JMBR wrote:Hello again. If I can remember it well, one of the steps of enumerating johnson solids involved the vertex angle defect.
In 2 dimensions, the exterior angles of every non-self-intersecting polygon sum up to 2π because it seems like they can be deformed into a circle. The case in 3 dimensions sums up to 4π, equivalent to a sphere. My question is: is there an equivalent of that in 4 dimensions? The complete theorem is the Gauss-Bonnet theorem, but when I tried to find such relation, I drowned in heavy calculus. So I was curious to know if someone found this result.
Marek14 wrote:The last three are generally valid and they all correspond to n-gonal antiprism joined to n-gonal prism with cupolas and truncated polyhedra filling the gaps.
333n-6644: n-gonal antiprism, two triangular cupolas, truncated tetrahedron, n-gonal prism.
333n-8844: n-gonal antiprism, two square cupolas, truncated cube, n-gonal prism.
333n-0044: n-gonal antiprism, two pentagonal cupolas, truncated dodecahedron, n-gonal prism.
Marek14 wrote:Unfortunately, now I can't find any solutions for these despite the fact that I know that 3333-4444 must always have a solution (it's a prism) and 3333-3464 should have as well. Until someone helps me here, I'll assume that the non-quadratic polyhedra only have solutions that all others have...
So, what do we get? Well, pentagonal dipyramid (and presumably snub disphenoid and sphenomegacorona) have the same three solutions as triangular dipyramid: A solution for 3333-4444 (prism) and two solutions for 3333-3464 (surrounding of degree-4 vertex with two square pyramids and two triangular cupolas with two variants based on which pair of edges has squares built on them and which has triangle and hexagon).
johannes@itap.physik.uni-stuttgart.de wrote:The other solutions are:
33333-(10)5335 for the 10.row (J90), 12. row (J86) and the 13.row (icosahedron).
33333-34(10)(10)4 exist for the icosahedron only.
Marek14 wrote:As for your solutions, they look very interesting. 33333-05335 (two pentagonal rotundas, two pentagonal pyramids and one tetrahedron around an edge) fitting to J86 or J90 is unexpected. I presume only one specific arrangement of these cells work?
mr_e_man wrote:http://hi.gher.space/forum/viewtopic.php?p=22993#p22993Marek14 wrote:Unfortunately, now I can't find any solutions for these despite the fact that I know that 3333-4444 must always have a solution (it's a prism) and 3333-3464 should have as well. Until someone helps me here, I'll assume that the non-quadratic polyhedra only have solutions that all others have...
So, what do we get? Well, pentagonal dipyramid (and presumably snub disphenoid and sphenomegacorona) have the same three solutions as triangular dipyramid: A solution for 3333-4444 (prism) and two solutions for 3333-3464 (surrounding of degree-4 vertex with two square pyramids and two triangular cupolas with two variants based on which pair of edges has squares built on them and which has triangle and hexagon).
How did you conclude that 3464 works in general? It's not obvious to me.
I don't see the difference. A vertex in 3D is not fully determined by the polygons around it; and an edge in 4D is not fully determined by the polyhedra around it. You can build out from a vertex in 3D to complete the polyhedron and make the vertex rigid; and you can surround a vertex in 4D with polyhedra to make the edges rigid.Marek14 wrote:This would have to be expanded a lot (especially step 3), but theoretically, it could allow for complete enumeration. Note that this needs 4D to work: it won't enumerate Johnson solids in 3D because the vertices in 3D are not fully determined by the polygons around them; in 4D, the cells around a vertex DO determine that vertex completely since there can be no continuous changes of angles.
We need a program that finds all convex polyhedra with faces from a given set of polygons, perhaps with an input multihedron (incomplete polyhedron) that any output polyhedron should contain as a subset.
quickfur wrote:We need a program that finds all convex polyhedra with faces from a given set of polygons, perhaps with an input multihedron (incomplete polyhedron) that any output polyhedron should contain as a subset.
I have built a database of all 3D CRF polyhedra (excluding members of known infinite families, from which only a select number of the most common members are included). It contains the full face lattice of every polyhedron, and distinguishes between orientations of the chiral ones. As such, it's suitable for querying facts about 3D CRFs via SQL, and can probably serve as the base upon which you could write such a program.
If you're interested I can post the file somewhere, either in the native SQLite3 format or in a text-based export dump that can be imported into any other RDBMS that supports SQL.
johannes@itap.physik.uni-stuttgart.de wrote:quickfur wrote:We need a program that finds all convex polyhedra with faces from a given set of polygons, perhaps with an input multihedron (incomplete polyhedron) that any output polyhedron should contain as a subset.
I have built a database of all 3D CRF polyhedra (excluding members of known infinite families, from which only a select number of the most common members are included). It contains the full face lattice of every polyhedron, and distinguishes between orientations of the chiral ones. As such, it's suitable for querying facts about 3D CRFs via SQL, and can probably serve as the base upon which you could write such a program.
If you're interested I can post the file somewhere, either in the native SQLite3 format or in a text-based export dump that can be imported into any other RDBMS that supports SQL.
I would certainly be interested. In both formats. How can I get that database?
mr_e_man wrote:The program can output a complete polyhedron (just the basic connective structure, not coordinates or metrical values) where all vertices have angle sums less than 360°; I think this happens rarely enough that we can further analyze the results by hand.
o3x5x (-x)3x5x x3o5x
x3f5o o3f5o o3f5o
F3o5x <- f3o5x <- f3o5x
x3x5x o3x5x o3x5x
o3x5o
o3o5f
o3x5x o3x5x
x3f5o x3f5o
F3o5x F3o5x /- D4.5.1
f3f5o <-> x3x5x --< this gets replaced in the thawroing process
o3F5o o3F5o \- D4.5.3+
f3x5x f3x5x
x3o5f x3o5f
F3o5f F3o5f --\
V3x5o V3x5o |-- remove for D4.5.4 (bilbiro-ing)
o3o5V o3o5V |
F3o5f F3o5f --/
x3o5f x3o5f
f3x5o f3x5o
o3F5o o3F5o
f3f5o x3x5x --< this gets replaced in the thawroing process
x3x5x
o3F5o
f3x5x
x3o5F
x3o5o
o3x5o
x3o5x <-
F3o5o
o3f5o <-
f3o5x <-
o3x5x <-
...
xofo3ofox5xoxx&#xt
x5o3x
o5o3F
f5o3o
o5o3F
x5o3x
student5 wrote:Sooooo..... It's been a looong time!
[...] So far a lot of polychora discovered in 2014 turned out to be EKFs:
[...]
which makes me wonder ifis CRF? There was also talk of possibly finding a wedge shape like this that has thawros.
- Code: Select all
xofo3ofox5xoxx&#xt
D4.3.1 and D4.3.2 are really cool: I couldn't find anything EKF-like from them.
[...]
Anyways, these crown jewels are thus far always found using a subsymmetry, I think a program should also do that, but I think that's obvious
quickfur wrote:[...]
I'd say that practically all of the discoveries in 2014 are EKFs or some modification/combination of them.
quickfur wrote:D4.3.1 and D4.3.2 are really cool: I couldn't find anything EKF-like from them.
I see it as an EKF (i.e., modified Stott expansion) of some faceted prism of an icosahedral family polyhedron.
x2x x2o x2o
o2F o2f o2f
f2o <- f2(-x) <- f2o
o2F o2f o2f
x2x x2o x2o
x5o3x x5o3o x5o3o
o5o3F o5o3f o5o3f
f5o3o <- f5o3(-x) <- f5(-x)3x <- o5x3o
o5o3F o5o3f o5o3f
x5o3x x5o3o x5o3o
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