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:
“The truth is that a Pigmy and a Patagonian, a Mouse and a Mammoth, derive their dimensions from the same nutritive juices.... [A]ll the manna of heaven would never raise the Mouse to the bulk of the Mammoth.”
—Thomas Jefferson (1743–1826)
“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)