PWrong wrote:What exactly do you mean by strictly convex? If all the angles are <180 degrees then you can't have any lines or planes in your shape.
It sounds cool anyway. What sort of shapes have you found so far?
wintersolstice wrote:Can I ask a question about your comment?
How can a shape be convex but self-intersecting?
And I thought convex meant the angles were "less than or equal than 180 degrees" (given that all the diagonals lie within the shape)
These shapes are based on the "Johnson solids"
"Johnsonian" was an name suggested by Bowers, I spoke to him about these shapes. And he seemed to understand the definition I created.
thanks (and sorry if I've upset you!)
Ok I'm a bit confused. The wiki article says that a square pyramid ad a pentagonal pyramid are Johnson solids, but not a hexagonal pyramid. Surely none of these are vertex transitive?
wintersolstice wrote:I've found about 193 so far and was wondering if anyone would be interested in helping me.
Keijj wrote:Could you list those 193?
Keiji wrote:Vertex-transitive means that you can pick any two vertices in the shape, and the facets joined to them will all be equivalent. So a cuboctahedron for example is vertex-transitive because on every vertex there are 2 triangles and 2 squares.
Keiji wrote:[...]1. are these two shapes indeed CRF polychora? (i.e. can they be built with all their (two-dimensional) faces regular?)
2. do these appear in wintersolstice's list above? IIRC, his list was mainly composed of prismatoid-like forms (3+1) whereas these are more duoprismatic-like forms (2+2)
3. is the "gyro" form the first explicitly constructed polychoron to include the square antiprism as a cell (other than trivial things like the square antiprism prism/pyramid)? if not, what came before it?
4. any name suggestions for these two?
5. could quickfur render them, pretty-please?
I'm aware that my diagrams have the black square faces rotated 22.5 degrees (I think) out of line with where they should be so the square faces do not appear planar even though they are supposed to be
quickfur wrote:Keiji wrote:[...]1. are these two shapes indeed CRF polychora? (i.e. can they be built with all their (two-dimensional) faces regular?)
Yess! And the listed cells are all correct.
3. is the "gyro" form the first explicitly constructed polychoron to include the square antiprism as a cell (other than trivial things like the square antiprism prism/pyramid)? if not, what came before it?
I suppose you regard Cartesian products of the square antiprism trivial too?
4. any name suggestions for these two?
That's a hard one. There are so many ways you can attach cells together in 4D, it's hard to think of a name that adequately conveys how this thing is constructed. Are there existing naming conventions for these things, that we can generalize from? (Yes I'm too lazy to actually look it up. )
5. could quickfur render them, pretty-please?
Do you have coordinates for them? I can do it a lot faster if you do. Otherwise it may take a while for me to calculate the coordinates.
I'm aware that my diagrams have the black square faces rotated 22.5 degrees (I think) out of line with where they should be so the square faces do not appear planar even though they are supposed to be
Plus you didn't fold the square cupolae enough to make the cube/square antiprism uniform.
But speaking of cupolae... you can actually make native 4D equivalents of cupolae.
Not all of them have gyro forms, though; the only one is the tetrahedral cupola (because the cuboctahedron has two different tetrahedral orientations, even though as a cubic truncate it's not self-dual). However, you can attach different cupola with the same base: for example, a cubic cupola can attach to an octahedral cupola to form a cubic-octahedral bicupola, and inserting a rhombicuboctahedral prism gives you a cubic-octahedral elongated bicupola.
Keiji wrote:[...]
Indeed, I would call the cube-octahedron and icosahedron-dodecahedron bicupolae gyro forms though, because they are bicupolae of a pair of duals - just like the tetrahedral gyrobicupola is of a pair of duals (two tetrahedra in dual orientations), and so are the n-gonal cupolae.
Mrrl wrote:Looks like there is 12-vertex polychoron with two 4-antiprism cells: you take two 4-antiprisms, one cube and form a tube from them (connecting them by square faces). Then close it with another tube of 4 tetrahedra and 4 square pyramids. So it's a kind of skew {4,3} douprism. All edges may be made equal in this object.
Keiji wrote:[...]
* "various augmented tesseracts", whatever they might be.
My question of whether my polychora were included in his list is more a question of whether they "accidentally" happen to overlap with something of a simpler construction which would be in that list, like how the two mentioned cases overlap with particular uniform polychora.
I would imagine that your "pseudo-antiprisms" (which I would just call antiprisms, for the same reason as calling the dual-based bicupolae gyro- even though you're doing more than just rotating them) are not in that list.
Would a "square pyramid antiprism" (note that 2->3D pyramids are self-dual) overlap? I have a feeling it may overlap with something containing an octahedron, but I'm not sure, mainly because I haven't bothered to calculate the facets of such an antiprism (yet).
As for the "reverse ortho" form, I like it Perhaps it should be called magna-something (in place of ortho/gyro-something), since we are inserting a larger polytope (octahedral prism as opposed to square (anti)prism), and magna is Latin for "great".
Marek14 wrote:Another class of shapes should be doable by cutting vertices from icositetrachoron. You can cut a vertex and replace it by a cube, cutting six octahedra in half so they become square pyramids, right?
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