Keiji wrote:Err, is this anything new/different to CRFP4DP/Augmentations#Augmented_duoprisms..?
3,3-duoprism:
3 augmentations
3,4-duoprism:
5 augmentations
3,5-duoprism:
11 augmentations
3,6-duoprism:
9 augmentations
3,8-duoprism:
5 augmentations
3,10-duoprism:
5 augmentations
4,4-duoprism:
20 augmentations
4,5-duoprism:
17 augmentations
4,6-duoprism:
7 augmentations
4,7-duoprism:
4 augmentations
4,8-duoprism:
103 augmentations
4,10-duoprism:
12 augmentations
5,5-duoprism:
35 augmentations
5,6-duoprism:
51 augmentations
5,7-duoprism:
17 augmentations
5,8-duoprism:
119 augmentations
5,9-duoprism:
45 augmentations
5,10-duoprism:
1715 augmentations
5,11-duoprism:
15 augmentations
5,12-duoprism:
25 augmentations
5,13-duoprism:
30 augmentations
5,14-duoprism:
48 augmentations
5,15-duoprism:
63 augmentations
5,16-duoprism:
98 augmentations
5,17-duoprism:
132 augmentations
5,18-duoprism:
208 augmentations
5,19-duoprism:
290 augmentations
5,20-duoprism:
454 augmentations
6,6-duoprism:
7 augmentations
6,8-duoprism:
7 augmentations
6,10-duoprism:
51 augmentations
7,8-duoprism:
8 augmentations
7,10-duoprism:
105 augmentations
8,8-duoprism:
170 augmentations
8,10-duoprism:
265 augmentations
9,10-duoprism:
629 augmentations
10,10-duoprism:
1365377 augmentations
10,11-duoprism:
62 augmentations
10,12-duoprism:
121 augmentations
10,13-duoprism:
189 augmentations
10,14-duoprism:
361 augmentations
10,15-duoprism:
611 augmentations
10,16-duoprism:
1161 augmentations
10,17-duoprism:
2055 augmentations
10,18-duoprism:
3913 augmentations
10,19-duoprism:
7154 augmentations
3,3-duoprism:
ring 1 intra: 60.000° aug=triangular prism pyramid
aug intra 52.239° inter +65.905° x48.190°
adj_augs
ring 2 intra: 60.000° aug=triangular prism pyramid
aug intra 52.239° inter +65.905° x48.190°
adj_augs
can only augment one ring at a time
3 augmentations
3,4-duoprism:
ring 1 intra: 60.000° aug=cubical pyramid
aug intra 45.000° inter +45.000° x35.264°
adj_augs
ring 2 intra: 90.000° aug=triangular prism pyramid
aug intra 52.239° inter +65.905° x48.190°
no_adj
can only augment one ring at a time
5 augmentations
3,5-duoprism:
ring 1 intra: 60.000° aug=pentagonal prism pyramid
aug intra 18.000° inter +13.283° x10.812°
adj_augs
ring 2 intra: 108.000° aug=triangular prism pyramid
aug intra 52.239° inter +65.905° x48.190°
no_adj
can augment both rings simultaneously
11 augmentations
3,6-duoprism:
ring 1 intra: 60.000° aug=triangular magnabicupolic ring
aug intra 52.239° inter +65.905° x48.190°
adj_augs has_gyro
ring 2 intra: 120.000° aug=triangular prism pyramid
aug intra 52.239° inter +65.905° x48.190°
no_adj
can only augment one ring at a time
9 augmentations
3,8-duoprism:
ring 1 intra: 60.000° aug=square magnabicupolic ring
aug intra 45.000° inter +45.000° x35.264°
adj_augs has_gyro
ring 2 intra: 135.000° no augs
can only augment one ring at a time
5 augmentations
3,10-duoprism:
ring 1 intra: 60.000° aug=pentagonal magnabicupolic ring
aug intra 18.000° inter +13.283° x10.812°
adj_augs has_gyro
ring 2 intra: 144.000° no augs
can only augment one ring at a time
5 augmentations
4,4-duoprism:
ring 1 intra: 90.000° aug=cubical pyramid
aug intra 45.000° inter +45.000° x35.264°
adj_augs
ring 2 intra: 90.000° aug=cubical pyramid
aug intra 45.000° inter +45.000° x35.264°
adj_augs
can augment both rings simultaneously
20 augmentations
4,5-duoprism:
ring 1 intra: 90.000° aug=pentagonal prism pyramid
aug intra 18.000° inter +13.283° x10.812°
adj_augs
ring 2 intra: 108.000° aug=cubical pyramid
aug intra 45.000° inter +45.000° x35.264°
no_adj
can augment both rings simultaneously
17 augmentations
4,6-duoprism:
ring 1 intra: 90.000° aug=triangular magnabicupolic ring
aug intra 52.239° inter +65.905° x48.190°
no_adj has_gyro
ring 2 intra: 120.000° aug=cubical pyramid
aug intra 45.000° inter +45.000° x35.264°
no_adj
can only augment one ring at a time
7 augmentations
4,7-duoprism:
ring 1 intra: 90.000° no augs
ring 2 intra: 128.571° aug=cubical pyramid
aug intra 45.000° inter +45.000° x35.264°
no_adj
can only augment one ring at a time
4 augmentations
4,8-duoprism:
ring 1 intra: 90.000° aug=square magnabicupolic ring
aug intra 45.000° inter +45.000° x35.264°
adj_augs has_gyro
ring 2 intra: 135.000° aug=cubical pyramid
aug intra 45.000° inter +45.000° x35.264°
no_adj
can augment both rings simultaneously
123 augmentations
4,10-duoprism:
ring 1 intra: 90.000° aug=pentagonal magnabicupolic ring
aug intra 18.000° inter +13.283° x10.812°
adj_augs has_gyro
ring 2 intra: 144.000° no augs
can only augment one ring at a time
12 augmentations
5,5-duoprism:
ring 1 intra: 108.000° aug=pentagonal prism pyramid
aug intra 18.000° inter +13.283° x10.812°
adj_augs
ring 2 intra: 108.000° aug=pentagonal prism pyramid
aug intra 18.000° inter +13.283° x10.812°
adj_augs
can augment both rings simultaneously
35 augmentations
5,6-duoprism:
ring 1 intra: 108.000° aug=triangular magnabicupolic ring
aug intra 52.239° inter +65.905° x48.190°
no_adj has_gyro
ring 2 intra: 120.000° aug=pentagonal prism pyramid
aug intra 18.000° inter +13.283° x10.812°
adj_augs
can augment both rings simultaneously
64 augmentations
5,7-duoprism:
ring 1 intra: 108.000° no augs
ring 2 intra: 128.571° aug=pentagonal prism pyramid
aug intra 18.000° inter +13.283° x10.812°
adj_augs
can only augment one ring at a time
17 augmentations
5,8-duoprism:
ring 1 intra: 108.000° aug=square magnabicupolic ring
aug intra 45.000° inter +45.000° x35.264°
no_adj has_gyro
ring 2 intra: 135.000° aug=pentagonal prism pyramid
aug intra 18.000° inter +13.283° x10.812°
adj_augs
can augment both rings simultaneously
166 augmentations
5,9-duoprism:
ring 1 intra: 108.000° no augs
ring 2 intra: 140.000° aug=pentagonal prism pyramid
aug intra 18.000° inter +13.283° x10.812°
adj_augs
can only augment one ring at a time
45 augmentations
5,10-duoprism:
ring 1 intra: 108.000° aug=pentagonal magnabicupolic ring
aug intra 18.000° inter +13.283° x10.812°
adj_augs has_gyro
ring 2 intra: 144.000° aug=pentagonal prism pyramid
aug intra 18.000° inter +13.283° x10.812°
adj_augs
can augment both rings simultaneously
3013 augmentations
5,11-duoprism:
ring 1 intra: 108.000° no augs
ring 2 intra: 147.273° aug=pentagonal prism pyramid
aug intra 18.000° inter +13.283° x10.812°
no_adj
can only augment one ring at a time
15 augmentations
5,12-duoprism:
ring 1 intra: 108.000° no augs
ring 2 intra: 150.000° aug=pentagonal prism pyramid
aug intra 18.000° inter +13.283° x10.812°
no_adj
can only augment one ring at a time
25 augmentations
5,13-duoprism:
ring 1 intra: 108.000° no augs
ring 2 intra: 152.308° aug=pentagonal prism pyramid
aug intra 18.000° inter +13.283° x10.812°
no_adj
can only augment one ring at a time
30 augmentations
5,14-duoprism:
ring 1 intra: 108.000° no augs
ring 2 intra: 154.286° aug=pentagonal prism pyramid
aug intra 18.000° inter +13.283° x10.812°
no_adj
can only augment one ring at a time
48 augmentations
5,15-duoprism:
ring 1 intra: 108.000° no augs
ring 2 intra: 156.000° aug=pentagonal prism pyramid
aug intra 18.000° inter +13.283° x10.812°
no_adj
can only augment one ring at a time
63 augmentations
5,16-duoprism:
ring 1 intra: 108.000° no augs
ring 2 intra: 157.500° aug=pentagonal prism pyramid
aug intra 18.000° inter +13.283° x10.812°
no_adj
can only augment one ring at a time
98 augmentations
5,17-duoprism:
ring 1 intra: 108.000° no augs
ring 2 intra: 158.824° aug=pentagonal prism pyramid
aug intra 18.000° inter +13.283° x10.812°
no_adj
can only augment one ring at a time
132 augmentations
5,18-duoprism:
ring 1 intra: 108.000° no augs
ring 2 intra: 160.000° aug=pentagonal prism pyramid
aug intra 18.000° inter +13.283° x10.812°
no_adj
can only augment one ring at a time
208 augmentations
5,19-duoprism:
ring 1 intra: 108.000° no augs
ring 2 intra: 161.053° aug=pentagonal prism pyramid
aug intra 18.000° inter +13.283° x10.812°
no_adj
can only augment one ring at a time
290 augmentations
5,20-duoprism:
ring 1 intra: 108.000° no augs
ring 2 intra: 162.000° aug=pentagonal prism pyramid
aug intra 18.000° inter +13.283° x10.812°
no_adj
can only augment one ring at a time
454 augmentations
6,6-duoprism:
ring 1 intra: 120.000° aug=triangular magnabicupolic ring
aug intra 52.239° inter +65.905° x48.190°
no_adj has_gyro
ring 2 intra: 120.000° aug=triangular magnabicupolic ring
aug intra 52.239° inter +65.905° x48.190°
no_adj has_gyro
can only augment one ring at a time
7 augmentations
6,8-duoprism:
ring 1 intra: 120.000° aug=square magnabicupolic ring
aug intra 45.000° inter +45.000° x35.264°
no_adj has_gyro
ring 2 intra: 135.000° no augs
can only augment one ring at a time
7 augmentations
6,10-duoprism:
ring 1 intra: 120.000° aug=pentagonal magnabicupolic ring
aug intra 18.000° inter +13.283° x10.812°
adj_augs has_gyro
ring 2 intra: 144.000° no augs
can only augment one ring at a time
51 augmentations
7,8-duoprism:
ring 1 intra: 128.571° aug=square magnabicupolic ring
aug intra 45.000° inter +45.000° x35.264°
no_adj has_gyro
ring 2 intra: 135.000° no augs
can only augment one ring at a time
8 augmentations
7,10-duoprism:
ring 1 intra: 128.571° aug=pentagonal magnabicupolic ring
aug intra 18.000° inter +13.283° x10.812°
adj_augs has_gyro
ring 2 intra: 144.000° no augs
can only augment one ring at a time
105 augmentations
8,8-duoprism:
ring 1 intra: 135.000° aug=square magnabicupolic ring
aug intra 45.000° inter +45.000° x35.264°
no_adj has_gyro
ring 2 intra: 135.000° aug=square magnabicupolic ring
aug intra 45.000° inter +45.000° x35.264°
no_adj has_gyro
can augment both rings simultaneously
416 augmentations
8,10-duoprism:
ring 1 intra: 135.000° aug=pentagonal magnabicupolic ring
aug intra 18.000° inter +13.283° x10.812°
adj_augs has_gyro
ring 2 intra: 144.000° no augs
can only augment one ring at a time
265 augmentations
9,10-duoprism:
ring 1 intra: 140.000° aug=pentagonal magnabicupolic ring
aug intra 18.000° inter +13.283° x10.812°
adj_augs has_gyro
ring 2 intra: 144.000° no augs
can only augment one ring at a time
629 augmentations
10,10-duoprism:
ring 1 intra: 144.000° aug=pentagonal magnabicupolic ring
aug intra 18.000° inter +13.283° x10.812°
adj_augs has_gyro
ring 2 intra: 144.000° aug=pentagonal magnabicupolic ring
aug intra 18.000° inter +13.283° x10.812°
adj_augs has_gyro
can augment both rings simultaneously
11921273 augmentations
10,11-duoprism:
ring 1 intra: 144.000° no augs
ring 2 intra: 147.273° aug=pentagonal magnabicupolic ring
aug intra 18.000° inter +13.283° x10.812°
no_adj has_gyro
can only augment one ring at a time
62 augmentations
10,12-duoprism:
ring 1 intra: 144.000° no augs
ring 2 intra: 150.000° aug=pentagonal magnabicupolic ring
aug intra 18.000° inter +13.283° x10.812°
no_adj has_gyro
can only augment one ring at a time
121 augmentations
10,13-duoprism:
ring 1 intra: 144.000° no augs
ring 2 intra: 152.308° aug=pentagonal magnabicupolic ring
aug intra 18.000° inter +13.283° x10.812°
no_adj has_gyro
can only augment one ring at a time
189 augmentations
10,14-duoprism:
ring 1 intra: 144.000° no augs
ring 2 intra: 154.286° aug=pentagonal magnabicupolic ring
aug intra 18.000° inter +13.283° x10.812°
no_adj has_gyro
can only augment one ring at a time
361 augmentations
10,15-duoprism:
ring 1 intra: 144.000° no augs
ring 2 intra: 156.000° aug=pentagonal magnabicupolic ring
aug intra 18.000° inter +13.283° x10.812°
no_adj has_gyro
can only augment one ring at a time
611 augmentations
10,16-duoprism:
ring 1 intra: 144.000° no augs
ring 2 intra: 157.500° aug=pentagonal magnabicupolic ring
aug intra 18.000° inter +13.283° x10.812°
no_adj has_gyro
can only augment one ring at a time
1161 augmentations
10,17-duoprism:
ring 1 intra: 144.000° no augs
ring 2 intra: 158.824° aug=pentagonal magnabicupolic ring
aug intra 18.000° inter +13.283° x10.812°
no_adj has_gyro
can only augment one ring at a time
2055 augmentations
10,18-duoprism:
ring 1 intra: 144.000° no augs
ring 2 intra: 160.000° aug=pentagonal magnabicupolic ring
aug intra 18.000° inter +13.283° x10.812°
no_adj has_gyro
can only augment one ring at a time
3913 augmentations
10,19-duoprism:
ring 1 intra: 144.000° no augs
ring 2 intra: 161.053° aug=pentagonal magnabicupolic ring
aug intra 18.000° inter +13.283° x10.812°
no_adj has_gyro
can only augment one ring at a time
7154 augmentations
10,20-duoprism:
ring 1 intra: 144.000° no augs
ring 2 intra: 162.000° aug=pentagonal magnabicupolic ring
aug intra 18.000° inter +13.283° x10.812°
no_adj has_gyro
can only augment one ring at a time
13647 augmentations
Total number of duoprism augmentations: 11956959
Deedlit wrote:To bring the topic back to diminishings of the 600-cell, I was thinking of taking up the project of enumerating these. But first, I have a question: for a diminishing, can we take any subset of the vertices such that the connected components are either isolated vertices or a pair of adjacent vertices? Or does diminishing a pair of adjacent vertices remove the possibility of some vertices not adjacent to the pair?
THE SPECIAL CUTS OF THE 600-CELL
Mathieu Dutour Sikirić, Wendy Myrvold
Abstract. A polytope is called regular-faced if every one of its
facets is a regular polytope. The 4-dimensional regular-faced polytopes
were determined by G. Blind and R. Blind [2, 3, 4]. The last
class of such polytopes is the one which consists of polytopes obtained
by removing a set of non-adjacent vertices (an independent
set) of the 600-cell. These independent sets are enumerated up to
isomorphism and it is determined that the number of polytopes in
this last class is 314, 248, 344.
(Submitted on 25 Aug 2007 (v1), last revised 22 Nov 2007 (this version, v2))
Klitzing wrote:Deedlit wrote:To bring the topic back to diminishings of the 600-cell, I was thinking of taking up the project of enumerating these. But first, I have a question: for a diminishing, can we take any subset of the vertices such that the connected components are either isolated vertices or a pair of adjacent vertices? Or does diminishing a pair of adjacent vertices remove the possibility of some vertices not adjacent to the pair?
Well, there is a paper with title / authors / abstractTHE SPECIAL CUTS OF THE 600-CELL
Mathieu Dutour Sikirić, Wendy Myrvold
Abstract. A polytope is called regular-faced if every one of its
facets is a regular polytope. The 4-dimensional regular-faced polytopes
were determined by G. Blind and R. Blind [2, 3, 4]. The last
class of such polytopes is the one which consists of polytopes obtained
by removing a set of non-adjacent vertices (an independent
set) of the 600-cell. These independent sets are enumerated up to
isomorphism and it is determined that the number of polytopes in
this last class is 314, 248, 344.
(Submitted on 25 Aug 2007 (v1), last revised 22 Nov 2007 (this version, v2))
There is web access to that on too, cf. either
http://arxiv.org/abs/0708.3443v2 or
http://www.academia.edu/745766/The_special_cuts_of_600-cell.
--- rk
Deedlit wrote:I know, but that paper only counted cuts of non-adjacent vertices. I was looking into counting the number of ways to cut a set consisting of nonadjacent groups, where each group is either a single vertex or a pair of adjacent vertices. But first I wanted to confirm that this always led to a CRF polychora.
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o5o x5x o5o
o5f o5f
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quickfur wrote:I would not be surprised if the actual number of 600-cell CRF diminishings are in the vicinity of 2^100 or so. Besides the "surface" diminishings, which include the adjacent and non-adjacent diminishings, there are also the deeper cuts, producing dodecahedral cross-sections (flanked by pentagonal pyramids) and icosidodecahedral cross-sections (also flanked by pentagonal pyramids). It's possible to make multiple such cuttings on the 600-cell, and the resulting shapes may also have their tetrahedral sections diminished combinatorially too. Besides that, there are also the 600-cell wedges, produced by two non-parallel bisections of the 600-cell, which I've posted before; each of these wedges also have a large number of possible diminishings.
1:120 1
1:30 2:90 60
1:10 5:110 288
1:4 2:116 450
1:2 2:8 10:110 1440
2:10 5:30 10:80 1440
2:6 3:30 6:84 1200
1:6 3:114 400
2:4 4:116 1800
1:2 2:4 6:114 1200
30:120 960
20:120 1440
15:120 960
12:120 1200
10:120 336
1:2 2:118 60
2:10 10:110 288
6:120 40
2:6 6:114 400
5:120 336
4:120 60
3:120 40
2:120 1
1/14400*(2^(120/1)*1+2^(30/1+90/2)*60+2^(10/1+110/5)*288+2^(4/1+116/2)*450+2^(2/1+8/2+110/10)*1440+2^(10/2+30/5+80/10)*1440+2^(6/2+30/3+84/6)*1200+2^(6/1+114/3)*400+2^(4/2+116/4)*1800+2^(2/1+4/2+114/6)*1200+2^(120/30)*960+2^(120/20)*1440+2^(120/15)*960+2^(120/12)*1200+2^(120/10)*336+2^(2/1+118/2)*60+2^(10/2+110/10)*288+2^(120/6)*40+2^(6/2+114/6)*400+2^(120/5)*336+2^(120/4)*60+2^(120/3)*40+2^(120/2)*1)
=92307499707443390526727850063504, approximately 2^106.186
quickfur wrote:Ah I see, so it's just a fancy name for that "partial expansion" thingy we found back then. Got it. Thanks!
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Klitzing wrote:Still am wrangling around with these - esp. on the level of classification of elements. Cause the swirl symmetry does not bow to that easily. But for the setup of their incidence matrices it is needed to distinguish between alike elements whenever they are to be used in different orbits of symmetry...
The 2 rings of tids are quite clear. That these are in mutual orthogonal orientation, too. That the peroes are attached to the decagons also is obvious. But then it runs very fast out of easy visualization. - Quickfur, do you have some clues on how these peroes of the two rings are to be connected? Which classes of tetrahedra are to be distinguished? How look the surroundings of the tid-to-tid connections, i.e. the ones of the sides of these decagons? And how do look the souroundings of the equatorial zick-zacks of each tid? How do these more or less parallel stripes of one ring correlate to the orthogonal ones of the other ring?
Moreover: I think, we so far have not even settled down on the question whether these 2 figures are chiral or not...
--- rk
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