Sistem kristal
Sistem kristal dalam ruang empat dimensi

Sel satuan empat dimensi didefinisikan oleh empat sisi panjang (a, b, c, d) dan enam sudut interaksial (α, β, γ, δ, ε, ζ). Kondisi berikut untuk parameter kisi menentukan 23 sistem kristal:
| No>  | Sistem kristal | Panjang tepi | Sudut interaksial |
| --- | --- | --- | --- |
| 1 | Heksaklinik | a ≠ b ≠ c ≠ d | α ≠ β ≠ γ ≠ δ ≠ ε ≠ ζ ≠ 90° |
| 2 | Triklinik | a ≠ b ≠ c ≠ d | α ≠ β ≠ γ ≠ 90° δ = ε = ζ = 90° |
| 3 | Diklinik | a ≠ b ≠ c ≠ d | α ≠ 90° β = γ = δ = ε = 90° ζ ≠ 90° |
| 4 | Monoklinik | a ≠ b ≠ c ≠ d | α ≠ 90° β = γ = δ = ε = ζ = 90° |
| 5 | Ortogonal | a ≠ b ≠ c ≠ d | α = β = γ = δ = ε = ζ = 90° |
| 6 | Tetragonal monoklinik | a ≠ b = c ≠ d | α ≠ 90° β = γ = δ = ε = ζ = 90° |
| 7 | Heksagonal monoklinik | a ≠ b = c ≠ d | α ≠ 90° β = γ = δ = ε = 90° ζ = 120° |
| 8 | Ditetragonal diklinik | a = d ≠ b = c | α = ζ = 90° β = ε ≠ 90° γ ≠ 90° δ = 180° − γ |
| 9 | Ditrigonal (diheksagonal) diklinik | a = d ≠ b = c | α = ζ = 120° β = ε ≠ 90° γ ≠ δ ≠ 90° cos δ = cos β − cos γ |
| 10 | Tetragonal ortogonal | a ≠ b = c ≠ d | α = β = γ = δ = ε = ζ = 90° |
| 11 | Heksagonal ortogonal | a ≠ b = c ≠ d | α = β = γ = δ = ε = 90°, ζ = 120° |
| 12 | Ditetragonal monoklinik | a = d ≠ b = c | α = γ = δ = ζ = 90° β = ε ≠ 90° |
| 13 | Ditrigonal (diheksagonal) monoklinik | a = d ≠ b = c | α = ζ = 120° β = ε ≠ 90° γ = δ ≠ 90° cos γ = −12cos β |
| 14 | Ditetragonal ortogonal | a = d ≠ b = c | α = β = γ = δ = ε = ζ = 90° |
| 15 | Heksagonal tetragonal | a = d ≠ b = c | α = β = γ = δ = ε = 90° ζ = 120° |
| 16 | Diheksagonal ortogonal | a = d ≠ b = c | α = ζ = 120° β = γ = δ = ε = 90° |
| 17 | Kubik ortogonal | a = b = c ≠ d | α = β = γ = δ = ε = ζ = 90° |
| 18 | Oktagonal | a = b = c = d | α = γ = ζ ≠ 90° β = ε = 90° δ = 180° − α |
| 19 | Dekagonal | a = b = c = d | α = γ = ζ ≠ β = δ = ε cos β = −12 − cos α |
| 20 | Dodekagonal | a = b = c = d | α = ζ = 90° β = ε = 120° γ = δ ≠ 90° |
| 21 | Diisoheksagonal ortogonal | a = b = c = d | α = ζ = 120° β = γ = δ = ε = 90° |
| 22 | Ikosagonal (ikosahedral) | a = b = c = d | α = β = γ = δ = ε = ζ cos α = −14 |
| 23 | Hiperkubik | a = b = c = d | α = β = γ = δ = ε = ζ = 90° |
Nama-nama tersebut diberikan menurut Whittaker. Mereka hampir sama seperti dalam Brown et al, dengan pengecualian untuk nama keluarga kristal 9, 13, dan 22. Nama untuk ketiga keluarga ini menurut Brown et al Diberikan dalam kurung.
Hubungan antara keluarga kristal empat dimensi, sistem kristal, dan sistem kisi ditunjukkan pada tabel berikut. Sistem enansiomorfis ditandai dengan tanda bintang. Jumlah pasangan enansiomorfis diberikan dalam tanda kurung. Disini istilah "enansiomorfis" memiliki arti yang berbeda daripada tabel untuk kelas kristal tiga dimensi. Yang terakhir berarti, bahwa kelompok titik enansiomorfis menggambarkan struktur kiral (enansiomorfis). Pada tabel saat ini, "enansiomorfis" berarti bahwa kelompok itu sendiri (dianggap sebagai objek geometris) adalah enansiomorfis, seperti pasangan enansiomorfis grup ruang tiga dimensi. P31 dan P32, P4122 dan P4322. Dimulai dari ruang empat dimensi, grup titik juga dapat enansiomorfis dalam pengertian ini.
| No>  keluarga kristal | Keluarga kristal | Sistem kristal | No>  sistem kristal | Grup titik | Grup ruang | Kisi Bravais | Sistem kisi |
| --- | --- | --- | --- | --- | --- | --- | --- |
| I | Heksaklinik | Heksaklinik | 1 | 2 | 2 | 1 | Heksaklinik P |
| II | Triklinik | Triklinik | 2 | 3 | 13 | 2 | Triklinik P, S |
| III | Diklinik | Diklinik | 3 | 2 | 12 | 3 | Diklinik P, S, D |
| IV | Monoklinik | Monoklinik | 4 | 4 | 207 | 6 | Monoklinik P, S, S, I, D, F |
| V | Ortogonal | Non-aksial ortogonal | 5 | 2 | 2 | 1 | Ortogonal KU |
| V | Ortogonal | Non-aksial ortogonal | 5 | 2 | 112 | 8 | Ortogonal P, S, I, Z, D, F, G, U |
| V | Ortogonal | Aksial ortogonal | 6 | 3 | 887 | 8 | Ortogonal P, S, I, Z, D, F, G, U |
| VI | Tetragonal monoklinik | Tetragonal monoklinik | 7 | 7 | 88 | 2 | Tetragonal monoklinik P, I |
| VII | Heksagonal monoklinik | Trigonal monoklinik | 8 | 5 | 9 | 1 | Heksagonal monoklinik R |
| VII | Heksagonal monoklinik | Trigonal monoklinik | 8 | 5 | 15 | 1 | Heksagonal monoklinik P |
| VII | Heksagonal monoklinik | Heksagonal monoklinik | 9 | 7 | 25 | 1 | Heksagonal monoklinik P |
| VIII | Ditetragonal diklinik* | Ditetragonal diklinik* | 10 | 1 (+1) | 1 (+1) | 1 (+1) | Ditetragonal diklinik P* |
| IX | Ditrigonal diklinik* | Ditrigonal diklinik* | 11 | 2 (+2) | 2 (+2) | 1 (+1) | Ditrigonal diklinik P* |
| X | Tetragonal ortogonal | Invers tetragonal ortogonal | 12 | 5 | 7 | 1 | Tetragonal ortogonal KG |
| X | Tetragonal ortogonal | Invers tetragonal ortogonal | 12 | 5 | 351 | 5 | Tetragonal ortogonal P, S, I, Z, G |
| X | Tetragonal ortogonal | Proper tetragonal ortogonal | 13 | 10 | 1312 | 5 | Tetragonal ortogonal P, S, I, Z, G |
| XI | Heksagonal ortogonal | Trigonal ortogonal | 14 | 10 | 81 | 2 | Heksagonal ortogonal R, RS |
| XI | Heksagonal ortogonal | Trigonal ortogonal | 14 | 10 | 150 | 2 | Heksagonal ortogonal P, S |
| XI | Heksagonal ortogonal | Heksagonal ortogonal | 15 | 12 | 240 | 2 | Heksagonal ortogonal P, S |
| XII | Ditetragonal monoklinik* | Ditetragonal monoklinik* | 16 | 1 (+1) | 6 (+6) | 3 (+3) | Ditetragonal monoklinik P*, S*, D* |
| XIII | Ditrigonal monoklinik* | Ditrigonal monoklinik* | 17 | 2 (+2) | 5 (+5) | 2 (+2) | Ditrigonal monoklinik P*, RR* |
| XIV | Ditetragonal ortogonal | Kripto-ditetragonal ortogonal | 18 | 5 | 10 | 1 | Ditetragonal ortogonal D |
| XIV | Ditetragonal ortogonal | Kripto-ditetragonal ortogonal | 18 | 5 | 165 (+2) | 2 | Ditetragonal ortogonal P, Z |
| XIV | Ditetragonal ortogonal | Ditetragonal ortogonal | 19 | 6 | 127 | 2 | Ditetragonal ortogonal P, Z |
| XV | Heksagonal tetragonal | Heksagonal tetragonal | 20 | 22 | 108 | 1 | Hexagonal tetragonal P |
| XVI | Diheksagonal ortogonal | Kripto-ditrigonal ortogonal* | 21 | 4 (+4) | 5 (+5) | 1 (+1) | Diheksagonal ortogonal G* |
| XVI | Diheksagonal ortogonal | Kripto-ditrigonal ortogonal* | 21 | 4 (+4) | 5 (+5) | 1 | Diheksagonal ortogonal P |
| XVI | Diheksagonal ortogonal | Diheksagonal ortogonal | 23 | 11 | 20 | 1 | Diheksagonal ortogonal P |
| XVI | Diheksagonal ortogonal | Ditrigonal ortogonal | 22 | 11 | 41 | 1 | Diheksagonal ortogonal P |
| XVI | Diheksagonal ortogonal | Ditrigonal ortogonal | 22 | 11 | 16 | 1 | Diheksagonal ortogonal RR |
| XVII | Kubik ortogonal | Kubik ortogonal sederhana | 24 | 5 | 9 | 1 | Kubik ortogonal KU |
| XVII | Kubik ortogonal | Kubik ortogonal sederhana | 24 | 5 | 96 | 5 | Kubik ortogonal P, I, Z, F, U |
| XVII | Kubik ortogonal | Kubic ortogonal kompleks | 25 | 11 | 366 | 5 | Kubik ortogonal P, I, Z, F, U |
| XVIII | Oktagonal* | Oktagonal* | 26 | 2 (+2) | 3 (+3) | 1 (+1) | Oktagonal P* |
| XIX | Dekagonal | Dekagonal | 27 | 4 | 5 | 1 | Dekagonal P |
| XX | Dodekagonal* | Dodekagonal* | 28 | 2 (+2) | 2 (+2) | 1 (+1) | Dodekagonal P* |
| XXI | Diisoheksagonal ortogonal | Diisoheksagonal ortogonal sederhana | 29 | 9 (+2) | 19 (+5) | 1 | Diisoheksagonal ortogonal RR |
| XXI | Diisoheksagonal ortogonal | Diisoheksagonal ortogonal sederhana | 29 | 9 (+2) | 19 (+3) | 1 | Diisoheksagonal ortogonal P |
| XXI | Diisoheksagonal ortogonal | Diisoheksagonal ortogonal kompleks | 30 | 13 (+8) | 15 (+9) | 1 | Diisoheksagonal ortogonal P |
| XXII | Ikosagonal | Ikosagonal | 31 | 7 | 20 | 2 | Ikosagonal P, SN |
| XXIII | Hiperkubik | Oktagonal hiperkubik | 32 | 21 (+8) | 73 (+15) | 1 | Hiperkubik P |
| XXIII | Hiperkubik | Oktagonal hiperkubik | 32 | 21 (+8) | 107 (+28) | 1 | Hiperkubik Z |
| XXIII | Hiperkubik | Dodekagonal hiperkubik | 33 | 16 (+12) | 25 (+20) | 1 | Hiperkubik Z |
| Total | 23 (+6) | 33 (+7) |  | 227 (+44) | 4783 (+111) | 64 (+10) | 33 (+7) |