Four Dimensions
The four-dimensional point groups, limiting to purely reflectional groups, can be listed by their Coxeter group, and like the polyhedral groups of 3D, can be named by their related convex regular 4-polytopes. 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 three 3-fold gyration points and symmetry order 60. Front-back symmetric groups like and can be doubled, shown as double brackets in Coxeter's notation, for example ] with its order doubled to 240.
| Coxeter group/notation | Coxeter diagram | Order | Related polytopes | ||
|---|---|---|---|---|---|
| A4 | 120 | 5-cell | |||
| A4×2 | ] | 240 | 5-cell dual compound | ||
| BC4 | 384 | 16-cell/Tesseract | |||
| D4 | 192 | Demitesseractic | |||
| D4×2 = BC4 | <> = | = | 384 | ||
| D4×6 = F4 | ] = | = | 1152 | ||
| F4 | 1152 | 24-cell | |||
| F4×2 | ] | 2304 | 24-cell dual compound | ||
| H4 | 14400 | 120-cell/600-cell | |||
| A3×A1 | 48 | Tetrahedral prism | |||
| A3×A1×2 | ,2] = | = | 96 | Octahedral prism | |
| BC3×A1 | 96 | ||||
| H3×A1 | 240 | Icosahedral prism | |||
| A2×A2 | 36 | Duoprism | |||
| A2×BC2 | 48 | ||||
| A2×H2 | 60 | ||||
| A2×G2 | 72 | ||||
| BC2×BC2 | 64 | ||||
| BC22×2 | ] | 128 | |||
| BC2×H2 | 80 | ||||
| BC2×G2 | 96 | ||||
| H2×H2 | 100 | ||||
| H2×G2 | 120 | ||||
| G2×G2 | 144 | ||||
| I2(p)×I2(q) | 4pq | ||||
| I2(2p)×I2(q) | ,2,q] = | = | 8pq | ||
| I2(2p)×I2(2q) | ],2,] = | = | 16pq | ||
| I2(p)2×2 | ] | 8p2 | |||
| I2(2p)2×2 | ,2,]] = ] | = | 32p2 | ||
| A2×A1×A1 | 24 | ||||
| BC2×A1×A1 | 32 | ||||
| H2×A1×A1 | 40 | ||||
| G2×A1×A1 | 48 | ||||
| I2(p)×A1×A1 | 8p | ||||
| I2(2p)×A1×A1×2 | ,2,2] = | = | 16p | ||
| I2(p)×A12×2 | ] = | = | 16p | ||
| I2(2p)×A12×4 | ],2,] = | = | 32p | ||
| A1×A1×A1×A1 | 16 | 4-orthotope | |||
| A12×A1×A1×2 | ,2,2] = | = | 32 | ||
| A12×A12×4 | ],2,] = | = | 64 | ||
| A13×A1×6 | ,2] = | = | 96 | ||
| A14×24 | ] = | = | 384 | ||
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)
“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)