Demo: Examples

This is a set of examples and test cases, roughly ordered in sections. Click on a test case to explore the OD family (at the moment still very rudimentary!).

These are fringe cases, but they should work

Input to OD Explorer Comment
[r,s] p11(1)
p11(1) {- - -} [r,s]
p11(1) {1 1 1} [r,s]
p11(1) {- - (-)} [r,s]
p11(1) {1 1 (1)} [r,s]
{- - -} [r,s] p11(1)
{1 1 1} [r,s] p11(1)
{- - (-)} [r,s] p11(1)
{1 1 (1)} [r,s] p11(1)

Testing of layer lattice restrictions

Input to OD Explorer Comment
p{a'=a-6b,b'=6a+b}11(1)[0,0]p{a'=5a+3b,b'=5b+3a}11(1)[0,0]
p{a'=3a+2b,b'=6b+a}11(1)[0,0]p{a'=2b+4a,b'=5a+3b}11(1)[0,0]
p(4) 1 1 [0,0] p (6) 1 1 [0,0] Tetragonal and hexagonal layers are incompatible, sorry
p{a'=a-b,b'=a+2b}1 2 (1){1 2_r 1} Possible, fixes the global lattice
p{a'=a-b,b'=a+2b}1 2/m (1){1 2_r 1} Same
op p{b'=1/5a+b}-1 -1 (-1) {2 1 (1)}
op p{b'=2/5a+b}-1 -1 (-1) {2 1 (1)}
p 2 1 (1) [r,s] p{a'=a-b,b'=a+b} 2 1 (1) Global lattice must be square (OD family is tetragonal!)
tp p 2 1 (1) [r,s] p{a'=a-b,b'=a+b} 2 1 (1) Global lattice must be square (Specify redundant restriction)
op p 2 1 (1) [r,s] p{a'=a-b,b'=a+b} 2 1 (1) Global lattice must be square (Orthogonal system compatible with square system)
hp p 2 1 (1) [r,s] p{a'=a-b,b'=a+b} 2 1 (1) Global lattice must be square (Error! Hexagonal and square systems are incompatible)
p 2 1 (1) [r,s] p{a'=-a-b,b'=a-b} 2 1 (1) Global lattice must be square (OD family is tetragonal!)
p 2 1 (1) [r,s] p{a'=2a+b,b'=-a+2b} 2 1 (1) Global lattice must be square (OD family is non-crystallographic)
p 2 1 (1) [r,s] p{a'=2a+b,b'=-a+b} 2 1 (1) Global lattice must be orthogonal with |b|=√2|a|
op p 2 1 (1) [r,s] p{a'=2a+b,b'=-a+b} 2 1 (1) Global lattice must be orthogonal with |b|=√2|a| (Orthogonal restriction is OK)
tp p 2 1 (1) [r,s] p{a'=2a+b,b'=-a+b} 2 1 (1) Global lattice must be orthogonal with |b|=√2|a| (Error! square incompatible with |b|=√2|a|)
<b^2=a^2>p11(m)11 {- - (-) 2_s n_(2,s)} An unusual way of describing this orthorhombic OD family

The "ambiguous" K2HAsO4·2.5H2O

Input to OD Explorer Comment
p-1(-1)-1{1 c_(1/2) 1} in terms of layers of one kind
p-1(-1)-1{1 c_(s) 1} more general
p-1(-1)-1{1 n_(r,s) 1} even more general
p{a'=a-c,c'=a+c}-1(-1)-1{1 c_(s) 1} The same in a different coordinate system should give same result!
p-1(-1)-1 [0,0] p{a'=a/2}1(c)1 in terms of layers of two kinds

KOH·2H2O

Input to OD Explorer Comment
tp p{a'=1/2a}bm(a){1 1 4_4 1 1} in terms of layers of one kind
tp p{a'=1/2a}bm(a){1 1 1 2_(r') 1}
tp p{a'=1/2a}bm(a){1 1 1 1 2_(s')}
tp p{a'=1/2a}bm(a){[001]: 4_4}
tp p{a'=1/2a}bm(a){[001]: 4_4 [2-10]: 2_r}
tp p{a'=1/2a}bm(a){[001]: 4_4 [2-10]: 2_r [210]: 2_s}
p{a'=1/2a}bm(a)[1/8,0]p{a'=1/2a,b'=1/2b}mm(-4)22

General reverse continuation

Input to OD Explorer Comment
pmm(m){11n_(r',s')} Has general reverse continuation

Not all parameters listed in σ-POs

Input to OD Explorer Comment
p1 1 (2){1 1 2_2}[r,s] No parameters in σ-POs!
[r,s]{-1 -1 -1}p1 1 (1){-1 -1 -1}[r',s'] No parameters in σ-POs!
op p1 1 (m){1 2_s 1}[r,s/2] Only one parameters in σ-POs!
op p1 1 (m){1 2_s 1}[r,s] Only one parameters in σ-POs! Can only work if s=s/2=0
[r,s]{-1 -1 -1}p1 a (1){-1 -1 -1}[r',s'] Only one parameter in σ-list!

Floating origins

Input to OD Explorer Comment
p21(1) {2_1 1 1} Only s relevant. r can be chosen arbitrarily (e.g. 0)
{2_1 - - } pm1(1) {2_1 - -} Only one r can be removed
{- 2_1 - } pm1(1) {2_1 - -} Like above, but here r in second σ-PO set is removed
p11(1) {- - 2_2} r and s can be chosen arbitrarily (e.g. 0)
{- - -4 - -} p11(1)11 {- - -4 - -} This is a bit tricky: we can only remove the parameters from one σ-PO list

Rectangular OD-Families differentiated by the lattice restriction

Input to OD Explorer Comment
op p{b'=a/5+b/5}11(1) {c_2 - -}
op p{b'=2a/5+b/5}11(1) {c_2 - -}
op p{b'=a/5+b/5}11(m) {2 - -}
op p{b'=2a/5+b/5}11(m) {2 - -}

Arcusfunctions (mixing not yet supported!)

Input to OD Explorer Comment
p 1 1 (4) 1 1 { 1 1 (2*pi/atan(4/3))_(2*pi/atan(4/3)) 1 1}
pmm(4)mm { 1 1 (2*pi/atan(4/3))_(2*pi/atan(4/3)) 1 1}
pmm(4)mm { 1 1 (2*pi/acos(3/5))_(2*pi/atan(4/3)) 1 1} System should recognize that acos(3/5) = atan(4/3)
pmm(4)mm { 1 1 (2*pi/acos(3/5))_(2*pi/atan(-4/3)) 1 1} Bug! Interpreted incorrectly: acos(3/5)/atan(4/3) = -1 !
pmm(4)mm { 1 1 (2*pi/acos(3/5))_(-2*pi/atan(-4/3)) 1 1} Bug! Interpreted incorrectly: acos(3/5)/atan(4/3) = -1 !
pmm(4/m)mm { 1 1 (2*pi/atan(4/3))_(2*pi/atan(4/3)) 1 1}

Specify arbitrary directions

Input to OD Explorer Comment
p(4)mm { [130]: n_(2,r) } Simpler way to express the above
p11(1) { [130]: n_(2,r) } Underspecified
tp p11(1) { [130]: n_(2,r) } OK
pbm(a)11 { - - -4 c_2 c_2 }
cmm(e)11 { - - -4 c_2 c_2 }
tp cmm(e) { [110]: c_2 }
p11(m)mm { c_2 c_2 -4 - - }

Same theme, this time with hexagonal lattices

Input to OD Explorer Comment
p(6)mm { [130]: n_(2,r) }
p(6)mm { [1-30]: n_(2,r) }
p1 1 1 (6) 1 1 1 {1 1 1 (2*pi/acos(1/7))_(2*pi/acos(1/7)) 1 1 1}
pm11(m)m11 { - c_2 - -6 - c_2 - }
pm11(m)m11 { - - c_2 -6- - - c_2 } Equivalent to the above, but given in other direction
pm11(m)m11 { c_2 - - - c_2 - - } This is silly - it could be given using 3-placed symbols and should be output as such
p1m1(m)1m1 { - c_2 - - - c_2 - } Likewise silly - but with other relation to global lattice
p11m(m)11m { - - c_2 - - - c_2 } And a third silly way to denote the same OD groupoid family!
pm11(1)111 { - - - 6_6 - - - } This is fully ordered, no matter what
p111(m)111 { - - - 6_6 - - - } This is fully ordered, no matter what
{ - - - (-) 2_s - - } p111(1)111 { - - - (-) - - 2_(s')} A trigonal OD family
oha cmm(m) { [310]: c_2 } Should give 7-placed symbol
ohb cmm(m) { [130]: c_2 } And this as well!
<ab=0,b^2=3a^2> cmm(m) { [310]: c_2 } Like above with spelled out lattice restriction
<ab=0,b^2/3=a^2> cmm(m) { [310]: c_2 } Like above with spelled out lattice restriction
<ab=0,b=sqrt(3)a> cmm(m) { [310]: c_2 } Like above with spelled out lattice restriction
<ab=0,b/sqrt(3)=a> cmm(m) { [310]: c_2 } Like above with spelled out lattice restriction
<ab=0,3b^2=a^2> cmm(m) { [130]: c_2 } Like above with spelled out lattice restriction
<ab=0,b=sqrt(3)/3a> cmm(m) { [130]: c_2 } Like above with spelled out lattice restriction
<ab=0,sqrt(3)b=a> cmm(m) { [130]: c_2 } Like above with spelled out lattice restriction
oha cmm(m) { [210]: c_2 } This involves layers with different lattices

Centred lattices

Input to OD Explorer Comment
cmm(m) { - - n_(1,1) }
op c-1-1(-1) { c_2 - - }
<ab=a^2/2> p-1-1(-1) {c_2 - -} Same as above! Output should ideally be formatted the same way!
<ab=a^2/2>p11(m){[1-20]:2_s} Another play on the same theme
op c11(e) { - n_(2,s) - } A a-glide and a centred lattice becomes an e-glide
op c11(d+) { - n_(2,s) - } We need to support d+-glides along (1/4,1/4)
op c11(d-) { - n_(2,s) - } We need to support d--glides along (1/4,-1/4)
op c11(d) { - n_(2,s) - } d without sign defaults to d+

OD families described in the literature

Input to OD Explorer Comment
pc(a)a { - n_(1/2,1/2) - } β-KAsO3
p1 2_1 (1) [0,0] p1 2_1/m (1) γ-CaTeO3
p(c)mb { - a_2 - } Aragonite (E. Makovicky, Mineral. Petrol., 106 (2012), 19-24)
p (1) 2_1/c 1 { n_(1,s) - - } Nitro-Isoxazol
a(e)mm [1/4,1/4] p(c)cb KAgCO3

The seven categories of OD structures. Category II with layers of M>1 kinds is especially nasty; See the example from Acta. Cryst. A38, 49-54

Input to OD Explorer Comment
pma(b) { c_2 n_(2,1/2) n_(-1/2,1) } Cat I, M=1 (spurrite)
{ c_2 n_(2,1/2) n_(3/2,1) } pma(b) Cat I, M=1 (spurrite) (other way 'round)
p12/a(1) {2_(1/2)/n_(1,2) - (2_2/n_(-1/2,1)) } Cat II, M=1 (decaborane)
{2_(1/2) 2 (a_(1/2))} cmm(2) {2_(1/2) 2 (a_(1/2))} Cat III, M=1 (γ-Hg3S2Cl2)
pmm(m) [r1,s1] pmm(2) { 2_(r2) 2_(s2) n_(r2,s2)} Cat I, M>1
{ 2_(r2) 2_(s2) n_(r2,s2)} pmm(2) [r1,s1] pmm(m) Cat I, M>1 (other way 'round)
p11(2) [r1,s1] p11(2) [r2,s2] Cat II, M>1, All layers b1 are translational equivalent
[r1,s1] p11(2) [r2,s2] p11(2) Same as above, but written the other way round
op p11(2) {n_(r3,4) n_(4,s3) (-) } [r1,s1] p11(2) [r2,s2] Cat II, M>1, Layers b12n and b12n+1 are not related by translation.Note the factor 4 in the n-glides, which means a translation along 2 layer thicknesses.
op [r1,s1] p11(2) {n_(s3,4) n_(4,s4) (-) } [r2,s2] p11(2) Same as above, but written the other way round
{ 2_(r1) 2_(s1) n_(r1,s1)} pmm(2) [r2,s2] p11(2) { 2_(r3) 2_(s3) - } Cat III, M>1
pmm(m) [r1,s1] pmm(2) [r2,s2] p22(2) Cat IV, M>1

OD families that can be simplified to other OD families

Input to OD Explorer Comment
op { -1 -1 (-1)} c11(1) { - 2_(s) (-)} Cat III, M=1 to Cat I, M=1
op { - 2_(s) (-) } c11(1) {-1 -1 (-1)} Cat III, M=1 to Cat I, M=1
p-1-1(-1) [r1,s1] c11(1) { -1 -1 (-1) } Cat I, M>1 to Cat IV, M>1
{ -1 -1 (-1) } c11(1) [r1,s1] p-1-1(-1) Cat I, M>1 to Cat IV, M>1
c-1-1(-1) [r1,s1] c11(1) { - - (n_(r2,s2)) } Cat I, M>1 to Cat I, M=1
{ - - (n_(r2,s2)) } c11(1) [r1,s1] c-1-1(-1) Cat I, M>1 to Cat I, M=1
c-1-1(-1) [r1,s1] c11(1) [r2,s2] c11(1) [r3,s3] c11(1) { - - (n_(r4,s4)) } Cat I, M>1 to Cat I, M=1
{ - - (n_(r1,s1)) } c11(1) [r2,s2] c11(1) [r3,s3] c11(1) [r4,s4] c-1-1(-1) Cat I, M>1 to Cat I, M=1
c-1-1(-1) [r1,s1] p11(1) [r2,s2] p11(1) [r3,s3] p11(1) { - - (n_(r4,s4)) } Cat I, M>1 to Cat I, M>1
{ - - (n_(r1,s1)) } p11(1) [r2,s2] p11(1) [r3,s3] p11(1) [r4,s4] c-1-1(-1) Cat I, M>1 to Cat I, M=1
c11(1) [r1,s1] c11(1) [r2,s2] p11(1) [r3,s3] Cat II, M>1 to Cat II, M>1
[r3,s3] c11(1) [r1,s1] c11(1) [r2,s2] p11(1) Cat II, M>1 to Cat II, M>1
c11(1) [r1,s1] c11(1) [r2,s2] p11(1) [r3,s3] c11(1) [r4,s4] Cat II, M>1 to Cat II, M>1
[r4,s4] p11(1) [r3,s3] c11(1) [r1,s1] c11(1) [r2,s2] p11(1) Cat II, M>1 to Cat II, M>1
c11(1) {- - 2_6 } [r1,s1] c11(1) [r2,s2] p11(1) [r3,s3] Cat II, M>1 to Cat II, M>1
[r3,s3] c11(1) [r1,s1] c11(1) {- - 2_(6) } [r2,s2] p11(1) Cat II, M>1 to Cat II, M>1
c11(1) {- - 2_8 } [r1,s1] c11(1) [r2,s2] p11(1) [r3,s3] c11(1) [r4,s4] Cat II, M>1 to Cat II, M>1
[r4,s4] p11(1) [r3,s3] c11(1) [r1,s1] c11(1) {- - 2_(8) } [r2,s2] p11(1) Cat II, M>1 to Cat II, M>1
{ - - (n_(r1,s1)) } p11(1) [r2,s2] p11(1) [r3,s3] p11(1) {- - (n_(r4,s4))} Cat III, M>1 to Cat III, M=1
{ - - (n_(r1,s1)) } p11(1) [r2,s2] c11(1) [r3,s3] c11(1) {- - (n_(r4,s4))} Cat III, M>1 to Cat III, M>1
{ -1 -1 -1 } p11(1) [r2,s2] p11(1) [r3,s3] p11(1) {- - (n_(r4,s4))} Cat III, M>1 to Cat I, M=1
{ - - (n_(r1,s1)) } p11(1) [r2,s2] p11(1) [r3,s3] p11(1) {-1 -1 -1} Cat III, M>1 to Cat I, M=1
{-1 -1 -1 } p11(1) [r1,s1] c11(1) [r2,s2] c11(1) [r3,s3] c11(1) {-1 -1 -1} Cat III, M>1 to Cat IV, M>1
c-1-1(-1) [r1,s1] c11(1) [r2,s2] p11(1) [r3,s3] p-1-1(-1) Cat IV, M>1 to Cat IV, M>1

OD families that can be simplified to space groups

Input to OD Explorer Comment
p-1-1(-1) { -1 -1 -1 } Cat I, M=1
op p11(1) { c_2 - - } Cat II, M=1
op { c_2 - - } p11(1) Cat II, M=1
op { -1 -1 (-1) } c11(1) {-1 -1 (-1)} Cat III, M=1
c-1-1(-1) [r1,s1] c11(1) [r2,s2] c11(1) [r3,s3] c11(1) { -1 -1 -1 } Cat I, M>1 to Cat I, M=1
op c11(1) {c_8 - - } [r1,s1] c11(1) [r2,s2] c11(1) [r3,s3] c11(1) [r4,s4] Cat II, M>1
op [r4,s4] c11(1) [r1,s1] c11(1) [r2,s2] c11(1) [r3,s3] {c_8 - - } c11(1) Cat II, M>1
{ -1 -1 -1 } p11(1) [r2,s2] p11(1) [r3,s3] p11(1) {-1 -1 -1} Cat III, M>1
p-1-1(-1) [r1,s1] p11(1) [r2,s2] p11(1) [r3,s3] p-1-1(-1) Cat IV, M>1