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:
“Words are finite organs of the infinite mind. They cannot cover the dimensions of what is in truth. They break, chop, and impoverish it.”
—Ralph Waldo Emerson (1803–1882)
“I was surprised by Joe’s asking me how far it was to the Moosehorn. He was pretty well acquainted with this stream, but he had noticed that I was curious about distances, and had several maps. He and Indians generally, with whom I have talked, are not able to describe dimensions or distances in our measures with any accuracy. He could tell, perhaps, at what time we should arrive, but not how far it was.”
—Henry David Thoreau (1817–1862)