Eight Dimensions
The eight-dimensional point groups, limiting to purely reflectional groups, can be listed by their Coxeter group. Related pure rotational groups exist for each with half the order, defined by an even number of reflections, and can be represented by the bracket Coxeter notation with a '+' exponent, for example + has seven 3-fold gyration points and symmetry order 181440.
| Coxeter group | Coxeter diagram | Order | Related polytopes | |
|---|---|---|---|---|
| A8 | 362880 (9!) | 8-simplex | ||
| A8×2 | ] | 725760 (2x9!) | 8-simplex dual compound | |
| BC8 | 10321920 (288!) | 8-cube,8-orthoplex | ||
| D8 | 5160960 (278!) | 8-demicube | ||
| E8 | 696729600 | 421, 241, 142 | ||
| A7×A1 | 80640 | 7-simplex prism | ||
| BC7×A1 | 645120 | 7-cube prism | ||
| D7×A1 | 322560 | 7-demicube prism | ||
| E7×A1 | 5806080 | 321 prism, 231 prism, 142 prism | ||
| A6×I2(p) | 10080p | duoprism | ||
| BC6×I2(p) | 92160p | |||
| D6×I2(p) | 46080p | |||
| E6×I2(p) | 103680p | |||
| A6×A12 | 20160 | |||
| BC6×A12 | 184320 | |||
| D6×A12 | 92160 | |||
| E6×A12 | 207360 | |||
| A5×A3 | 17280 | |||
| BC5×A3 | 92160 | |||
| D5×A3 | 46080 | |||
| A5×BC3 | 34560 | |||
| BC5×BC3 | 184320 | |||
| D5×BC3 | 92160 | |||
| A5×H3 | ||||
| BC5×H3 | ||||
| D5×H3 | ||||
| A5×I2(p)×A1 | ||||
| BC5×I2(p)×A1 | ||||
| D5×I2(p)×A1 | ||||
| A5×A13 | ||||
| BC5×A13 | ||||
| D5×A13 | ||||
| A4×A4 | ||||
| BC4×A4 | ||||
| D4×A4 | ||||
| F4×A4 | ||||
| H4×A4 | ||||
| BC4×BC4 | ||||
| D4×BC4 | ||||
| F4×BC4 | ||||
| H4×BC4 | ||||
| D4×D4 | ||||
| F4×D4 | ||||
| H4×D4 | ||||
| F4×F4 | ||||
| H4×F4 | ||||
| H4×H4 | ||||
| A4×A3×A1 | duoprism prisms | |||
| A4×BC3×A1 | ||||
| A4×H3×A1 | ||||
| BC4×A3×A1 | ||||
| BC4×BC3×A1 | ||||
| BC4×H3×A1 | ||||
| H4×A3×A1 | ||||
| H4×BC3×A1 | ||||
| H4×H3×A1 | ||||
| F4×A3×A1 | ||||
| F4×BC3×A1 | ||||
| F4×H3×A1 | ||||
| D4×A3×A1 | ||||
| D4×BC3×A1 | ||||
| D4×H3×A1 | ||||
| A4×I2(p)×I2(q) | triaprism | |||
| BC4×I2(p)×I2(q) | ||||
| F4×I2(p)×I2(q) | ||||
| H4×I2(p)×I2(q) | ||||
| D4×I2(p)×I2(q) | ||||
| A4×I2(p)×A12 | ||||
| BC4×I2(p)×A12 | ||||
| F4×I2(p)×A12 | ||||
| H4×I2(p)×A12 | ||||
| D4×I2(p)×A12 | ||||
| A4×A14 | ||||
| BC4×A14 | ||||
| F4×A14 | ||||
| H4×A14 | ||||
| D4×A14 | ||||
| A3×A3×I2(p) | ||||
| BC3×A3×I2(p) | ||||
| H3×A3×I2(p) | ||||
| BC3×BC3×I2(p) | ||||
| H3×BC3×I2(p) | ||||
| H3×H3×I2(p) | ||||
| A3×A3×A12 | ||||
| BC3×A3×A12 | ||||
| H3×A3×A12 | ||||
| BC3×BC3×A12 | ||||
| H3×BC3×A12 | ||||
| H3×H3×A12 | ||||
| A3×I2(p)×I2(q)×A1 | ||||
| BC3×I2(p)×I2(q)×A1 | ||||
| H3×I2(p)×I2(q)×A1 | ||||
| A3×I2(p)×A13 | ||||
| BC3×I2(p)×A13 | ||||
| H3×I2(p)×A13 | ||||
| A3×A15 | ||||
| BC3×A15 | ||||
| H3×A15 | ||||
| I2(p)×I2(q)×I2(r)×I2(s) | 16pqrs | |||
| I2(p)×I2(q)×I2(r)×A12 | 32pqr | |||
| I2(p)×I2(q)×A14 | 64pq | |||
| I2(p)×A16 | 128p | |||
| A18 | 256 | |||
Read more about this topic: Point Group
Famous quotes containing the word dimensions:
“Why is it that many contemporary male thinkers, especially men of color, repudiate the imperialist legacy of Columbus but affirm dimensions of that legacy by their refusal to repudiate patriarchy?”
—bell hooks (b. c. 1955)
“It seems to me that we do not know nearly enough about ourselves; that we do not often enough wonder if our lives, or some events and times in our lives, may not be analogues or metaphors or echoes of evolvements and happenings going on in other people?—or animals?—even forests or oceans or rocks?—in this world of ours or, even, in worlds or dimensions elsewhere.”
—Doris Lessing (b. 1919)