Product metric
In nine-dimensional geometry, a polyyotton (or 9-polytope) is a polytope contained by 8-polytope facets. Each 7-polytope ridge being shared by exactly two 8-polytope facets.
A uniform polyyotton is one which is vertex-transitive, and constructed from uniform facets.
A proposed name for 9-polytope is polyyotton (plural: polyyotta), created from poly-, yotta- (a variation on octa, meaning eight) and -on.
Regular 9-polytopes
Regular 9-polytopes can be represented by the Schläfli symbol {p,q,r,s,t,u,v,w}, with w {p,q,r,s,t,u,v} 8-polytope facets around each peak.
There are exactly three such convex regular 9-polytopes:
- {3,3,3,3,3,3,3,3} - 9-simplex
- {4,3,3,3,3,3,3,3} - 9-cube
- {3,3,3,3,3,3,3,4} - 9-orthoplex
There are no nonconvex regular 9-polytopes.
Euler characteristic
The Euler characteristic for 9-polytopes that are topological 8-spheres (including all convex 9-polytopes) is zero. χ=V-E+F-C+f4-f5+f6-f7+f8=2.
Uniform 9-polytopes by fundamental Coxeter groups
Uniform 9-polytopes with reflective symmetry can be generated by these three Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams:
Coxeter group | Coxeter-Dynkin diagram | |
---|---|---|
A9 | [38] | Template:CDD |
B9 | [4,37] | Template:CDD |
D9 | [36,1,1] | Template:CDD |
Selected regular and uniform 9-polytopes from each family include:
- Simplex family: A9 [38] - Template:CDD
- 271 uniform 9-polytopes as permutations of rings in the group diagram, including one regular:
- {38} - 9-simplex or deca-9-tope or decayotton - Template:CDD
- 271 uniform 9-polytopes as permutations of rings in the group diagram, including one regular:
- Hypercube/orthoplex family: B9 [4,38] - Template:CDD
- 511 uniform 9-polytopes as permutations of rings in the group diagram, including two regular ones:
- {4,37} - 9-cube or enneract - Template:CDD
- {37,4} - 9-orthoplex or enneacross - Template:CDD
- 511 uniform 9-polytopes as permutations of rings in the group diagram, including two regular ones:
- Demihypercube D9 family: [36,1,1] - Template:CDD
- 383 uniform 9-polytope as permutations of rings in the group diagram, including:
- {31,6,1} - 9-demicube or demienneract, 16,1 - Template:CDD; also as h{4,38} Template:CDD.
- {36,1,1} - 9-orthoplex, 61,1 - Template:CDD
- 383 uniform 9-polytope as permutations of rings in the group diagram, including:
The A9 family
The A9 family has symmetry of order 3628800 (10 factorial).
There are 256+16-1=271 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. These are all enumerated below. Bowers-style acronym names are given in parentheses for cross-referencing.
# | Graph | Coxeter-Dynkin diagram Schläfli symbol Name |
Element counts | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
8-faces | 7-faces | 6-faces | 5-faces | 4-faces | Cells | Faces | Edges | Vertices | |||
1 |
Template:CDD |
10 | 45 | 120 | 210 | 252 | 210 | 120 | 45 | 10 | |
2 |
Template:CDD |
360 | 45 | ||||||||
3 |
Template:CDD |
1260 | 120 | ||||||||
4 |
Template:CDD |
2520 | 210 | ||||||||
5 |
Template:CDD |
3150 | 252 | ||||||||
6 |
Template:CDD |
405 | 90 | ||||||||
7 |
Template:CDD |
2880 | 360 | ||||||||
8 |
Template:CDD |
1620 | 360 | ||||||||
9 |
Template:CDD |
8820 | 840 | ||||||||
10 |
Template:CDD |
10080 | 1260 | ||||||||
11 |
Template:CDD |
3780 | 840 | ||||||||
12 |
Template:CDD |
15120 | 1260 | ||||||||
13 |
Template:CDD |
26460 | 2520 | ||||||||
14 |
Template:CDD |
20160 | 2520 | ||||||||
15 |
Template:CDD |
5670 | 1260 | ||||||||
16 |
Template:CDD |
15750 | 1260 | ||||||||
17 |
Template:CDD |
37800 | 3150 | ||||||||
18 |
Template:CDD |
44100 | 4200 | ||||||||
19 |
Template:CDD |
25200 | 3150 | ||||||||
20 |
Template:CDD |
10080 | 840 | ||||||||
21 |
Template:CDD |
31500 | 2520 | ||||||||
22 |
Template:CDD |
50400 | 4200 | ||||||||
23 |
Template:CDD |
3780 | 360 | ||||||||
24 |
Template:CDD |
15120 | 1260 | ||||||||
25 |
Template:CDD |
720 | 90 | ||||||||
26 |
Template:CDD |
3240 | 720 | ||||||||
27 |
Template:CDD |
18900 | 2520 | ||||||||
28 |
Template:CDD |
12600 | 2520 | ||||||||
29 |
Template:CDD |
11340 | 2520 | ||||||||
30 |
Template:CDD |
47880 | 5040 | ||||||||
31 |
Template:CDD |
60480 | 7560 | ||||||||
32 |
Template:CDD |
52920 | 7560 | ||||||||
33 |
Template:CDD |
27720 | 5040 | ||||||||
34 |
Template:CDD |
41580 | 7560 | ||||||||
35 |
Template:CDD |
22680 | 5040 | ||||||||
36 |
Template:CDD |
66150 | 6300 | ||||||||
37 |
Template:CDD |
126000 | 12600 | ||||||||
38 |
Template:CDD |
107100 | 12600 | ||||||||
39 |
Template:CDD |
107100 | 12600 | ||||||||
40 |
Template:CDD |
151200 | 18900 | ||||||||
41 |
Template:CDD |
81900 | 12600 | ||||||||
42 |
Template:CDD |
37800 | 6300 | ||||||||
43 |
Template:CDD |
81900 | 12600 | ||||||||
44 |
Template:CDD |
75600 | 12600 | ||||||||
45 |
Template:CDD |
28350 | 6300 | ||||||||
46 |
Template:CDD |
52920 | 5040 | ||||||||
47 |
Template:CDD |
138600 | 12600 | ||||||||
48 |
Template:CDD |
113400 | 12600 | ||||||||
49 |
Template:CDD |
176400 | 16800 | ||||||||
50 |
Template:CDD |
239400 | 25200 | ||||||||
51 |
Template:CDD |
126000 | 16800 | ||||||||
52 |
Template:CDD |
113400 | 12600 | ||||||||
53 |
Template:CDD |
226800 | 25200 | ||||||||
54 |
Template:CDD |
201600 | 25200 | ||||||||
55 |
Template:CDD |
32760 | 5040 | ||||||||
56 |
Template:CDD |
94500 | 12600 | ||||||||
57 |
Template:CDD |
23940 | 2520 | ||||||||
58 |
Template:CDD |
83160 | 7560 | ||||||||
59 |
Template:CDD |
64260 | 7560 | ||||||||
60 |
Template:CDD |
144900 | 12600 | ||||||||
61 |
Template:CDD |
189000 | 18900 | ||||||||
62 |
Template:CDD |
138600 | 12600 | ||||||||
63 |
Template:CDD |
264600 | 25200 | ||||||||
64 |
Template:CDD |
71820 | 7560 | ||||||||
65 |
Template:CDD |
17640 | 2520 | ||||||||
66 |
Template:CDD |
5400 | 720 | ||||||||
67 |
Template:CDD |
25200 | 2520 | ||||||||
68 |
Template:CDD |
57960 | 5040 | ||||||||
69 |
Template:CDD |
75600 | 6300 | ||||||||
70 |
Template:CDD |
22680 | 5040 | ||||||||
71 |
Template:CDD |
105840 | 15120 | ||||||||
72 |
Template:CDD |
75600 | 15120 | ||||||||
73 |
Template:CDD |
75600 | 15120 | ||||||||
74 |
Template:CDD |
68040 | 15120 | ||||||||
75 |
Template:CDD |
214200 | 25200 | ||||||||
76 |
Template:CDD |
283500 | 37800 | ||||||||
77 |
Template:CDD |
264600 | 37800 | ||||||||
78 |
Template:CDD |
245700 | 37800 | ||||||||
79 |
Template:CDD |
138600 | 25200 | ||||||||
80 |
Template:CDD |
226800 | 37800 | ||||||||
81 |
Template:CDD |
189000 | 37800 | ||||||||
82 |
Template:CDD |
138600 | 25200 | ||||||||
83 |
Template:CDD |
207900 | 37800 | ||||||||
84 |
Template:CDD |
113400 | 25200 | ||||||||
85 |
Template:CDD |
226800 | 25200 | ||||||||
86 |
Template:CDD |
453600 | 50400 | ||||||||
87 |
Template:CDD |
403200 | 50400 | ||||||||
88 |
Template:CDD |
378000 | 50400 | ||||||||
89 |
Template:CDD |
403200 | 50400 | ||||||||
90 |
Template:CDD |
604800 | 75600 | ||||||||
91 |
Template:CDD |
529200 | 75600 | ||||||||
92 |
Template:CDD |
352800 | 50400 | ||||||||
93 |
Template:CDD |
529200 | 75600 | ||||||||
94 |
Template:CDD |
302400 | 50400 | ||||||||
95 |
Template:CDD |
151200 | 25200 | ||||||||
96 |
Template:CDD |
352800 | 50400 | ||||||||
97 |
Template:CDD |
277200 | 50400 | ||||||||
98 |
Template:CDD |
352800 | 50400 | ||||||||
99 |
Template:CDD |
491400 | 75600 | ||||||||
100 |
Template:CDD |
252000 | 50400 | ||||||||
101 |
Template:CDD |
151200 | 25200 | ||||||||
102 |
Template:CDD |
327600 | 50400 | ||||||||
103 |
Template:CDD |
128520 | 15120 | ||||||||
104 |
Template:CDD |
359100 | 37800 | ||||||||
105 |
Template:CDD |
302400 | 37800 | ||||||||
106 |
Template:CDD |
283500 | 37800 | ||||||||
107 |
Template:CDD |
478800 | 50400 | ||||||||
108 |
Template:CDD |
680400 | 75600 | ||||||||
109 |
Template:CDD |
604800 | 75600 | ||||||||
110 |
Template:CDD |
378000 | 50400 | ||||||||
111 |
Template:CDD |
567000 | 75600 | ||||||||
112 |
Template:CDD |
321300 | 37800 | ||||||||
113 |
Template:CDD |
680400 | 75600 | ||||||||
114 |
Template:CDD |
567000 | 75600 | ||||||||
115 |
Template:CDD |
642600 | 75600 | ||||||||
116 |
Template:CDD |
907200 | 113400 | ||||||||
117 |
Template:CDD |
264600 | 37800 | ||||||||
118 |
Template:CDD |
98280 | 15120 | ||||||||
119 |
Template:CDD |
302400 | 37800 | ||||||||
120 |
Template:CDD |
226800 | 37800 | ||||||||
121 |
Template:CDD |
428400 | 50400 | ||||||||
122 |
Template:CDD |
302400 | 37800 | ||||||||
123 |
Template:CDD |
98280 | 15120 | ||||||||
124 |
Template:CDD |
35280 | 5040 | ||||||||
125 |
Template:CDD |
136080 | 15120 | ||||||||
126 |
Template:CDD |
105840 | 15120 | ||||||||
127 |
Template:CDD |
252000 | 25200 | ||||||||
128 |
Template:CDD |
340200 | 37800 | ||||||||
129 |
Template:CDD |
176400 | 25200 | ||||||||
130 |
Template:CDD |
252000 | 25200 | ||||||||
131 |
Template:CDD |
504000 | 50400 | ||||||||
132 |
Template:CDD |
453600 | 50400 | ||||||||
133 |
Template:CDD |
136080 | 15120 | ||||||||
134 |
Template:CDD |
378000 | 37800 | ||||||||
135 |
Template:CDD |
35280 | 5040 | ||||||||
136 |
Template:CDD |
136080 | 30240 | ||||||||
137 |
Template:CDD |
491400 | 75600 | ||||||||
138 |
Template:CDD |
378000 | 75600 | ||||||||
139 |
Template:CDD |
378000 | 75600 | ||||||||
140 |
Template:CDD |
378000 | 75600 | ||||||||
141 |
Template:CDD |
340200 | 75600 | ||||||||
142 |
Template:CDD |
756000 | 100800 | ||||||||
143 |
Template:CDD |
1058400 | 151200 | ||||||||
144 |
Template:CDD |
982800 | 151200 | ||||||||
145 |
Template:CDD |
982800 | 151200 | ||||||||
146 |
Template:CDD |
907200 | 151200 | ||||||||
147 |
Template:CDD |
554400 | 100800 | ||||||||
148 |
Template:CDD |
907200 | 151200 | ||||||||
149 |
Template:CDD |
831600 | 151200 | ||||||||
150 |
Template:CDD |
756000 | 151200 | ||||||||
151 |
Template:CDD |
554400 | 100800 | ||||||||
152 |
Template:CDD |
907200 | 151200 | ||||||||
153 |
Template:CDD |
756000 | 151200 | ||||||||
154 |
Template:CDD |
554400 | 100800 | ||||||||
155 |
Template:CDD |
831600 | 151200 | ||||||||
156 |
Template:CDD |
453600 | 100800 | ||||||||
157 |
Template:CDD |
567000 | 75600 | ||||||||
158 |
Template:CDD |
1209600 | 151200 | ||||||||
159 |
Template:CDD |
1058400 | 151200 | ||||||||
160 |
Template:CDD |
1058400 | 151200 | ||||||||
161 |
Template:CDD |
982800 | 151200 | ||||||||
162 |
Template:CDD |
1134000 | 151200 | ||||||||
163 |
Template:CDD |
1701000 | 226800 | ||||||||
164 |
Template:CDD |
1587600 | 226800 | ||||||||
165 |
Template:CDD |
1474200 | 226800 | ||||||||
166 |
Template:CDD |
982800 | 151200 | ||||||||
167 |
Template:CDD |
1587600 | 226800 | ||||||||
168 |
Template:CDD |
1360800 | 226800 | ||||||||
169 |
Template:CDD |
982800 | 151200 | ||||||||
170 |
Template:CDD |
1474200 | 226800 | ||||||||
171 |
Template:CDD |
453600 | 75600 | ||||||||
172 |
Template:CDD |
1058400 | 151200 | ||||||||
173 |
Template:CDD |
907200 | 151200 | ||||||||
174 |
Template:CDD |
831600 | 151200 | ||||||||
175 |
Template:CDD |
1058400 | 151200 | ||||||||
176 |
Template:CDD |
1587600 | 226800 | ||||||||
177 |
Template:CDD |
1360800 | 226800 | ||||||||
178 |
Template:CDD |
907200 | 151200 | ||||||||
179 |
Template:CDD |
453600 | 75600 | ||||||||
180 |
Template:CDD |
1058400 | 151200 | ||||||||
181 |
Template:CDD |
1058400 | 151200 | ||||||||
182 |
Template:CDD |
453600 | 75600 | ||||||||
183 |
Template:CDD |
196560 | 30240 | ||||||||
184 |
Template:CDD |
604800 | 75600 | ||||||||
185 |
Template:CDD |
491400 | 75600 | ||||||||
186 |
Template:CDD |
491400 | 75600 | ||||||||
187 |
Template:CDD |
856800 | 100800 | ||||||||
188 |
Template:CDD |
1209600 | 151200 | ||||||||
189 |
Template:CDD |
1134000 | 151200 | ||||||||
190 |
Template:CDD |
655200 | 100800 | ||||||||
191 |
Template:CDD |
1058400 | 151200 | ||||||||
192 |
Template:CDD |
655200 | 100800 | ||||||||
193 |
Template:CDD |
604800 | 75600 | ||||||||
194 |
Template:CDD |
1285200 | 151200 | ||||||||
195 |
Template:CDD |
1134000 | 151200 | ||||||||
196 |
Template:CDD |
1209600 | 151200 | ||||||||
197 |
Template:CDD |
1814400 | 226800 | ||||||||
198 |
Template:CDD |
491400 | 75600 | ||||||||
199 |
Template:CDD |
196560 | 30240 | ||||||||
200 |
Template:CDD |
604800 | 75600 | ||||||||
201 |
Template:CDD |
856800 | 100800 | ||||||||
202 |
Template:CDD |
680400 | 151200 | ||||||||
203 |
Template:CDD |
1814400 | 302400 | ||||||||
204 |
Template:CDD |
1512000 | 302400 | ||||||||
205 |
Template:CDD |
1512000 | 302400 | ||||||||
206 |
Template:CDD |
1512000 | 302400 | ||||||||
207 |
Template:CDD |
1512000 | 302400 | ||||||||
208 |
Template:CDD |
1360800 | 302400 | ||||||||
209 |
Template:CDD |
1965600 | 302400 | ||||||||
210 |
Template:CDD |
2948400 | 453600 | ||||||||
211 |
Template:CDD |
2721600 | 453600 | ||||||||
212 |
Template:CDD |
2721600 | 453600 | ||||||||
213 |
Template:CDD |
2721600 | 453600 | ||||||||
214 |
Template:CDD |
2494800 | 453600 | ||||||||
215 |
Template:CDD |
1663200 | 302400 | ||||||||
216 |
Template:CDD |
2721600 | 453600 | ||||||||
217 |
Template:CDD |
2494800 | 453600 | ||||||||
218 |
Template:CDD |
2494800 | 453600 | ||||||||
219 |
Template:CDD |
2268000 | 453600 | ||||||||
220 |
Template:CDD |
1663200 | 302400 | ||||||||
221 |
Template:CDD |
2721600 | 453600 | ||||||||
222 |
Template:CDD |
2494800 | 453600 | ||||||||
223 |
Template:CDD |
2268000 | 453600 | ||||||||
224 |
Template:CDD |
1663200 | 302400 | ||||||||
225 |
Template:CDD |
2721600 | 453600 | ||||||||
226 |
Template:CDD |
1663200 | 302400 | ||||||||
227 |
Template:CDD |
907200 | 151200 | ||||||||
228 |
Template:CDD |
2116800 | 302400 | ||||||||
229 |
Template:CDD |
1814400 | 302400 | ||||||||
230 |
Template:CDD |
1814400 | 302400 | ||||||||
231 |
Template:CDD |
1814400 | 302400 | ||||||||
232 |
Template:CDD |
2116800 | 302400 | ||||||||
233 |
Template:CDD |
3175200 | 453600 | ||||||||
234 |
Template:CDD |
2948400 | 453600 | ||||||||
235 |
Template:CDD |
2948400 | 453600 | ||||||||
236 |
Template:CDD |
1814400 | 302400 | ||||||||
237 |
Template:CDD |
2948400 | 453600 | ||||||||
238 |
Template:CDD |
2721600 | 453600 | ||||||||
239 |
Template:CDD |
1814400 | 302400 | ||||||||
240 |
Template:CDD |
907200 | 151200 | ||||||||
241 |
Template:CDD |
2116800 | 302400 | ||||||||
242 |
Template:CDD |
1814400 | 302400 | ||||||||
243 |
Template:CDD |
2116800 | 302400 | ||||||||
244 |
Template:CDD |
3175200 | 453600 | ||||||||
245 |
Template:CDD |
907200 | 151200 | ||||||||
246 |
Template:CDD |
2721600 | 604800 | ||||||||
247 |
Template:CDD |
4989600 | 907200 | ||||||||
248 |
Template:CDD |
4536000 | 907200 | ||||||||
249 |
Template:CDD |
4536000 | 907200 | ||||||||
250 |
Template:CDD |
4536000 | 907200 | ||||||||
251 |
Template:CDD |
4536000 | 907200 | ||||||||
252 |
Template:CDD |
4536000 | 907200 | ||||||||
253 |
Template:CDD |
4082400 | 907200 | ||||||||
254 |
Template:CDD |
3326400 | 604800 | ||||||||
255 |
Template:CDD |
5443200 | 907200 | ||||||||
256 |
Template:CDD |
4989600 | 907200 | ||||||||
257 |
Template:CDD |
4989600 | 907200 | ||||||||
258 |
Template:CDD |
4989600 | 907200 | ||||||||
259 |
Template:CDD |
4989600 | 907200 | ||||||||
260 |
Template:CDD |
3326400 | 604800 | ||||||||
261 |
Template:CDD |
5443200 | 907200 | ||||||||
262 |
Template:CDD |
4989600 | 907200 | ||||||||
263 |
Template:CDD |
4989600 | 907200 | ||||||||
264 |
Template:CDD |
3326400 | 604800 | ||||||||
265 |
Template:CDD |
5443200 | 907200 | ||||||||
266 |
Template:CDD |
8164800 | 1814400 | ||||||||
267 |
Template:CDD |
9072000 | 1814400 | ||||||||
268 |
Template:CDD |
9072000 | 1814400 | ||||||||
269 |
Template:CDD |
9072000 | 1814400 | ||||||||
270 |
Template:CDD |
9072000 | 1814400 | ||||||||
271 |
Template:CDD |
16329600 | 3628800 |
The B9 family
There are 511 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings.
Eleven cases are shown below: Nine rectified forms and 2 truncations. Bowers-style acronym names are given in parentheses for cross-referencing. Bowers-style acronym names are given in parentheses for cross-referencing.
# | Graph | Coxeter-Dynkin diagram Schläfli symbol Name |
Element counts | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
8-faces | 7-faces | 6-faces | 5-faces | 4-faces | Cells | Faces | Edges | Vertices | ||||
1 | Template:CDD t0{4,3,3,3,3,3,3,3} 9-cube (enne) |
18 | 144 | 672 | 2016 | 4032 | 5376 | 4608 | 2304 | 512 | ||
2 | Template:CDD t0,1{4,3,3,3,3,3,3,3} Truncated 9-cube (ten) |
2304 | 4608 | |||||||||
3 | Template:CDD t1{4,3,3,3,3,3,3,3} Rectified 9-cube (ren) |
18432 | 2304 | |||||||||
4 | Template:CDD t2{4,3,3,3,3,3,3,3} Birectified 9-cube (barn) |
64512 | 4608 | |||||||||
5 | Template:CDD t3{4,3,3,3,3,3,3,3} Trirectified 9-cube (tarn) |
96768 | 5376 | |||||||||
6 | Template:CDD t4{4,3,3,3,3,3,3,3} Quadrirectified 9-cube (nav) (Quadrirectified 9-orthoplex) |
80640 | 4032 | |||||||||
7 | Template:CDD t3{3,3,3,3,3,3,3,4} Trirectified 9-orthoplex (tarv) |
40320 | 2016 | |||||||||
8 | Template:CDD t2{3,3,3,3,3,3,3,4} Birectified 9-orthoplex (brav) |
12096 | 672 | |||||||||
9 | Template:CDD t1{3,3,3,3,3,3,3,4} Rectified 9-orthoplex (riv) |
2016 | 144 | |||||||||
10 | Template:CDD t0,1{3,3,3,3,3,3,3,4} Truncated 9-orthoplex (tiv) |
2160 | 288 | |||||||||
11 | Template:CDD t0{3,3,3,3,3,3,3,4} 9-orthoplex (vee) |
512 | 2304 | 4608 | 5376 | 4032 | 2016 | 672 | 144 | 18 |
The D9 family
The D9 family has symmetry of order 92,897,280 (9 factorial × 28).
This family has 3×128−1=383 Wythoffian uniform polytopes, generated by marking one or more nodes of the D9 Coxeter-Dynkin diagram. Of these, 255 (2×128−1) are repeated from the B9 family and 128 are unique to this family, with the eight 1 or 2 ringed forms listed below. Bowers-style acronym names are given in parentheses for cross-referencing.
# | Coxeter plane graphs | Coxeter-Dynkin diagram Schläfli symbol |
Base point (Alternately signed) |
Element counts | Circumrad | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
B9 | D9 | D8 | D7 | D6 | D5 | D4 | D3 | A7 | A5 | A3 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 | ||||
1 | Template:CDD 9-demicube (henne) |
(1,1,1,1,1,1,1,1,1) | 274 | 2448 | 9888 | 23520 | 36288 | 37632 | 21404 | 4608 | 256 | 1.0606601 | |||||||||||
2 | Template:CDD Truncated 9-demicube (thenne) |
(1,1,3,3,3,3,3,3,3) | 69120 | 9216 | 2.8504384 | ||||||||||||||||||
3 | Template:CDD Cantellated 9-demicube |
(1,1,1,3,3,3,3,3,3) | 225792 | 21504 | 2.6692696 | ||||||||||||||||||
4 | Template:CDD Runcinated 9-demicube |
(1,1,1,1,3,3,3,3,3) | 419328 | 32256 | 2.4748735 | ||||||||||||||||||
5 | Template:CDD Stericated 9-demicube |
(1,1,1,1,1,3,3,3,3) | 483840 | 32256 | 2.2638462 | ||||||||||||||||||
6 | Template:CDD Pentellated 9-demicube |
(1,1,1,1,1,1,3,3,3) | 354816 | 21504 | 2.0310094 | ||||||||||||||||||
7 | Template:CDD Hexicated 9-demicube |
(1,1,1,1,1,1,1,3,3) | 161280 | 9216 | 1.7677668 | ||||||||||||||||||
8 | Template:CDD Heptellated 9-demicube |
(1,1,1,1,1,1,1,1,3) | 41472 | 2304 | 1.4577379 |
Regular and uniform honeycombs
There are five fundamental affine Coxeter groups that generate regular and uniform tessellations in 8-space:
# | Coxeter group | Coxeter diagram | Forms | |
---|---|---|---|---|
1 | [3[9]] | Template:CDD | 45 | |
2 | [4,36,4] | Template:CDD | 271 | |
3 | h[4,36,4] [4,35,31,1] |
Template:CDD | 383 (128 new) | |
4 | q[4,36,4] [31,1,34,31,1] |
Template:CDD | 155 (15 new) | |
5 | [35,2,1] | Template:CDD | 511 |
Regular and uniform tessellations include:
- 45 uniquely ringed forms
- 8-simplex honeycomb: {3[9]} Template:CDD
- 271 uniquely ringed forms
- Regular 8-cube honeycomb: {4,36,4}, Template:CDD
- : 383 uniquely ringed forms, 255 shared with , 128 new
- 8-demicube honeycomb: h{4,36,4} or {31,1,35,4}, Template:CDD or Template:CDD
- , [31,1,34,31,1]: 155 unique ring permutations, and 15 are new, the first, Template:CDD, Coxeter called a quarter 8-cubic honeycomb, representing as q{4,36,4}, or qδ9.
- 511 forms
Regular and uniform hyperbolic honeycombs
There are no compact hyperbolic Coxeter groups of rank 9, groups that can generate honeycombs with all finite facets, and a finite vertex figure. However there are 4 noncompact hyperbolic Coxeter groups of rank 9, each generating uniform honeycombs in 8-space as permutations of rings of the Coxeter diagrams.
= [3,3[8]]: Template:CDD |
= [31,1,33,32,1]: Template:CDD |
= [4,34,32,1]: Template:CDD |
= [34,3,1]: Template:CDD |
References
- T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
- A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
- H.S.M. Coxeter:
- H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
- Template:KlitzingPolytopes
External links
- Polytope names
- Polytopes of Various Dimensions, Jonathan Bowers
- Multi-dimensional Glossary
- Template:PolyCell
Template:Polytopes
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