Seven Dimensions
The seven-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 six 3-fold gyration points and symmetry order 20160.
| Coxeter group | Coxeter diagram | Order | Related polytopes | |
|---|---|---|---|---|
| A7 | 40320 (8!) | 7-simplex | ||
| A7×2 | ] | 80640 (2×8!) | 7-simplex dual compound | |
| BC7 | 645120 (27×7!) | 7-cube, 7-orthoplex | ||
| D7 | 322560 (26×7!) | 7-demicube | ||
| E7 | 2903040 (8×9!) | 321, 231, 132 | ||
| A6×A1 | 10080 (2×7!) | |||
| BC6×A1 | 92160 (27×6!) | |||
| D6×A1 | 46080 (26×6!) | |||
| E6×A1 | 103680 (144×6!) | |||
| A5×I2(p) | 1440p | |||
| BC5×I2(p) | 7680p | |||
| D5×I2(p) | 3840p | |||
| A5×A12 | 2880 | |||
| BC5×A12 | 15360 | |||
| D5×A12 | 7680 | |||
| A4×A3 | 2880 | |||
| A4×BC3 | 5760 | |||
| A4×H3 | 14400 | |||
| BC4×A3 | 9216 | |||
| BC4×BC3 | 18432 | |||
| BC4×H3 | 46080 | |||
| H4×A3 | 345600 | |||
| H4×BC3 | 691200 | |||
| H4×H3 | 1728000 | |||
| F4×A3 | 27648 | |||
| F4×BC3 | 55296 | |||
| F4×H3 | 138240 | |||
| D4×A3 | 4608 | |||
| D4×BC3 | 9216 | |||
| D4×H3 | 23040 | |||
| A4×I2(p)×A1 | 480p | |||
| BC4×I2(p)×A1 | 1536p | |||
| D4×I2(p)×A1 | 768p | |||
| F4×I2(p)×A1 | 4608p | |||
| H4×I2(p)×A1 | 57600p | |||
| A4×A13 | 960 | |||
| BC4×A13 | 3072 | |||
| F4×A13 | 9216 | |||
| H4×A13 | 115200 | |||
| D4×A13 | 1536 | |||
| A32×A1 | 1152 | |||
| A3×BC3×A1 | 2304 | |||
| A3×H3×A1 | 5760 | |||
| BC32×A1 | 4608 | |||
| BC3×H3×A1 | 11520 | |||
| H32×A1 | 28800 | |||
| A3×I2(p)×I2(q) | 96pq | |||
| BC3×I2(p)×I2(q) | 192pq | |||
| H3×I2(p)×I2(q) | 480pq | |||
| A3×I2(p)×A12 | 192p | |||
| BC3×I2(p)×A12 | 384p | |||
| H3×I2(p)×A12 | 960p | |||
| A3×A14 | 384 | |||
| BC3×A14 | 768 | |||
| H3×A14 | 1920 | |||
| I2(p)×I2(q)×I2(r)×A1 | 16pqr | |||
| I2(p)×I2(q)×A13 | 32pq | |||
| I2(p)×A15 | 64p | |||
| A17 | 128 | |||
Read more about this topic: Point Group
Famous quotes containing the word dimensions:
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—J.L. (John Langshaw)
“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)