer of ex is just a tau-scaled icosahedron!
Keiji wrote:We all really should be posting results to the wiki [...] so that everyone knows what's the up-to-date information.
Yes, this is what I am doing now. Who was it that brought this topic from the wiki into the forum?
We should be spending time on new discoveries instead of redoing what has already been done (except for verification of previous results, of course, but that shouldn't be happening by accident!).
On the contrary, trying to verify a previous incorrect result means you're more likely to repeat the same mistake because you think it's right. Independently rediscovering the same thing is far more likely to imply correctness.
One good tool to validate the structure of a candidate polytope is to calculate its incidence matrix and put it into the Polytope Explorer. At the moment I'm the only one able to enter things into that though, but feel free to send me imats to check. (Plus, it needs more polytopes! It doesn't even have the uniform polychora yet!)
[...]
Keiji wrote:Thanks wintersolstice.
Is there any chance you can post the K4.x numbers for each polychoron you've classified?
(Only the ones listed under the two "as-yet-unclassified" categories at Segmentochoron#Classification are necessary)
quickfur wrote:[wiki versus forums]
[independently rediscovering everything]
As for the uniform polychora, I have coordinates for all of them (the convex ones, anyway). I think I'll start posting them to the wiki. As well as the CRFs I've rendered so far, since I have coordinates for those. I'm going to start putting up the Stella4D .off files that I made (I think I neglected to convert my polytopes for Marek for my recent CRF renders, I'll just put them up on the wiki from now on).
Keiji wrote:Klitzing wrote:That crown-jewel, cube || ike, as Wendy lately pointed out, in fact is not so especial after all. It is a diminished element from ex: cube is a faceted dodecahedron, and the next vertex layer of ex is just a tau-scaled icosahedron!
Okay, I've been staring at projections of the hydrochoron for a while, and the icosahedron is obvious, but I can't see how you can get a cube out of it. I assume you're saying it's a parallel cross-section somehow?
I can see a dodecahedron, but that is bigger than the icosahedron, so surely wouldn't give equal edge lengths?
Keiji wrote:Klitzing wrote:That crown-jewel, cube || ike, as Wendy lately pointed out, in fact is not so especial after all. It is a diminished element from ex: cube is a faceted dodecahedron, and the next vertex layer of ex is just a tau-scaled icosahedron!
Okay, I've been staring at projections of the hydrochoron for a while, and the icosahedron is obvious, but I can't see how you can get a cube out of it. I assume you're saying it's a parallel cross-section somehow?
I can see a dodecahedron, but that is bigger than the icosahedron, so surely wouldn't give equal edge lengths?
wintersolstice wrote:[...]
according to wikipedia a 4D cupola is a platonic solid to it's cantellate. and if that is the case the icosahedron case isn't CRF
[...]
quickfur wrote:[...]
I'm surprised the icosahedron case isn't CRF, though. Is it because edge length < difference in circumradius?
quickfur wrote:wintersolstice wrote:[...]
according to wikipedia a 4D cupola is a platonic solid to it's cantellate. and if that is the case the icosahedron case isn't CRF
[...]
Well, I was proposing that we generalize the term "cupola" to cover all cases of A || B where A and B are of the same symmetry group, A ≠ B, and the difference in circumradius between A and B < edge length. There are some cases for which the difference in circumradius is exactly equal to the edge length, in which case you get a 3D tiling, not a full-dimensioned segmentochoron. If the edge length < difference in circumradius, then the shape cannot be made CRF.
I'm surprised the icosahedron case isn't CRF, though. Is it because edge length < difference in circumradius?
quickfur wrote:I found this very counterintuitive, so I did a little mental experiment where I attached triangular prisms to the faces of the icosahedron, as if I was going to construct the cupola, and then I realized that the reason this doesn't give a CRF cupola is because the dihedral angle of the icosahedron is too large: once you attach the triangular prisms to its faces, the angle between adjacent prisms is too narrow to fit in another triangular prism (which is necessary to construct the cupola).
wintersolstice wrote:quickfur wrote:wintersolstice wrote:[...]
according to wikipedia a 4D cupola is a platonic solid to it's cantellate. and if that is the case the icosahedron case isn't CRF
[...]
Well, I was proposing that we generalize the term "cupola" to cover all cases of A || B where A and B are of the same symmetry group, A ≠ B, and the difference in circumradius between A and B < edge length. There are some cases for which the difference in circumradius is exactly equal to the edge length, in which case you get a 3D tiling, not a full-dimensioned segmentochoron. If the edge length < difference in circumradius, then the shape cannot be made CRF.
I'm surprised the icosahedron case isn't CRF, though. Is it because edge length < difference in circumradius?
when you say "symmetry groups" I'm guessing you mean a regular polytope and it's hypertruncates?:D) for these though I use the term "cupola relatives" and have come up with various names (containing the word "cupola" have you seen them above?
and a "polytope||expanded polytope" the word "Cupola" is used by itself:D
This is just how I would do it though, what do you think? (there still classed as cupola there's just a "true cupola" that's all:D)
quickfur wrote:I found this very counterintuitive, so I did a little mental experiment where I attached triangular prisms to the faces of the icosahedron, as if I was going to construct the cupola, and then I realized that the reason this doesn't give a CRF cupola is because the dihedral angle of the icosahedron is too large: once you attach the triangular prisms to its faces, the angle between adjacent prisms is too narrow to fit in another triangular prism (which is necessary to construct the cupola).
according to an observation I made, in order for a cupola (of the base being an expanded top) to be CRF the "top" needs to have ditope angles of less than 120 degrees, regardless of dimension!!!
quickfur wrote:In the end, I think I decided to lean towards a more general category that includes all cupola-like shapes, which IMO is a cleaner definition that doesn't make an arbitrary choice to treat a certain subclass of objects in a special way, unless they stand out geometrically.
So in my new definition, a segmentotope A||B is:
- a prism if A=B;
- a pyramid if A=point (and B≠point);
- a wedge if A is subdimensional ((n-2)-dimensions or less) and B is full-dimensional ((n-1)-D);
- a cupola otherwise.
So a pyramid is just a subclass of a wedge where the tip is a point (as opposed to a line or a polygon, etc.). This gives a clean division of segmentotopes into prisms, wedges, and cupolae. Prisms take care of the special case where A and B are the same shape, wedges take care of the case where one of them is subdimensional, and everything else is lumped into the general category of cupolae.
Keiji wrote:To be blunt, I really can't see how this could be a good idea.
There's no point defining something as being "any X that is not Y nor Z nor W", because you can just call it an otherwise unclassified X.
I forget exactly how I defined cupolae when I made my list, but I do recall coming up with a wider definition than the "pure" one. I'll have a look at that this evening though.
I also don't agree with picking out wedges, writing something as a wedge often seems to hide its real symmetry (I struggled to construct the K4.8 from its being written as a wedge, when really it's a much more interesting shape than that, and your later observation that it's a diminishing of a certain uniform polychoron is a much easier way to understand it).
Keiji wrote: Perhaps enumerating the 5D segmentotopes would give more reason to pick out "wedges", but so far I don't think it's appropriate
wintersolstice wrote:[...]there also seems to be an issue with the cupolae.
according to wikipedia a 4D cupola is a platonic solid to it's cantellate. and if that is the case the icosahedron case isn't CRF
now Gyrobicupolae are made by taking the dual of the base so the "cube gyrobicupola" is actually a "cube cupola" and a "octahedron cupola" joined together. (it also means that "cube gyrobicupola" = "octahedron gyrobicupola")
since the icosahedron cupola isn't CRF it means that dodecahedron gyrobicupola (or the icosahedron gyrobicupola) isn't CRF
However on Klizting paper a cupola is a "platonic solid to rectate" and "rectate to cantellate" (this is meaning currently being used on here http://teamikaria.com/hddb/wiki/Segment ... sification)
it also says under the CRF article about the cupolae that there is only one for each Platonic solid (which appears to mean Platonic to cantellate but it isn't clear) and it counts ortho and gyro forms for all 5 even though some are the same and some not CRF
But it does say that the ability to construct them CRF needs to be checked.
quickfur wrote:wintersolstice wrote:[...]
according to wikipedia a 4D cupola is a platonic solid to it's cantellate. and if that is the case the icosahedron case isn't CRF
[...]
Well, I was proposing that we generalize the term "cupola" to cover all cases of A || B where A and B are of the same symmetry group, A ≠ B, and the difference in circumradius between A and B < edge length. There are some cases for which the difference in circumradius is exactly equal to the edge length, in which case you get a 3D tiling, not a full-dimensioned segmentochoron. If the edge length < difference in circumradius, then the shape cannot be made CRF.
I'm surprised the icosahedron case isn't CRF, though. Is it because edge length < difference in circumradius?
Klitzing wrote:quickfur wrote:[...]
Well, I was proposing that we generalize the term "cupola" to cover all cases of A || B where A and B are of the same symmetry group, A ≠ B, and the difference in circumradius between A and B < edge length. There are some cases for which the difference in circumradius is exactly equal to the edge length, in which case you get a 3D tiling, not a full-dimensioned segmentochoron. If the edge length < difference in circumradius, then the shape cannot be made CRF.
[...]
No need for getting even deeper into troubles of what should be or should not be considered a cupola. There is already a well established term for that subset of segmentotopes (those with the same symmetry group for both bases): the lace prisms! (Or, if you would like: the unit-edged ones.)
[...]
quickfur wrote:wintersolstice wrote:quickfur wrote:wintersolstice wrote:[...]
according to wikipedia a 4D cupola is a platonic solid to it's cantellate. and if that is the case the icosahedron case isn't CRF
[...]
Well, I was proposing that we generalize the term "cupola" to cover all cases of A || B where A and B are of the same symmetry group, A ≠ B, and the difference in circumradius between A and B < edge length. There are some cases for which the difference in circumradius is exactly equal to the edge length, in which case you get a 3D tiling, not a full-dimensioned segmentochoron. If the edge length < difference in circumradius, then the shape cannot be made CRF.
I'm surprised the icosahedron case isn't CRF, though. Is it because edge length < difference in circumradius?
when you say "symmetry groups" I'm guessing you mean a regular polytope and it's hypertruncates?:D) for these though I use the term "cupola relatives" and have come up with various names (containing the word "cupola" have you seen them above?
and a "polytope||expanded polytope" the word "Cupola" is used by itself:D
This is just how I would do it though, what do you think? (there still classed as cupola there's just a "true cupola" that's all:D)
I kinda prefer to just class all of them the same way, especially after Klitzing pointed out that his definition of cupola (or was it antiprism?) is different from the one I always assumed would be most obvious. That caused me to rethink my definition of cupola, and why I should prefer that definition and not Klitzing's, or some other altogether. In the end, I think I decided to lean towards a more general category that includes all cupola-like shapes, which IMO is a cleaner definition that doesn't make an arbitrary choice to treat a certain subclass of objects in a special way, unless they stand out geometrically.
So in my new definition, a segmentotope A||B is:
- a prism if A=B;
- a pyramid if A=point (and B≠point);
- a wedge if A is subdimensional ((n-2)-dimensions or less) and B is full-dimensional ((n-1)-D);
- a cupola otherwise.
So a pyramid is just a subclass of a wedge where the tip is a point (as opposed to a line or a polygon, etc.). This gives a clean division of segmentotopes into prisms, wedges, and cupolae. Prisms take care of the special case where A and B are the same shape, wedges take care of the case where one of them is subdimensional, and everything else is lumped into the general category of cupolae.
In higher dimensions, having a general definition of cupola is much more useful, because the number of ways to put two things together just increases exponentially as the dimension increases. So rather than having to invent brand new categories for every dimension, might as well group them together.
[...]
quickfur wrote:Klitzing wrote:quickfur wrote:[...]
Well, I was proposing that we generalize the term "cupola" to cover all cases of A || B where A and B are of the same symmetry group, A ≠ B, and the difference in circumradius between A and B < edge length. There are some cases for which the difference in circumradius is exactly equal to the edge length, in which case you get a 3D tiling, not a full-dimensioned segmentochoron. If the edge length < difference in circumradius, then the shape cannot be made CRF.
[...]
No need for getting even deeper into troubles of what should be or should not be considered a cupola. There is already a well established term for that subset of segmentotopes (those with the same symmetry group for both bases): the lace prisms! (Or, if you would like: the unit-edged ones.)
[...]
You're right, lace-prisms is a better name for the catch-all category. In this case, it would be the monostratic lace-prisms.
As for what is/isn't a cupola, I think that may be splitting hairs. As Klitzing says, there's no unique generalization of the 3D notion of cupola. IMO the real solution is to recognize that there is no unique 4D analogue of the 3D cupolae instead of trying to decide which of the equally-valid possibilities should be designated as "true" cupolae.
Keiji wrote:[...] I also don't agree with picking out wedges, writing something as a wedge often seems to hide its real symmetry (I struggled to construct the K4.8 from its being written as a wedge, when really it's a much more interesting shape than that, and your later observation that it's a diminishing of a certain uniform polychoron is a much easier way to understand it).
IIRC almost all the wedges in Klitzing's list are bicupolic or biantiprismatic rings and there are only three or four left over. Whereas there are infinitely many prisms and a decent number of pyramids.
Perhaps enumerating the 5D segmentotopes would give more reason to pick out "wedges", but so far I don't think it's appropriate.
Klitzing wrote:quickfur wrote:[...]
You're right, lace-prisms is a better name for the catch-all category. In this case, it would be the monostratic lace-prisms.
No you are wrong here! All segmentotopes with a common non-trivial symmetrygroup in both bases are lace prisms. In fact those are exactly the unit-edged ones. And, a lace prism allways is monostratic (else it would be a lace tower)!
As for what is/isn't a cupola, I think that may be splitting hairs. As Klitzing says, there's no unique generalization of the 3D notion of cupola. IMO the real solution is to recognize that there is no unique 4D analogue of the 3D cupolae instead of trying to decide which of the equally-valid possibilities should be designated as "true" cupolae.
Well, a cupola in 3d has full symmetrical bases. It has triangles and squares connecting those.
So to extrapolate these figures into 4d, you would have to say what each of those components would become. Top-base polygons (the smaller ones) might become platonic solids. (But I extended that even to quasiregular cases after all.) The triangles most obviously generalize to pyramids. The squares should become somthing with axial symmetry again. So they either could become antiprisms (my choice, as there are more segmentochora which can be classified by that case) or prisms. In the latter case you would have additional things occuring within that new dimension: triangular prisms (kind as line atop square). Those cases do not even apply to ike (it would be hyperbolic!), nor to the quasiregulars. But those OTOH are caps again...
[...]
Klitzing wrote:You should be careful when applying names to kind of property-extrapolations into 4d, which do not conform with the very meaning of the word itself. This is a great deal esp. of Wendys polygloss, to try to cut all that historically wrong applied even. - The very word gyro just means rotated. Sure a rotated polygon looks like its dual, thus for segmentohedra this would be the same. But in 4d a rotated cube does not become an octahedron!
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