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:
“Is it true or false that Belfast is north of London? That the galaxy is the shape of a fried egg? That Beethoven was a drunkard? That Wellington won the battle of Waterloo? There are various degrees and dimensions of success in making statements: the statements fit the facts always more or less loosely, in different ways on different occasions for different intents and purposes.”
—J.L. (John Langshaw)
“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)