student5 wrote:Hmmm... Even though a lot of them turned out to be EKF, I'd say that bilbiro-ing and thawro-ing are still techniques that could produce further families of CRFpolychora. Like creating two "eggshells" that a brute force program could further try to connect. But then again, this will go into ex-family territory and thus get a combinatorial explosion.
Anyways, Quickfur, could you maybe share your program? I'd be interested to take a look at it and maybe make it more user-friendly (I'm into GUI's these days )
student5 wrote: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.
In this lace tower configuration here, I found it difficult to "see" the bilibiro's, but they are on the sides:
- Code: Select all
x2x x2o x2o
o2F o2f o2f
f2o <- f2(-x) <- f2o
o2F o2f o2f
x2x x2o x2o
Ah yes, now I see it: The bilbs are expanded along the square, so the original shape is a dodecahedral-prism kind of thing:Notice that we have to swap twice to get back to a non-faceted something. However, this is not anything as far as I can see... The ikes don't show up, because the faceting doesn't work out... I can't find it as an ekf now...
- Code: Select all
x5o3x x5o3o x5o3o
o5o3F o5o3f o5o3f
f5o3o <- f5o3(-x) <- f5(-x)3x <- o5x3o
o5o3F o5o3f o5o3f
x5o3x x5o3o x5o3o
x5x3x x5x3o x5x3o
o5x3F o5x3f o5x3f
f5x3o <- f5x3(-x) <- f5o3x
o5x3F o5x3f o5x3f
x5x3x x5x3o x5x3o
x3o5o
o3x5o
F3o5o
o3f5o
f3o5x
o3x5x <-
f3x5o <-
V3o5o (V=2f)
x3o5f <-
f3x5o <-
o3x5x <-
...
x2o3o x2o3x
f2x3o f2x3x
o2f3o <- o2f3x
x2o3x x2o3u
mibdi:
x2o
f2x
o2f
x2o
lace city:
o o
x x
f
o o
x2o
o2f
f2x
o2f
x2o
o3o o3o
x3o x3o
f3o
o3x o3x <- where does this x come from?
o3x o3x
x3x x3x
f3x
x3o x3o <- what to do here?
x3o
o3x x3f F3o x3x
x3o x3f F3x o3F
F3o x3F f3x o3x
x3x o3F f3x o3x
o3x
student5 wrote:D4.7: Can be augmented undocumented but probably some EKF?
ABCDEFGHIJ ABCDEFGHIJ ABCDEFGHIJ ABCDEFGHIJ
-------------------------------------------------
fFoxffooxo 3 foFfxofxoo 2 oxofofxFof 3 ooxofxfoFf &#zx (ex)
fFo(-x)ffoo(-x)o 3 foFFxofxxo 2 oxofofxFof 3 ooxofxfoFf &#zx (D1+I1 rewrite)
FAxoFFxxox 3 foFFxofxxo 2 oxofofxFof 3 ooxofxfoFf &#zx (first node expanded -> already contained EKF)
FAxoFFxxo. 3 foFFxofxx. 2 oxofofxFo. 3 ooxofxfoF. &#zx (J-level diminished = student5's/Quickfur's above mentioned, but so far missing find)
.AxoFFxxo. 3 .oFFxofxx. 2 .xofofxFo. 3 .oxofxfoF. &#zx (A+J-levels diminished = todays new find!)
x5x x5x
F5o
o5F o5F
x5x x5x
F5x
x5F x5F
F5o F5o
F5x F5x
x5x x5x
x5F x5F
o5F o5F
F5x F5x
x5F
x5x x5x
F5o F5o
o5F
x5x x5x
FxFox5xFoFx xofox5oxofx&#zx
o....5o.... o....5o.... & | 100 * * | 1 2 2 1 0 0 0 | 2 1 3 1 2 0 0 0 0 0 | 1 1 3 0 0
..o..5..o.. ..o..5..o.. & | * 50 * | 0 0 0 2 4 0 0 | 0 0 0 1 4 2 2 0 0 0 | 0 0 4 1 0
....o5....o ....o5....o | * * 100 | 0 0 0 0 2 2 2 | 0 0 0 0 1 2 2 1 2 2 | 0 0 2 2 2
------------------------------+------------+----------------------------+-----------------------------------+---------------
..... x.... ..... ..... & | 2 0 0 | 50 * * * * * * | 2 0 0 1 0 0 0 0 0 0 | 0 1 2 0 0
..... ..... x.... ..... & | 2 0 0 | * 100 * * * * * | 1 1 1 0 0 0 0 0 0 0 | 1 1 1 0 0
oo...5oo... oo...5oo...&#x | 2 0 0 | * * 100 * * * * | 0 0 2 0 1 0 0 0 0 0 | 1 0 2 0 0
o.o..5o.o.. o.o..5o.o..&#x & | 1 1 0 | * * * 100 * * * | 0 0 0 1 2 0 0 0 0 0 | 0 0 3 0 0
..o.o5..o.o ..o.o5..o.o&#x & | 0 1 1 | * * * * 200 * * | 0 0 0 0 1 1 1 0 0 0 | 0 0 2 1 0
....x ..... ..... ..... & | 0 0 2 | * * * * * 100 * | 0 0 0 0 0 1 0 1 1 1 | 0 0 1 1 2
..... ..... ....x ..... & | 0 0 2 | * * * * * * 100 | 0 0 0 0 0 0 1 0 1 1 | 0 0 1 1 1
------------------------------+------------+----------------------------+-----------------------------------+---------------
..... x.... x.... ..... & | 4 0 0 | 2 2 0 0 0 0 0 | 50 * * * * * * * * * | 0 1 1 0 0
..... ..... x....5o.... & | 5 0 0 | 0 5 0 0 0 0 0 | * 20 * * * * * * * * | 1 1 0 0 0
..... ..... xo... .....&#x & | 3 0 0 | 0 1 2 0 0 0 0 | * * 100 * * * * * * * | 1 0 1 0 0
..... x.o.. ..... .....&#x & | 2 1 0 | 1 0 0 2 0 0 0 | * * * 50 * * * * * * | 0 0 2 0 0
ooooo5ooooo ooooo5ooooo&#xr | 2 2 1 | 0 0 1 2 2 0 0 | * * * * 100 * * * * * | 0 0 2 0 0
..... ..o.x ..... .....&#x & | 0 1 2 | 0 0 0 0 2 1 0 | * * * * * 100 * * * * | 0 0 1 1 0
..... ..... ..... ..o.x&#x & | 0 1 2 | 0 0 0 0 2 0 1 | * * * * * * 100 * * * | 0 0 1 1 0
....x5....x ..... ..... | 0 0 10 | 0 0 0 0 0 10 0 | * * * * * * * 10 * * | 0 0 0 0 2
....x ..... ....x ..... & | 0 0 4 | 0 0 0 0 0 2 2 | * * * * * * * * 50 * | 0 0 0 1 1
....x ..... ..... ....x & | 0 0 4 | 0 0 0 0 0 2 2 | * * * * * * * * * 50 | 0 0 1 0 1
------------------------------+------------+----------------------------+-----------------------------------+---------------
..... ..... xo...5ox...&#x | 10 0 0 | 0 10 10 0 0 0 0 | 0 2 10 0 0 0 0 0 0 0 | 10 * * * * pap
..... x.... x....5o.... & | 10 0 0 | 5 10 0 0 0 0 0 | 5 2 0 0 0 0 0 0 0 0 | * 10 * * * pip
FxFox ..... ..... oxofx&#xt & | 6 4 4 | 2 2 4 6 8 2 2 | 1 0 2 2 4 2 2 0 0 1 | * * 50 * * bilbiro (tower: 21435)
..... ..o.x ..... ..o.x&#x & | 0 1 4 | 0 0 0 0 4 2 2 | 0 0 0 0 0 2 2 0 1 0 | * * * 50 * squippy
....x5....x ....x ..... & | 0 0 20 | 0 0 0 0 0 20 10 | 0 0 0 0 0 0 0 2 5 5 | * * * * 10 dip
mr_e_man wrote:Assuming we've enumerated all CRH polytopes in the previous dimension
Marek14 wrote: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.
If it's equivalent to hypersphere, then it would be 2π^2.
JMBR wrote:But if that's the case, there's some kind of angle defect which has this sum. What would it be? Dihedral angles and vertex solid angles are not comensurable with π, except for the cube (which can tile the space alone) and, maybe, truncated octahedron. https://en.wikipedia.org/wiki/Platonic_solid#Geometric_properties
That led me to question if this invariant even exists, which it does because of that theorem.
If a polytope, in n-dimensional Euclidean or spheric space with n>2, has convex faces (the [-1, n-1]-spyts) and convex vertex figures (the [0, n]-spyts), then the polytope (the [-1, n]-spyt) is convex.
If a polytope, in n-dimensional Euclidean or spheric space with n>2, has convex [k, k+3]-subpolytopes for all k (including hedra with k = -1), then it is convex.
student5 wrote:Further develop exxact crate so it can:
Do division, multiplication etc
do cos(pi/n) easily using DeMoivre's theorem, cuz hard-coding is boring and ugly
I have to go now, please look at the stuff and/or add your own programs/snippets to it in a different folder so we can cooperate!
mr_e_man wrote:
- Try to add a new face at this edge. Go through the list of all orientations of all polygons, in order of the angles in the polygons:
- Make sure the edge lengths match.
[...]
- Add the angle sums at the corresponding vertices (unless they were already the same vertex). The resulting sum(s) must be less than 360°.
mr_e_man wrote:
- Optionally, calculate some dihedral angles. If there are several ways to get an angle, the results must agree with each other.
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