晶系
其它維度 > 四維

‌四維晶胞由四個邊長（a、b、c、d）和六個軸間角（α、β、γ、δ、ε、ζ）定義。以下晶格參數條件定義了23種晶系。
| No>   | 晶系（1985年Whittaker命名） | 邊長 | 軸間角 |
| --- | --- | --- | --- |
| 1 | Hexaclinic | a ≠ b ≠ c ≠ d | α ≠ β ≠ γ ≠ δ ≠ ε ≠ ζ ≠ 90° |
| 2 | Triclinic | a ≠ b ≠ c ≠ d | α ≠ β ≠ γ ≠ 90° δ = ε = ζ = 90° |
| 3 | Diclinic | a ≠ b ≠ c ≠ d | α ≠ 90° β = γ = δ = ε = 90° ζ ≠ 90° |
| 4 | Monoclinic | a ≠ b ≠ c ≠ d | α ≠ 90° β = γ = δ = ε = ζ = 90° |
| 5 | Orthogonal | a ≠ b ≠ c ≠ d | α = β = γ = δ = ε = ζ = 90° |
| 6 | Tetragonal monoclinic | a ≠ b = c ≠ d | α ≠ 90° β = γ = δ = ε = ζ = 90° |
| 7 | Hexagonal monoclinic | a ≠ b = c ≠ d | α ≠ 90° β = γ = δ = ε = 90° ζ = 120° |
| 8 | Ditetragonal diclinic | a = d ≠ b = c | α = ζ = 90° β = ε ≠ 90° γ ≠ 90° δ = 180° − γ |
| 9 | Ditrigonal (dihexagonal) diclinic | a = d ≠ b = c | α = ζ = 120° β = ε ≠ 90° γ ≠ δ ≠ 90° cos δ = cos β − cos γ |
| 10 | Tetragonal orthogonal | a ≠ b = c ≠ d | α = β = γ = δ = ε = ζ = 90° |
| 11 | Hexagonal orthogonal | a ≠ b = c ≠ d | α = β = γ = δ = ε = 90°, ζ = 120° |
| 12 | Ditetragonal monoclinic | a = d ≠ b = c | α = γ = δ = ζ = 90° β = ε ≠ 90° |
| 13 | Ditrigonal (dihexagonal) monoclinic | a = d ≠ b = c | α = ζ = 120° β = ε ≠ 90° γ = δ ≠ 90° cos γ = −1/2cos β |
| 14 | Ditetragonal orthogonal | a = d ≠ b = c | α = β = γ = δ = ε = ζ = 90° |
| 15 | Hexagonal tetragonal | a = d ≠ b = c | α = β = γ = δ = ε = 90° ζ = 120° |
| 16 | Dihexagonal orthogonal | a = d ≠ b = c | α = ζ = 120° β = γ = δ = ε = 90° |
| 17 | Cubic orthogonal | a = b = c ≠ d | α = β = γ = δ = ε = ζ = 90° |
| 18 | Octagonal | a = b = c = d | α = γ = ζ ≠ 90° β = ε = 90° δ = 180° − α |
| 19 | Decagonal | a = b = c = d | α = γ = ζ ≠ β = δ = ε cos β = −1/2 − cos α |
| 20 | Dodecagonal | a = b = c = d | α = ζ = 90° β = ε = 120° γ = δ ≠ 90° |
| 21 | Diisohexagonal orthogonal | a = b = c = d | α = ζ = 120° β = γ = δ = ε = 90° |
| 22 | Icosagonal (icosahedral) | a = b = c = d | α = β = γ = δ = ε = ζ cos α = −1/4 |
| 23 | Hypercubic | a = b = c = d | α = β = γ = δ = ε = ζ = 90° |
由1985年Whittaker命名。
名字幾乎與Brown等人的命名相同，只有9、13、22名稱不同。括號是他們命的名。
四維晶族、晶系、晶格系之間的關係如下表所示。
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Enantiomorphic systems are marked with an asterisk. The number of enantiomorphic pairs is given in parentheses. Here the term "enantiomorphic" has a different meaning than in the table for three-dimensional crystal classes. The latter means, that enantiomorphic point groups describe chiral (enantiomorphic) structures. In the current table, "enantiomorphic" means that a group  itself (considered as a geometric object) is enantiomorphic, like enantiomorphic pairs of three-dimensional space groups P3 and P3, P422 and P422. Starting from four-dimensional space, point groups also can be enantiomorphic in this sense.
| 晶族序 | 晶族（英文） | 晶系（英文） | 晶系序 | 點群 | 空間群 | 布拉菲晶格 | 晶格 |
| --- | --- | --- | --- | --- | --- | --- | --- |
| I | Hexaclinic | Hexaclinic | 1 | 2 | 2 | 1 | Hexaclinic P |
| II | Triclinic | Triclinic | 2 | 3 | 13 | 2 | Triclinic P, S |
| III | Diclinic | Diclinic | 3 | 2 | 12 | 3 | Diclinic P, S, D |
| IV | Monoclinic | Monoclinic | 4 | 4 | 207 | 6 | Monoclinic P, S, S, I, D, F |
| V | Orthogonal | Non-axial orthogonal | 5 | 2 | 2 | 1 | Orthogonal KU |
| V | Orthogonal | Non-axial orthogonal | 5 | 2 | 112 | 8 | Orthogonal P, S, I, Z, D, F, G, U |
| V | Orthogonal | Axial orthogonal | 6 | 3 | 887 | 8 | Orthogonal P, S, I, Z, D, F, G, U |
| VI | Tetragonal monoclinic | Tetragonal monoclinic | 7 | 7 | 88 | 2 | Tetragonal monoclinic P, I |
| VII | Hexagonal monoclinic | Trigonal monoclinic | 8 | 5 | 9 | 1 | Hexagonal monoclinic R |
| VII | Hexagonal monoclinic | Trigonal monoclinic | 8 | 5 | 15 | 1 | Hexagonal monoclinic P |
| VII | Hexagonal monoclinic | Hexagonal monoclinic | 9 | 7 | 25 | 1 | Hexagonal monoclinic P |
| VIII | Ditetragonal diclinic* | Ditetragonal diclinic* | 10 | 1 (+1) | 1 (+1) | 1 (+1) | Ditetragonal diclinic P* |
| IX | Ditrigonal diclinic* | Ditrigonal diclinic* | 11 | 2 (+2) | 2 (+2) | 1 (+1) | Ditrigonal diclinic P* |
| X | Tetragonal orthogonal | Inverse tetragonal orthogonal | 12 | 5 | 7 | 1 | Tetragonal orthogonal KG |
| X | Tetragonal orthogonal | Inverse tetragonal orthogonal | 12 | 5 | 351 | 5 | Tetragonal orthogonal P, S, I, Z, G |
| X | Tetragonal orthogonal | Proper tetragonal orthogonal | 13 | 10 | 1312 | 5 | Tetragonal orthogonal P, S, I, Z, G |
| XI | Hexagonal orthogonal | Trigonal orthogonal | 14 | 10 | 81 | 2 | Hexagonal orthogonal R, RS |
| XI | Hexagonal orthogonal | Trigonal orthogonal | 14 | 10 | 150 | 2 | Hexagonal orthogonal P, S |
| XI | Hexagonal orthogonal | Hexagonal orthogonal | 15 | 12 | 240 | 2 | Hexagonal orthogonal P, S |
| XII | Ditetragonal monoclinic* | Ditetragonal monoclinic* | 16 | 1 (+1) | 6 (+6) | 3 (+3) | Ditetragonal monoclinic P*, S*, D* |
| XIII | Ditrigonal monoclinic* | Ditrigonal monoclinic* | 17 | 2 (+2) | 5 (+5) | 2 (+2) | Ditrigonal monoclinic P*, RR* |
| XIV | Ditetragonal orthogonal | Crypto-ditetragonal orthogonal | 18 | 5 | 10 | 1 | Ditetragonal orthogonal D |
| XIV | Ditetragonal orthogonal | Crypto-ditetragonal orthogonal | 18 | 5 | 165 (+2) | 2 | Ditetragonal orthogonal P, Z |
| XIV | Ditetragonal orthogonal | Ditetragonal orthogonal | 19 | 6 | 127 | 2 | Ditetragonal orthogonal P, Z |
| XV | Hexagonal tetragonal | Hexagonal tetragonal | 20 | 22 | 108 | 1 | Hexagonal tetragonal P |
| XVI | Dihexagonal orthogonal | Crypto-ditrigonal orthogonal* | 21 | 4 (+4) | 5 (+5) | 1 (+1) | Dihexagonal orthogonal G* |
| XVI | Dihexagonal orthogonal | Crypto-ditrigonal orthogonal* | 21 | 4 (+4) | 5 (+5) | 1 | Dihexagonal orthogonal P |
| XVI | Dihexagonal orthogonal | Dihexagonal orthogonal | 23 | 11 | 20 | 1 | Dihexagonal orthogonal P |
| XVI | Dihexagonal orthogonal | Ditrigonal orthogonal | 22 | 11 | 41 | 1 | Dihexagonal orthogonal P |
| XVI | Dihexagonal orthogonal | Ditrigonal orthogonal | 22 | 11 | 16 | 1 | Dihexagonal orthogonal RR |
| XVII | Cubic orthogonal | Simple cubic orthogonal | 24 | 5 | 9 | 1 | Cubic orthogonal KU |
| XVII | Cubic orthogonal | Simple cubic orthogonal | 24 | 5 | 96 | 5 | Cubic orthogonal P, I, Z, F, U |
| XVII | Cubic orthogonal | Complex cubic orthogonal | 25 | 11 | 366 | 5 | Cubic orthogonal P, I, Z, F, U |
| XVIII | Octagonal* | Octagonal* | 26 | 2 (+2) | 3 (+3) | 1 (+1) | Octagonal P* |
| XIX | Decagonal | Decagonal | 27 | 4 | 5 | 1 | Decagonal P |
| XX | Dodecagonal* | Dodecagonal* | 28 | 2 (+2) | 2 (+2) | 1 (+1) | Dodecagonal P* |
| XXI | Diisohexagonal orthogonal | Simple diisohexagonal orthogonal | 29 | 9 (+2) | 19 (+5) | 1 | Diisohexagonal orthogonal RR |
| XXI | Diisohexagonal orthogonal | Simple diisohexagonal orthogonal | 29 | 9 (+2) | 19 (+3) | 1 | Diisohexagonal orthogonal P |
| XXI | Diisohexagonal orthogonal | Complex diisohexagonal orthogonal | 30 | 13 (+8) | 15 (+9) | 1 | Diisohexagonal orthogonal P |
| XXII | Icosagonal | Icosagonal | 31 | 7 | 20 | 2 | Icosagonal P, SN |
| XXIII | Hypercubic | Octagonal hypercubic | 32 | 21 (+8) | 73 (+15) | 1 | Hypercubic P |
| XXIII | Hypercubic | Octagonal hypercubic | 32 | 21 (+8) | 107 (+28) | 1 | Hypercubic Z |
| XXIII | Hypercubic | Dodecagonal hypercubic | 33 | 16 (+12) | 25 (+20) | 1 | Hypercubic Z |
| 共計 | 23 (+6) | 33 (+7) |  | 227 (+44) | 4783 (+111) | 64 (+10) | 33 (+7) |