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:
“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)
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—Thomas Jefferson (1743–1826)