Editing Crystal system (section)

==In other dimensions==
===Two-dimensional space===
In two-dimensional space, there are four crystal systems (oblique, rectangular, square, hexagonal), four crystal families (oblique, rectanguar, square, hexagonal), and four lattice systems ([[oblique lattice|oblique]], [[rectangular lattice|rectangular]], [[square lattice|square]], and [[hexagonal lattice|hexagonal]]).<ref name="Giacovazzo">{{cite book |last1=Giacovazzo |first1=Carmelo |title=Fundamentals of Crystallography |date=10 February 2011 |publisher=Oxford University Press |isbn=978-0-19-957366-0 |edition=3rd}}</ref><ref name="ITA">{{cite book |last1=Hahn |first1=Theo |title=International Tables for Crystallography Volume A: Space-Group Symmetry |date=2005 |publisher=Springer |location=Table 2.1.2.1 |edition=5th}}</ref>

{|class="wikitable" style="margin: 1em auto; text-align: center;"
|-
! Crystal family
! Crystal system
! Crystallographic point groups
! No. of plane groups
! Bravais lattices
|-
! Oblique (monoclinic)
| Oblique
| 1, 2
| 2
| ''mp''
|-
! Rectangular (orthorhombic)
| Rectangular
| ''m'', 2''mm''
| 7
| ''op'', ''oc''
|-
! Square (tetragonal)
| Square
| 4, 4''mm''
| 3
| ''tp''
|-
! Hexagonal
| Hexagonal
| 3, 6, 3''m'', 6''mm''
| 5
| ''hp''
|-
| '''Total'''
| 4
| 10
| 17
| 5
|}

===Four-dimensional space=== 

‌The four-dimensional unit cell is defined by four edge lengths (''a'', ''b'', ''c'', ''d'') and six interaxial angles (''α'', ''β'', ''γ'', ''δ'', ''ε'', ''ζ''). The following conditions for the lattice parameters define 23 crystal families
{| class="wikitable" style="text-align:center"
|+ Crystal families in 4D space
! No.
! Family
! Edge lengths
! Interaxial angles
|-
! 1
| Hexaclinic
| ''a'' ≠ ''b'' ≠ ''c'' ≠ ''d''
| ''α'' ≠ ''β'' ≠ ''γ'' ≠ ''δ'' ≠ ''ε'' ≠ ''ζ'' ≠ 90°
|-
! 2
| Triclinic
| ''a'' ≠ ''b'' ≠ ''c'' ≠ ''d''
| ''α'' ≠ ''β'' ≠ ''γ'' ≠ 90°<br>''δ'' = ''ε'' = ''ζ'' = 90°
|-
! 3
| Diclinic
| ''a'' ≠ ''b'' ≠ ''c'' ≠ ''d''
| ''α'' ≠ 90°<br>''β'' = ''γ'' = ''δ'' = ''ε'' = 90°<br>''ζ'' ≠ 90°
|-
! 4
| Monoclinic
| ''a'' ≠ ''b'' ≠ ''c'' ≠ ''d''
| ''α'' ≠ 90°<br>''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°<br>''β'' = ''γ'' = ''δ'' = ''ε'' = ''ζ'' = 90°
|-
! 7
| Hexagonal monoclinic
| ''a'' ≠ ''b'' = ''c'' ≠ ''d''
| ''α'' ≠ 90°<br>''β'' = ''γ'' = ''δ'' = ''ε'' = 90°<br>''ζ'' = 120°
|-
! 8
| Ditetragonal diclinic
| ''a'' = ''d'' ≠ ''b'' = ''c''
| ''α'' = ''ζ'' = 90°<br>''β'' = ''ε'' ≠ 90°<br>''γ'' ≠ 90°<br>''δ'' = 180° − ''γ''
|-
! 9
| Ditrigonal (dihexagonal) diclinic
| ''a'' = ''d'' ≠ ''b'' = ''c''
| ''α'' = ''ζ'' = 120°<br>''β'' = ''ε'' ≠ 90°<br>''γ'' ≠ ''δ'' ≠ 90°<br>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°<br>''β'' = ''ε'' ≠ 90°
|-
! 13
| Ditrigonal (dihexagonal) monoclinic
| ''a'' = ''d'' ≠ ''b'' = ''c''
| ''α'' = ''ζ'' = 120°<br>''β'' = ''ε'' ≠ 90°<br>''γ'' = ''δ'' ≠ 90°<br>cos ''γ'' = −{{sfrac|1|2}}cos ''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°<br>''ζ'' = 120°
|-
! 16
| Dihexagonal orthogonal
| ''a'' = ''d'' ≠ ''b'' = ''c''
| ''α'' = ''ζ'' = 120°<br>''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°<br>''β'' = ''ε'' = 90°<br>''δ'' = 180° − ''α''
|-
! 19
| Decagonal
| ''a'' = ''b'' = ''c'' = ''d''
| ''α'' = ''γ'' = ''ζ'' ≠ ''β'' = ''δ'' = ''ε''<br>cos ''β'' = −{{sfrac|1|2}} − cos ''α''
|-
! 20
| Dodecagonal
| ''a'' = ''b'' = ''c'' = ''d''
| ''α'' = ''ζ'' = 90°<br>''β'' = ''ε'' = 120°<br>''γ'' = ''δ'' ≠ 90°
|-
! 21
| Diisohexagonal orthogonal
| ''a'' = ''b'' = ''c'' = ''d''
| ''α'' = ''ζ'' = 120°<br>''β'' = ''γ'' = ''δ'' = ''ε'' = 90°
|-
! 22
| Icosagonal (icosahedral)
| ''a'' = ''b'' = ''c'' = ''d''
| ''α'' = ''β'' = ''γ'' = ''δ'' = ''ε'' = ''ζ''<br>cos ''α'' = −{{sfrac|1|4}}
|-
! 23
| Hypercubic
| ''a'' = ''b'' = ''c'' = ''d''
| ''α'' = ''β'' = ''γ'' = ''δ'' = ''ε'' = ''ζ'' = 90°
|}
The names here are given according to Whittaker.<ref name="Whittaker">{{cite book|last=Whittaker|first=E. J. W.|title=An Atlas of Hyperstereograms of the Four-Dimensional Crystal Classes|publisher=[[Oxford_University_Press#Clarendon_Press|Clarendon Press]]|year=1985|isbn=978-0-19-854432-6|location=[[Oxford]]|oclc=638900498}}</ref> They are almost the same as in Brown ''et al.'',<ref name="Brown">{{cite book|last1=Brown|first1=H.|title=Crystallographic Groups of Four-Dimensional Space|last2=Bülow|first2=R.|last3=Neubüser|first3=J.|last4=Wondratschek|first4=H.|last5=Zassenhaus|first5=H.|publisher=[[Wiley (publisher)|Wiley]]|year=1978|isbn=978-0-471-03095-9|location=[[New York City|New York]]|oclc=939898594}}</ref> with exception for names of the crystal families 9, 13, and 22. The names for these three families according to Brown ''et al.'' are given in parentheses.

The relation between four-dimensional crystal families, crystal systems, and lattice systems is shown in the following table.<ref name="Whittaker"/><ref name="Brown"/> 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<sub>1</sub> and P3<sub>2</sub>, P4<sub>1</sub>22 and P4<sub>3</sub>22. Starting from four-dimensional space, point groups also can be enantiomorphic in this sense.
{| class="wikitable" style="text-align:center"
|+ Crystal systems in 4D space
! No. of <br />crystal family
! Crystal family
! Crystal system
! Lattice system
! No. of <br>crystal system
! Point groups
! width=120| Space groups
! Bravais lattices
|-
! I
| colspan=2| Hexaclinic
| Hexaclinic P
| 1
| 2
| 2
| 1
|-
! II
| colspan=2| Triclinic
| Triclinic P, S
| 2
| 3
| 13
| 2
|-
! III
| colspan=2| Diclinic
| Diclinic P, S, D
| 3
| 2
| 12
| 3
|-
! IV
| colspan=2| Monoclinic
| Monoclinic P, S, S, I, D, F
| 4
| 4
| 207
| 6
|-
! rowspan=3| V
| rowspan=3| Orthogonal
| rowspan=2| Non-axial orthogonal
| Orthogonal KU
| rowspan="2" | 5
| rowspan=2| 2
| 2
| 1
|-
| rowspan="2" | Orthogonal P, S, I, Z, D, F, G, U
| 112
| rowspan=2| 8
|-
| Axial orthogonal
| 6
| 3
| 887
|-
! VI
| colspan=2| Tetragonal monoclinic
| Tetragonal monoclinic P, I
| 7
| 7
| 88
| 2
|-
! rowspan=3| VII
| rowspan=3| Hexagonal monoclinic
| rowspan=2| Trigonal monoclinic
| Hexagonal monoclinic R
| rowspan="2" | 8
| rowspan=2| 5
| 9
| 1
|-
| rowspan="2" | Hexagonal monoclinic P
| 15
| rowspan=2| 1
|-
| Hexagonal monoclinic
| 9
| 7
| 25
|-
! VIII
| colspan=2| Ditetragonal diclinic*
| Ditetragonal diclinic P*
| 10
| 1 (+1)
| 1 (+1)
| 1 (+1)
|-
! IX
| colspan=2| Ditrigonal diclinic*
| Ditrigonal diclinic P*
| 11
| 2 (+2)
| 2 (+2)
| 1 (+1)
|-
! rowspan=3| X
| rowspan=3| Tetragonal orthogonal
| rowspan=2| Inverse tetragonal orthogonal
| Tetragonal orthogonal KG
| rowspan="2" | 12
| rowspan=2| 5
| 7
| 1
|-
| rowspan="2" | Tetragonal orthogonal P, S, I, Z, G
| 351
| rowspan=2| 5
|-
| Proper tetragonal orthogonal
| 13
| 10
| 1312
|-
! rowspan=3| XI
| rowspan=3| Hexagonal orthogonal
| rowspan=2| Trigonal orthogonal
| Hexagonal orthogonal R, RS
| rowspan="2" | 14
| rowspan=2| 10
| 81
| 2
|-
| rowspan="2" | Hexagonal orthogonal P, S
| 150
| rowspan=2| 2
|-
|Hexagonal orthogonal
| 15
| 12
| 240
|-
! XII
| colspan=2| Ditetragonal monoclinic*
| Ditetragonal monoclinic P*, S*, D*
| 16
| 1 (+1)
| 6 (+6)
| 3 (+3)
|-
! XIII
| colspan=2| Ditrigonal monoclinic*
| Ditrigonal monoclinic P*, RR*
| 17
| 2 (+2)
| 5 (+5)
| 2 (+2)
|-
! rowspan=3| XIV
| rowspan=3| Ditetragonal orthogonal
| rowspan=2| Crypto-ditetragonal orthogonal
| Ditetragonal orthogonal D
| rowspan="2" | 18
| rowspan=2| 5
| 10
| 1
|-
| rowspan="2" | Ditetragonal orthogonal P, Z
| 165 (+2)
| rowspan=2| 2
|-
| Ditetragonal orthogonal
| 19
| 6
| 127
|-
! XV
| colspan=2| Hexagonal tetragonal
| Hexagonal tetragonal P
| 20
| 22
| 108
| 1
|-
! rowspan=5| XVI
| rowspan=5| Dihexagonal orthogonal
| rowspan=2| Crypto-ditrigonal orthogonal*
| Dihexagonal orthogonal G*
| rowspan="2" | 21
| rowspan=2| 4 (+4)
| 5 (+5)
| 1 (+1)
|-
| rowspan="3" | Dihexagonal orthogonal P
| 5 (+5)
| rowspan=3| 1
|-
| Dihexagonal orthogonal
| 23
| 11
| 20
|-
| rowspan=2| Ditrigonal orthogonal
| rowspan=2| 22
| rowspan=2| 11
| 41
|-
| Dihexagonal orthogonal RR
| 16
| 1
|-
! rowspan=3| XVII
| rowspan=3| Cubic orthogonal
| rowspan=2| Simple cubic orthogonal
| Cubic orthogonal KU
| rowspan="2" | 24
| rowspan=2| 5
| 9
| 1
|-
| rowspan="2" | Cubic orthogonal P, I, Z, F, U
| 96
| rowspan=2| 5
|-
| Complex cubic orthogonal
| 25
| 11
| 366
|-
! XVIII
| colspan=2| Octagonal*
| Octagonal P*
| 26
| 2 (+2)
| 3 (+3)
| 1 (+1)
|-
! XIX
| colspan=2| Decagonal
| Decagonal P
| 27
| 4
| 5
| 1
|-
! XX
| colspan=2| Dodecagonal*
| Dodecagonal P*
| 28
| 2 (+2)
| 2 (+2)
| 1 (+1)
|-
! rowspan=3| XXI
| rowspan=3| Diisohexagonal orthogonal
| rowspan=2| Simple diisohexagonal orthogonal
| Diisohexagonal orthogonal RR
| rowspan="2" | 29
| rowspan=2| 9 (+2)
| 19 (+5)
| 1
|-
| rowspan="2" | Diisohexagonal orthogonal P
| 19 (+3)
| rowspan=2| 1
|-
| Complex diisohexagonal orthogonal
| 30
| 13 (+8)
| 15 (+9)
|-
! XXII
| colspan=2| Icosagonal
| Icosagonal P, SN
| 31
| 7
| 20
| 2
|-
! rowspan=3| XXIII
| rowspan=3| Hypercubic
| rowspan=2| Octagonal hypercubic
| Hypercubic P
| rowspan="2" | 32
| rowspan=2| 21 (+8)
| 73 (+15)
| 1
|-
| rowspan="2" | Hypercubic Z
| 107 (+28)
| rowspan=2 | 1
|-
| Dodecagonal hypercubic
| 33
| 16 (+12)
| 25 (+20)
|- bgcolor=#e0e0e0
|'''Total'''
| 23 (+6)
| 33 (+7)
| 33 (+7)
|
| 227 (+44)
| 4783 (+111)
| 64 (+10)
|}