Cubic grid symmetries
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An enumeration of the 92 cubic grid symmetries.
Notation
We will use Conway notation for the underlying symmetry group on Euclidean three space. This means the 'subscript' corresponds to the order of the group unless the group name starts with ±, in which case the group order is twice the subscript.
We will use the letters c, f, e, and k to refer to a cell, face, edge, or vertex respectively.
There are six relevant axes, they will be referred to by the two types of facets through whose centers they pass.
For dihedral-type symmetries the first letter will refer to the fixed point, and if necessary to disambiguate the second will give the direction of the axis.
For some pyramidal symmetries o and d will refer to an orthogonal or diagonal respectively alignment of the base.
For the three relevant planes: c will refer to the orthogonal plane through the centers of cells, f will refer to the orthogonal plane through the faces, and / will refer to the diagonal plane.
Symmetries fixing a cell
Chiral symmetries
| Name | Description | Example | RLE3 |
|---|---|---|---|
| O24_c | chiral cubic symmetry centered on a cell | 
|
3D version=1 size=13 pos=3,3,3 #C O24_c x=7 y=7 z=7 rule=3D0..6/F $4bo$bo2$5bo$2bo/2bo2$3bo2bo$2bobo$o2bo2$4bo/5bo$o2bo2$bo3bo2$3bo2bo$bo/$2bobo$bo3bo2$bo3bo$2bobo/bo$3bo2bo2$bo 3bo2$o2bo$5bo/4bo2$o2bo$2bobo$3bo2bo2$2bo/$2bo$5bo2$bo$4bo! |
| T12_c | chiral tetrahedral symmetry centered on a cell | 
|
3D version=1 size=11 pos=3,3,3 #C T12_c x=5 y=5 z=5 rule=3D0..6/F $2bo2$2bo/$3bo$o3bo$bo/bobo2$2bo2$bobo/$bo$o3bo$3bo/$2bo2$2bo! |
| D8_c | chiral square prismatic symmetry centered on a cell | 
|
3D version=1 size=11 pos=3,3,4 #C D8_c x=5 y=5 z=3 rule=3D0..6/F bo$4bo2$o$3bo/$2bo$bobo$2bo/3bo$o2$4bo$bo! |
| C4_cf | chiral square pyramidal symmetry a.k.a. 90 degree rotation line goes through cells and faces |
|
3D version=1 size=11 pos=3,3,4 #C C4_cf x=5 y=5 z=2 rule=3D0..6/F $2bo$bobo$2bo/3bo$o2$4bo$bo! |
| D4_cf | chiral brick symmetry centered on a cell orthogonal orientation |
|
3D version=1 size=11 pos=4,3,4 #C D4_cf x=3 y=5 z=3 rule=3D0..6/F $2bo2$o/bo2$obo2$bo/$o2$2bo! |
| D4_ce | chiral brick symmetry centered on a cell diagonal orientation |
|
3D version=1 size=11 pos=3,3,4 #C D4_ce x=5 y=5 z=3 rule=3D0..6/F bo4$3bo/$bo$2bo$3bo/$o2$4bo! |
| D6_ck or simply D6_c |
chiral triangular prismatic symmetry centered on a cell | 
|
3D version=1 size=9 pos=3,3,3 #C D6_c x=3 y=3 z=3 rule=3D0..6/F 2$bo/2bo$bo/$o! |
| C3_ck or simply C3 |
chiral triangular pyramidal symmetry a.k.a. 120 degree rotation line goes through vertices and cells |
|
3D version=1 size=9 pos=3,3,3 #C C3 x=3 y=3 z=3 rule=3D0..6/F 2$bo/2bo$2bo$bo/$2o! |
| C2_cf | chiral digonal pyramidal symmetry a.k.a. 180 degree rotation line goes through cells and faces |
|
3D version=1 size=9 pos=3,3,3 #C C2_cf x=3 y=2 z=3 rule=3D0..6/F 2o/$bo/b2o! |
| C2_ce | chiral digonal pyramidal symmetry a.k.a. 180 degree rotation line goes through cells and edges |
|
3D version=1 size=9 pos=3,3,3 #C C2_ce x=2 y=2 z=3 rule=3D0..6/F o/$o/$bo! |
| C1 | no symmetry | 
|
3D version=1 size=9 pos=3,3,3 #C C1 x=2 y=3 z=3 rule=3D0..6/F o/$o/$bo$bo! |
Containing the inversion
| Name | Description | Example | RLE3 |
|---|---|---|---|
| ±O24_c | cubic symmetry centered on a cell | 
|
3D version=1 size=7 pos=3,3,3 #C ±O24_c x=1 y=1 z=1 rule=3D0..6/F o! |
| ±T12_c | pyritic symmetry centered on a cell | 
|
3D version=1 size=11 pos=3,3,3 #C ±T12_c x=5 y=5 z=5 rule=3D0..6/F 2$bobo/2bo2$2bo2$2bo/$obobo$bobo$obobo/2bo2$2bo2$2bo/2$bobo! |
| ±D8_c | square prismatic symmetry centered on a cell | 
|
3D version=1 size=9 pos=4,4,3 #C ±D8_c x=1 y=1 z=3 rule=3D0..6/F o/o/o! |
| ±C4_cf | square proprismatic symmetry centered on a cell | 
|
3D version=1 size=11 pos=3,3,5 #C ±C4_cf x=5 y=5 z=1 rule=3D0..6/F 3bo$obo$bobo$2bobo$bo! |
| ±D4_cf | brick symmetry centered on a cell orthogonal orientation |
|
3D version=1 size=9 pos=3,3,4 #C ±D4_cf x=3 y=3 z=1 rule=3D0..6/F obo$3o$obo! |
| ±D4_ce | brick symmetry centered on a cell diagonal orientation |
|
3D version=1 size=9 pos=3,3,4 #C ±D4_ce x=3 y=3 z=1 rule=3D0..6/F o$bo$2bo! |
| ±D6_ck or simply ±D6_c |
triangular antiprismatic symmetry centered on a cell | 
|
3D version=1 size=9 pos=3,3,3 #C ±D6_c x=3 y=3 z=3 rule=3D0..6/F o/$bo/2$2bo! |
| ±C3_ck or simply ±C3_c |
triangular proantiprismatic symmetry centered on a cell | 
|
3D version=1 size=11 pos=3,3,3 #C ±C3_c x=5 y=5 z=5 rule=3D0..6/F $2bo/$bo$o/bo2$2bo2$3bo/2$4bo$3bo/3$2bo! |
| ±C2_cf | espic symmetry centered on a cell axis goes through face |
|
3D version=1 size=9 pos=3,4,3 #C ±C2_cf x=3 y=1 z=3 rule=3D0..6/F o/3o/2bo! |
| ±C2_ce | espic symmetry centered on a cell axis goes through edge |
|
3D version=1 size=9 pos=3,3,3 #C ±C2_ce x=3 y=3 z=3 rule=3D0..6/F o/o$bo$2bo/2$2bo! |
| ±C1_c | central inversion symmetry centered on a cell | 
|
3D version=1 size=11 pos=4,4,3 #C ±C1_c x=3 y=3 z=5 rule=3D0..6/F o/$o/$bo/$2bo/2$2bo! |
Other achiral symmetries
| Name | Description | Example | RLE3 |
|---|---|---|---|
| TO24_c | tetrahedral symmetry centered on a cell | 
|
3D version=1 size=9 pos=3,3,3 #C TO24_c x=3 y=3 z=3 rule=3D0..6/F 2bo2$o/$bo/o2$2bo! |
| CD6_ck or simply CD6 |
triangular pyramidal symmetry line goes through vertices and cells |
|
3D version=1 size=8 pos=3,3,3 #C CD6 x=2 y=2 z=2 rule=3D0..6/F 2o$o/o! |
| CD8_cf | square pyramidal symmetry line goes through cells and faces |
|
3D version=1 size=9 pos=3,3,3 #C CD8_cf x=3 y=3 z=2 rule=3D0..6/F $bo/bo$obo$bo! |
| DD8_cf | digonal antiprismatic symmetry centered on a cell digon is diagonal |
|
3D version=1 size=9 pos=3,3,3 #C DD8_cf x=3 y=3 z=3 rule=3D0..6/F 2bo$bo$o/$bo/o$bo$2bo! |
| DD8_ce | digonal antiprismatic symmetry centered on a cell digon is orthogonal |
|
3D version=1 size=9 pos=3,3,3 #C DD8_ce x=3 y=3 z=3 rule=3D0..6/F bo2$bo/$bo/$obo! |
| CC4_cf | prodappic symmetry centered on a cell | 
|
3D version=1 size=11 pos=3,3,4 #C CC4_cf x=5 y=5 z=3 rule=3D0..6/F 3bo4$bo/$2bo$bobo$2bo/$o2$4bo! |
| CD4_cfo | digonal pyramidal symmetry line goes through cells and faces orthogonal orientation |
|
3D version=1 size=9 pos=3,3,4 #C CD4_cfo x=3 y=2 z=1 rule=3D0..6/F bo$obo! |
| CD4_cfd | digonal pyramidal symmetry line goes through cells and faces diagonal orientation |
|
3D version=1 size=9 pos=3,3,3 #C CD4_cfd x=3 y=3 z=2 rule=3D0..6/F $bo/o2$2bo! |
| CD4_ce | digonal pyramidal symmetry line goes through cells and edges |
|
3D version=1 size=8 pos=3,3,3 #C CD4_ce x=2 y=2 z=1 rule=3D0..6/F 2o$o! |
| CC2_c | reflection plane goes through cells, edges, and faces |
|
3D version=1 size=9 pos=4,3,3 #C CC2_c x=1 y=3 z=2 rule=3D0..6/F o$o/2$o! |
| CC2_/ | reflection plane goes through cells, faces, edges, and vertices |
|
3D version=1 size=9 pos=3,3,3 #C CC2_/ x=3 y=3 z=2 rule=3D0..6/F o/$bo$2bo! |
Symmetries fixing a face but not a cell
Chiral symmetries
| Name | Description | Example | RLE3 |
|---|---|---|---|
| D8_f | chiral square prismatic symmetry centered on a face | 
|
3D version=1 size=11 pos=3,3,4 #C D8_f x=5 y=5 z=2 rule=3D0..6/F bo$2bobo$bobo$obo$3bo/3bo$obo$bobo$2bobo$bo! |
| D4_fe | chiral brick symmetry centered on a face orthogonal orientation |
|
3D version=1 size=9 pos=3,3,3 #C D4_fe x=3 y=3 z=2 rule=3D0..6/F 2bo$3o$o/o$3o$2bo! |
| D4_fk | chiral brick symmetry centered on a face diagonal orientation |
|
3D version=1 size=11 pos=3,3,4 #C D4_fk x=5 y=5 z=2 rule=3D0..6/F $o$bobo$4bo/bo$2bo2$2bo$3bo! |
| C2_fe | chiral digonal pyramidal symmetry a.k.a. 180 degree rotation line goes through faces and edges |
|
3D version=1 size=9 pos=3,3,3 #C C2_fe x=3 y=2 z=2 rule=3D0..6/F bo$2bo/bo$o! |
| C2_fk | chiral digonal pyramidal symmetry a.k.a. 180 degree rotation line goes through faces and vertices |
|
3D version=1 size=8 pos=3,3,3 #C C2_fk x=2 y=2 z=2 rule=3D0..6/F 2o/o$o! |
Containing the inversion
| Name | Description | Example | RLE3 |
|---|---|---|---|
| ±D8_f | square prismatic symmetry centered on a face | 
|
3D version=1 size=8 pos=3,3,3 #C ±D8_f x=1 y=1 z=2 rule=3D0..6/F o/o! |
| ±D4_fe | brick symmetry centered on a face orthogonal orientation |
|
3D version=1 size=9 pos=3,4,3 #C ±D4_fe x=3 y=1 z=2 rule=3D0..6/F 3o/3o! |
| ±D4_fk | brick symmetry centered on a face diagonal orientation |
|
3D version=1 size=9 pos=3,3,3 #C ±D4_fk x=3 y=3 z=2 rule=3D0..6/F o$bo$2bo/o$bo$2bo! |
| ±C4_fc | square proprismatic symmetry centered on a face | 
|
3D version=1 size=11 pos=3,3,4 #C ±C4_fc x=5 y=5 z=2 rule=3D0..6/F 3bo$obo$bobo$2bobo$bo/3bo$obo$bobo$2bobo$bo! |
| ±C2_fc | espic symmetry centered on a face axis goes through cell |
|
3D version=1 size=9 pos=3,3,3 #C ±C2_fc x=3 y=3 z=2 rule=3D0..6/F o$3o$2bo/o$3o$2bo! |
| ±C2_fe | espic symmetry centered on a face axis goes through edge |
|
3D version=1 size=9 pos=4,3,3 #C ±C2_fe x=1 y=3 z=2 rule=3D0..6/F o$o/$o$o! |
| ±C2_fk | espic symmetry centered on a face axis goes through vertex |
|
3D version=1 size=9 pos=3,3,3 #C ±C2_fk x=3 y=3 z=2 rule=3D0..6/F $bo$2bo/o$bo! |
| ±C1_f | central inversion symmetry centered on a face | 
|
3D version=1 size=9 pos=3,3,3 #C ±C1_f x=3 y=3 z=2 rule=3D0..6/F $b2o$2bo/o$2o! |
Other achiral symmetries
| Name | Description | Example | RLE3 |
|---|---|---|---|
| DD8_fe | digonal antiprismatic symmetry centered on a face digon is diagonal |
|
3D version=1 size=9 pos=3,3,3 #C DD8_fe x=3 y=3 z=2 rule=3D0..6/F o$bo$2bo/2bo$bo$o! |
| DD8_fk | digonal antiprismatic symmetry centered on a face digon is orthogonal |
|
3D version=1 size=9 pos=3,3,3 #C DD8_fk x=3 y=3 z=2 rule=3D0..6/F $obo/bo2$bo! |
| CC4_fc | prodappic symmetry centered on a face | 
|
3D version=1 size=10 pos=3,3,3 #C CC4_fc x=3 y=3 z=4 rule=3D0..6/F 2bo2$o/$obo/bo2$bo/o2$2bo! |
| CD4_fe | digonal pyramidal symmetry line goes through faces and edges |
|
3D version=1 size=10 pos=4,4,3 #C CD4_fe x=1 y=2 z=4 rule=3D0..6/F $o/o/o/$o! |
| CD4_fk | digonal pyramidal symmetry line goes through faces and vertices |
|
3D version=1 size=10 pos=4,4,3 #C CD4_fk x=2 y=2 z=4 rule=3D0..6/F $bo/o/o/$bo! |
| CC2_f | reflection plane goes through faces, edges, and vertices |
|
3D version=1 size=9 pos=3,3,3 #C CC2_f x=3 y=2 z=2 rule=3D0..6/F 2o$2bo/2o$2bo! |
Symmetries fixing an edge but not a cell or face
Chiral symmetries
| Name | Description | Example | RLE3 |
|---|---|---|---|
| D8_e | chiral square prismatic symmetry centered on an edge | 
|
3D version=1 size=10 pos=3,3,3 #C D8_e x=3 y=4 z=4 rule=3D0..6/F $o$2bo/2bo$bo$bo$o/o$bo$bo$2bo/$2bo$o! |
| D4_ef | chiral brick symmetry centered on an edge orthogonal orientation |
|
3D version=1 size=10 pos=3,3,4 #C D4_ef x=3 y=4 z=2 rule=3D0..6/F 2bo$bo$bo$o/o$bo$bo$2bo! |
| D4_ec | chiral brick symmetry centered on an edge diagonal orientation |
|
3D version=1 size=10 pos=3,3,3 #C D4_ec x=3 y=4 z=4 rule=3D0..6/F $o/2bo$bo/2$bo$2bo/2$o! |
| C4_ek | chiral square pyramidal symmetry a.k.a. 90 degree rotation line goes through edges and vertices |
|
3D version=1 size=10 pos=4,3,3 #C C4_ek x=2 y=4 z=4 rule=3D0..6/F 2$bo/bo$o$o/$o$o$bo/$bo! |
| C2_ek | chiral digonal pyramidal symmetry a.k.a. 180 degree rotation line goes through edges and vertices |
|
3D version=1 size=10 pos=4,4,3 #C C2_ek x=2 y=2 z=4 rule=3D0..6/F o/bo/$bo/$o! |
Containing the inversion
| Name | Description | Example | RLE3 |
|---|---|---|---|
| ±D8_e | square prismatic symmetry centered on an edge | 
|
3D version=1 size=8 pos=3,3,3 #C ±D8_e x=1 y=2 z=2 rule=3D0..6/F o$o/o$o! |
| ±D4_ef | brick symmetry centered on an edge orthogonal orientation |
|
3D version=1 size=10 pos=4,3,4 #C ±D4_ef x=1 y=4 z=2 rule=3D0..6/F o$o$o$o/o$o$o$o! |
| ±D4_ec | brick symmetry centered on an edge diagonal orientation |
|
3D version=1 size=8 pos=3,3,3 #C ±D4_ec x=1 y=2 z=2 rule=3D0..6/F o/$o! |
| ±C4_ek | square proprismatic symmetry centered on an edge | 
|
3D version=1 size=10 pos=4,3,3 #C ±C4_ek x=1 y=4 z=4 rule=3D0..6/F 2$o/o$o$o/$o$o$o/$o! |
| ±C2_ek | espic symmetry centered on an edge axis goes through vertex |
|
3D version=1 size=10 pos=4,3,4 #C ±C2_ek x=1 y=4 z=2 rule=3D0..6/F o$o/2$o$o! |
| ±C2_ef | espic symmetry centered on an edge axis goes through face |
|
3D version=1 size=9 pos=3,3,3 #C ±C2_ef x=3 y=2 z=2 rule=3D0..6/F b2o$2o/b2o$2o! |
| ±C2_ec | espic symmetry centered on an edge axis goes through cell |
|
3D version=1 size=9 pos=3,3,3 #C ±C2_ec x=3 y=2 z=2 rule=3D0..6/F b2o/$2o! |
| ±C1_e | central inversion symmetry centered on an edge | 
|
3D version=1 size=10 pos=3,3,4 #C ±C1_e x=3 y=4 z=2 rule=3D0..6/F 2bo$bo/2$bo$o! |
Other achiral symmetries
| Name | Description | Example | RLE3 |
|---|---|---|---|
| DD8_ef | digonal antiprismatic symmetry centered on an edge digon is diagonal |
|
3D version=1 size=9 pos=3,3,3 #C DD8_ef x=3 y=2 z=2 rule=3D0..6/F b2o$2o/2o$b2o! |
| DD8_ec | digonal antiprismatic symmetry centered on an edge digon is orthogonal |
|
3D version=1 size=10 pos=3,3,3 #C DD8_ec x=3 y=4 z=4 rule=3D0..6/F $o$o/2bo$bo$bo$2bo/2bo$bo$bo$2bo/$o$o! |
| CD8_ek | square pyramidal symmetry line goes through edges and vertices |
|
3D version=1 size=10 pos=4,3,3 #C CD8_ek x=2 y=4 z=4 rule=3D0..6/F $bo$bo/bo$o$o$bo/bo$o$o$bo/$bo$bo! |
| CC4_ek | prodappic symmetry centered on an edge | 
|
3D version=1 size=10 pos=3,3,3 #C CC4_ek x=3 y=4 z=4 rule=3D0..6/F 2$o/2bo$bo$bo/$bo$bo$2bo/$o! |
| CD4_eko | digonal pyramidal symmetry line goes through edges and vertices orthogonal orientation |
|
3D version=1 size=10 pos=4,3,4 #C CD4_eko x=2 y=4 z=2 rule=3D0..6/F bo$o$o$bo/bo$o$o$bo! |
| CD4_ekd | digonal pyramidal symmetry line goes through edges and vertices diagonal orientation |
|
3D version=1 size=10 pos=4,3,3 #C CD4_ekd x=2 y=4 z=4 rule=3D0..6/F bo/$o/2$o/3$bo! |
Symmetries fixing a vertex but not an edge, face, or cell
Chiral symmetries
| Name | Description | Example | RLE3 |
|---|---|---|---|
| O24_k | chiral cubic symmetry centered on a vertex | 
|
3D version=1 size=12 pos=3,3,3 #C O24_k x=6 y=6 z=6 rule=3D0..6/F $2bo$4bo$bo$3bo/3bo$bo2bo$o$5bo$bo2bo$2bo/bo$5bo3$o$4bo/4bo$o3$5bo$bo/2bo$bo2bo$5bo$o$bo2bo$3bo/$3bo$bo$4bo$2bo! |
| T12_k | chiral tetrahedral symmetry centered on a vertex | 
|
3D version=1 size=12 pos=3,3,3 #C T12_k x=6 y=6 z=6 rule=3D0..6/F $3bo3$2bo/2$3bobo$obo/4bo$3bo$4bo$bo$2bo$bo/bo$2bo$bo$4bo$3bo$4bo/2$obo$3bobo/$2bo3$3bo! |
| D8_k | chiral square prismatic symmetry centered on a vertex | 
|
3D version=1 size=10 pos=3,3,3 #C D8_k x=4 y=4 z=4 rule=3D0..6/F 2bo$o$3bo$bo/$b2o$b2o/$b2o$b2o/bo$3bo$o$2bo! |
| D4_ke | chiral brick symmetry centered on a vertex orthogonal orientation |
|
3D version=1 size=12 pos=4,3,5 #C D4_ke x=4 y=6 z=2 rule=3D0..6/F bo$o$o$3bo$3bo$2bo/2bo$3bo$3bo$o$o$bo! |
| D4_kf | chiral brick symmetry centered on a vertex diagonal orientation |
|
3D version=1 size=10 pos=3,3,4 #C D4_kf x=4 y=4 z=2 rule=3D0..6/F $2o$2b2o/bo$bo$2bo$2bo! |
| D6_kc or simply D6_k |
chiral triangular prismatic symmetry centered on a vertex | 
|
3D version=1 size=10 pos=3,3,3 #C D6_k x=4 y=4 z=4 rule=3D0..6/F 2$bo/2bo$bo2$2bo/$o2bo$2bo/2$bo! |
Containing the inversion
| Name | Description | Example | RLE3 |
|---|---|---|---|
| ±O24_k | cubic symmetry centered on a vertex | 
|
3D version=1 size=8 pos=3,3,3 #C ±O24_k x=2 y=2 z=2 rule=3D0..6/F 2o$2o/2o$2o! |
| ±T12_k | pyritic symmetry centered on a vertex | 
|
3D version=1 size=12 pos=3,3,3 #C ±T12_k x=6 y=6 z=6 rule=3D0..6/F 2$bo2bo$bo2bo/2b2o$bo2bo3$bo2bo$2b2o/$o4bo3$o4bo/$o4bo3$o4bo/2b2o$bo2bo3$bo2bo$2b2o/2$bo2bo$bo2bo! |
| ±D8_k | square prismatic symmetry centered on a vertex | 
|
3D version=1 size=10 pos=4,4,3 #C ±D8_k x=2 y=2 z=4 rule=3D0..6/F 2o$2o/2o$2o/2o$2o/2o$2o! |
| ±C4_ke | square proprismatic symmetry centered on a vertex | 
|
3D version=1 size=10 pos=3,3,4 #C ±C4_ke x=4 y=4 z=2 rule=3D0..6/F 2bo$3o$b3o$bo/2bo$3o$b3o$bo! |
| ±D4_ke | brick symmetry centered on a vertex orthogonal orientation |
|
3D version=1 size=10 pos=3,3,4 #C ±D4_ke x=4 y=4 z=2 rule=3D0..6/F o2bo$4o$4o$o2bo/o2bo$4o$4o$o2bo! |
| ±D4_kf | brick symmetry centered on a vertex diagonal orientation |
|
3D version=1 size=8 pos=3,3,3 #C ±D4_kf x=2 y=2 z=2 rule=3D0..6/F o$bo/o$bo! |
| ±D6_kc or simply ±D6_k |
triangular antiprismatic symmetry centered on a vertex | 
|
3D version=1 size=8 pos=3,3,3 #C ±D6_k x=2 y=2 z=2 rule=3D0..6/F o/$bo! |
| ±C3_kc or simply ±C3_k |
triangular proantiprismatic symmetry centered on a vertex | 
|
3D version=1 size=10 pos=3,3,3 #C ±C3_k x=4 y=4 z=4 rule=3D0..6/F 2$bo/2bo$bo$3bo/$o$2bo$bo/$2bo! |
| ±C2_ke | espic symmetry centered on a vertex axis goes through edge |
|
3D version=1 size=10 pos=3,4,4 #C ±C2_ke x=4 y=2 z=2 rule=3D0..6/F 2o$2o/2b2o$2b2o! |
| ±C2_kf | espic symmetry centered on a vertex axis goes through face |
|
3D version=1 size=10 pos=4,4,3 #C ±C2_kf x=2 y=2 z=4 rule=3D0..6/F o/o/$bo/$bo! |
| ±C1_k | central inversion symmetry centered on a vertex | 
|
3D version=1 size=10 pos=3,4,4 #C ±C1_k x=4 y=2 z=2 rule=3D0..6/F o$bo/2bo$3bo! |
Other achiral symmetries
| Name | Description | Example | RLE3 |
|---|---|---|---|
| TO24_k | tetrahedral symmetry centered on a vertex | 
|
3D version=1 size=8 pos=3,3,3 #C TO24_k x=2 y=2 z=2 rule=3D0..6/F o$bo/bo$o! |
| DD8_ke | digonal antiprismatic symmetry centered on a vertex digon is diagonal |
|
3D version=1 size=10 pos=4,4,3 #C DD8_ke x=2 y=2 z=4 rule=3D0..6/F o$bo/o$bo/bo$o/bo$o! |
| DD8_kf | digonal antiprismatic symmetry centered on a vertex digon is orthogonal |
|
3D version=1 size=10 pos=3,3,4 #C DD8_kf x=4 y=4 z=2 rule=3D0..6/F $o2bo$o2bo/b2o3$b2o! |
| CC4_ke | prodappic symmetry centered on a vertex | 
|
3D version=1 size=10 pos=3,3,3 #C CC4_ke x=4 y=4 z=4 rule=3D0..6/F $o$3bo/$bo$2bo/$2bo$bo/2bo3$bo! |
Generators
Below is a list of generators for these groups.
Each row contains the group name, followed by the list of generators.
Each generator consists of a 3 × 3 matrix, M, followed by a length 3 vector, v. The transformation is M*x+v where x is the coordinates of the point.
Click on "Expand" to the right to view the generators data.
[ ['C1', []], ['C2_cf', [[[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 0]]]], ['C2_ce', [[[[0, -1, 0], [-1, 0, 0], [0, 0, -1]], [0, 0, 0]]]], ['CC2_/', [[[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]]]], ['±C1_c', [[[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 0, 0]]]], ['CC2_c', [[[[-1, 0, 0], [0, 1, 0], [0, 0, 1]], [0, 0, 0]]]], ['C3', [[[[0, 1, 0], [0, 0, 1], [1, 0, 0]], [0, 0, 0]]]], ['D4_cf', [[[[-1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 0, 0]], [[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 0]]]], ['C4_cf', [[[[0, -1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]]]], ['CC4_cf', [[[[0, 1, 0], [-1, 0, 0], [0, 0, -1]], [0, 0, 0]]]], ['CD4_cfd', [[[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[-1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 0, 0]]]], ['CD4_ce', [[[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 0]]]], ['±C2_ce', [[[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 0, 0]], [[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]]]], ['±C2_cf', [[[[1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 0, 0]], [[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 0]]]], ['CD4_cfo', [[[[1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 0]], [[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 0]]]], ['D4_ce', [[[[0, -1, 0], [-1, 0, 0], [0, 0, -1]], [0, 0, 0]], [[[-1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 0, 0]]]], ['CD6', [[[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[0, 1, 0], [0, 0, 1], [1, 0, 0]], [0, 0, 0]]]], ['D6_c', [[[[0, -1, 0], [-1, 0, 0], [0, 0, -1]], [0, 0, 0]], [[[0, 1, 0], [0, 0, 1], [1, 0, 0]], [0, 0, 0]]]], ['±C3_c', [[[[0, -1, 0], [0, 0, -1], [-1, 0, 0]], [0, 0, 0]]]], ['±D4_cf', [[[[-1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 0, 0]], [[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 0]], [[[-1, 0, 0], [0, 1, 0], [0, 0, 1]], [0, 0, 0]]]], ['±C4_cf', [[[[0, -1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 0, 0]]]], ['±D4_ce', [[[[1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 0]], [[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[-1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 0, 0]]]], ['DD8_ce', [[[[1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 0, 0]], [[[0, 1, 0], [-1, 0, 0], [0, 0, -1]], [0, 0, 0]]]], ['DD8_cf', [[[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 0]], [[[0, 1, 0], [-1, 0, 0], [0, 0, -1]], [0, 0, 0]]]], ['CD8_cf', [[[[1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 0, 0]], [[[0, -1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]]]], ['D8_c', [[[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 0]], [[[0, -1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]]]], ['±D6_c', [[[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[0, -1, 0], [0, 0, -1], [-1, 0, 0]], [0, 0, 0]]]], ['T12_c', [[[[0, 0, -1], [-1, 0, 0], [0, 1, 0]], [0, 0, 0]], [[[1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 0, 0]]]], ['±D8_c', [[[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 0]], [[[0, -1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 0, 0]]]], ['±T12_c', [[[[1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 0, 0]], [[[0, 1, 0], [0, 0, 1], [-1, 0, 0]], [0, 0, 0]]]], ['TO24_c', [[[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[-1, 0, 0], [0, 0, -1], [0, 1, 0]], [0, 0, 0]]]], ['O24_c', [[[[0, -1, 0], [-1, 0, 0], [0, 0, -1]], [0, 0, 0]], [[[1, 0, 0], [0, 0, 1], [0, -1, 0]], [0, 0, 0]]]], ['±O24_c', [[[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[1, 0, 0], [0, 0, 1], [0, -1, 0]], [0, 0, 0]]]], ['C2_fe', [[[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 1]]]], ['±C1_f', [[[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 0, 1]]]], ['C2_fk', [[[[0, 1, 0], [1, 0, 0], [0, 0, -1]], [0, 0, 1]]]], ['CC2_f', [[[[1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 1]]]], ['±C2_fe', [[[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 0, 1]], [[[-1, 0, 0], [0, 1, 0], [0, 0, 1]], [0, 0, 0]]]], ['CD4_fe', [[[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 1]], [[[1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 1]]]], ['CD4_fk', [[[[0, 1, 0], [1, 0, 0], [0, 0, -1]], [0, 0, 1]], [[[1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 1]]]], ['±C2_fk', [[[[0, -1, 0], [-1, 0, 0], [0, 0, -1]], [0, 0, 1]], [[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]]]], ['±C2_fc', [[[[1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 1]], [[[-1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 0, 0]]]], ['D4_fk', [[[[0, 1, 0], [1, 0, 0], [0, 0, -1]], [0, 0, 1]], [[[-1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 0, 0]]]], ['D4_fe', [[[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 1]], [[[-1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 0, 0]]]], ['CC4_fc', [[[[0, 1, 0], [-1, 0, 0], [0, 0, -1]], [0, 0, 1]]]], ['±C4_fc', [[[[0, -1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 0, 1]]]], ['DD8_fk', [[[[0, 1, 0], [1, 0, 0], [0, 0, -1]], [0, 0, 1]], [[[0, 1, 0], [-1, 0, 0], [0, 0, -1]], [0, 0, 1]]]], ['D8_f', [[[[0, 1, 0], [1, 0, 0], [0, 0, -1]], [0, 0, 1]], [[[0, -1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]]]], ['±D4_fk', [[[[0, -1, 0], [-1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 1]]]], ['±D4_fe', [[[[1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 1]], [[[-1, 0, 0], [0, 1, 0], [0, 0, 1]], [0, 0, 0]], [[[-1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 0, 0]]]], ['DD8_fe', [[[[0, -1, 0], [-1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[0, 1, 0], [-1, 0, 0], [0, 0, -1]], [0, 0, 1]]]], ['±D8_f', [[[[0, 1, 0], [1, 0, 0], [0, 0, -1]], [0, 0, 1]], [[[0, -1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 0, 1]]]], ['C2_ek', [[[[1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 1, 1]]]], ['±C1_e', [[[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 1, 1]]]], ['±C2_ef', [[[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 1, 1]], [[[1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 1]]]], ['±C2_ec', [[[[-1, 0, 0], [0, 0, -1], [0, -1, 0]], [0, 1, 1]], [[[1, 0, 0], [0, 0, 1], [0, 1, 0]], [0, 0, 0]]]], ['±C2_ek', [[[[-1, 0, 0], [0, 1, 0], [0, 0, 1]], [0, 0, 0]], [[[1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 1, 1]]]], ['D4_ec', [[[[-1, 0, 0], [0, 0, 1], [0, 1, 0]], [0, 0, 0]], [[[1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 1, 1]]]], ['CD4_eko', [[[[1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 1, 0]], [[[1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 1, 1]]]], ['C4_ek', [[[[1, 0, 0], [0, 0, 1], [0, -1, 0]], [0, 0, 1]]]], ['CD4_ekd', [[[[1, 0, 0], [0, 0, -1], [0, -1, 0]], [0, 1, 1]], [[[1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 1, 1]]]], ['D4_ef', [[[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 1]], [[[1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 1, 1]]]], ['CC4_ek', [[[[-1, 0, 0], [0, 0, -1], [0, 1, 0]], [0, 1, 0]]]], ['±C4_ek', [[[[1, 0, 0], [0, 0, 1], [0, -1, 0]], [0, 0, 1]], [[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 1, 1]]]], ['DD8_ec', [[[[-1, 0, 0], [0, 0, 1], [0, 1, 0]], [0, 0, 0]], [[[-1, 0, 0], [0, 0, -1], [0, 1, 0]], [0, 1, 0]]]], ['D8_e', [[[[-1, 0, 0], [0, 0, 1], [0, 1, 0]], [0, 0, 0]], [[[1, 0, 0], [0, 0, 1], [0, -1, 0]], [0, 0, 1]]]], ['±D4_ec', [[[[1, 0, 0], [0, 0, -1], [0, -1, 0]], [0, 1, 1]], [[[1, 0, 0], [0, 0, 1], [0, 1, 0]], [0, 0, 0]], [[[-1, 0, 0], [0, 1, 0], [0, 0, 1]], [0, 0, 0]]]], ['±D4_ef', [[[[-1, 0, 0], [0, 1, 0], [0, 0, 1]], [0, 0, 0]], [[[1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 1]], [[[1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 1, 1]]]], ['CD8_ek', [[[[1, 0, 0], [0, 0, -1], [0, -1, 0]], [0, 1, 1]], [[[1, 0, 0], [0, 0, 1], [0, -1, 0]], [0, 0, 1]]]], ['DD8_ef', [[[[1, 0, 0], [0, 0, -1], [0, -1, 0]], [0, 1, 1]], [[[-1, 0, 0], [0, 0, -1], [0, 1, 0]], [0, 1, 0]]]], ['±D8_e', [[[[-1, 0, 0], [0, 0, 1], [0, 1, 0]], [0, 0, 0]], [[[1, 0, 0], [0, 0, 1], [0, -1, 0]], [0, 0, 1]], [[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 1, 1]]]], ['±C1_k', [[[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [1, 1, 1]]]], ['D4_ke', [[[[-1, 0, 0], [0, -1, 0], [0, 0, 1]], [1, 1, 0]], [[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [1, 0, 1]]]], ['CC4_ke', [[[[0, 1, 0], [-1, 0, 0], [0, 0, -1]], [0, 1, 1]]]], ['±C2_kf', [[[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [1, 1, 1]], [[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]]]], ['±C2_ke', [[[[1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 1, 0]], [[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [1, 0, 1]]]], ['D4_kf', [[[[0, -1, 0], [-1, 0, 0], [0, 0, -1]], [1, 1, 1]], [[[-1, 0, 0], [0, -1, 0], [0, 0, 1]], [1, 1, 0]]]], ['D6_k', [[[[0, -1, 0], [-1, 0, 0], [0, 0, -1]], [1, 1, 1]], [[[0, 1, 0], [0, 0, 1], [1, 0, 0]], [0, 0, 0]]]], ['±C3_k', [[[[0, -1, 0], [0, 0, -1], [-1, 0, 0]], [1, 1, 1]]]], ['±D4_ke', [[[[-1, 0, 0], [0, -1, 0], [0, 0, 1]], [1, 1, 0]], [[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [1, 0, 1]], [[[-1, 0, 0], [0, 1, 0], [0, 0, 1]], [1, 0, 0]]]], ['±C4_ke', [[[[0, -1, 0], [1, 0, 0], [0, 0, 1]], [1, 0, 0]], [[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [1, 1, 1]]]], ['±D4_kf', [[[[1, 0, 0], [0, 1, 0], [0, 0, -1]], [0, 0, 1]], [[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[-1, 0, 0], [0, -1, 0], [0, 0, 1]], [1, 1, 0]]]], ['DD8_kf', [[[[1, 0, 0], [0, -1, 0], [0, 0, 1]], [0, 1, 0]], [[[0, 1, 0], [-1, 0, 0], [0, 0, -1]], [0, 1, 1]]]], ['DD8_ke', [[[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [1, 0, 1]], [[[0, 1, 0], [-1, 0, 0], [0, 0, -1]], [0, 1, 1]]]], ['D8_k', [[[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [1, 0, 1]], [[[0, -1, 0], [1, 0, 0], [0, 0, 1]], [1, 0, 0]]]], ['±D6_k', [[[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[0, -1, 0], [0, 0, -1], [-1, 0, 0]], [1, 1, 1]]]], ['T12_k', [[[[0, 0, -1], [-1, 0, 0], [0, 1, 0]], [1, 1, 0]], [[[1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 1, 1]]]], ['±D8_k', [[[[-1, 0, 0], [0, 1, 0], [0, 0, -1]], [1, 0, 1]], [[[0, -1, 0], [1, 0, 0], [0, 0, 1]], [1, 0, 0]], [[[-1, 0, 0], [0, -1, 0], [0, 0, -1]], [1, 1, 1]]]], ['±T12_k', [[[[1, 0, 0], [0, -1, 0], [0, 0, -1]], [0, 1, 1]], [[[0, 1, 0], [0, 0, 1], [-1, 0, 0]], [0, 0, 1]]]], ['TO24_k', [[[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[-1, 0, 0], [0, 0, -1], [0, 1, 0]], [1, 1, 0]]]], ['O24_k', [[[[0, -1, 0], [-1, 0, 0], [0, 0, -1]], [1, 1, 1]], [[[1, 0, 0], [0, 0, 1], [0, -1, 0]], [0, 0, 1]]]], ['±O24_k', [[[[0, 1, 0], [1, 0, 0], [0, 0, 1]], [0, 0, 0]], [[[1, 0, 0], [0, 0, 1], [0, -1, 0]], [0, 0, 1]]]] ]
And here we present the same generators in a more human-readable format, specifically where they send the vector [x,y,z].
Click on "Expand" to the right to view the generators data.
[ ['C1', []], ['C2_cf', [[-x, y, -z]]], ['C2_ce', [[-y, -x, -z]]], ['CC2_/', [[y, x, z]]], ['±C1_c', [[-x, -y, -z]]], ['CC2_c', [[-x, y, z]]], ['C3', [[y, z, x]]], ['D4_cf', [[-x, -y, z], [-x, y, -z]]], ['C4_cf', [[-y, x, z]]], ['CC4_cf', [[y, -x, -z]]], ['CD4_cfd', [[y, x, z], [-x, -y, z]]], ['CD4_ce', [[y, x, z], [x, y, -z]]], ['±C2_ce', [[-x, -y, -z], [y, x, z]]], ['±C2_cf', [[x, -y, z], [-x, y, -z]]], ['CD4_cfo', [[x, y, -z], [-x, y, -z]]], ['D4_ce', [[-y, -x, -z], [-x, -y, z]]], ['CD6', [[y, x, z], [y, z, x]]], ['D6_c', [[-y, -x, -z], [y, z, x]]], ['±C3_c', [[-y, -z, -x]]], ['±D4_cf', [[-x, -y, z], [-x, y, -z], [-x, y, z]]], ['±C4_cf', [[-y, x, z], [-x, -y, -z]]], ['±D4_ce', [[x, y, -z], [y, x, z], [-x, -y, z]]], ['DD8_ce', [[x, -y, z], [y, -x, -z]]], ['DD8_cf', [[-x, y, -z], [y, -x, -z]]], ['CD8_cf', [[x, -y, z], [-y, x, z]]], ['D8_c', [[-x, y, -z], [-y, x, z]]], ['±D6_c', [[y, x, z], [-y, -z, -x]]], ['T12_c', [[-z, -x, y], [x, -y, -z]]], ['±D8_c', [[-x, y, -z], [-y, x, z], [-x, -y, -z]]], ['±T12_c', [[x, -y, -z], [y, z, -x]]], ['TO24_c', [[y, x, z], [-x, -z, y]]], ['O24_c', [[-y, -x, -z], [x, z, -y]]], ['±O24_c', [[y, x, z], [x, z, -y]]], ['C2_fe', [[-x, y, -z + 1]]], ['±C1_f', [[-x, -y, -z + 1]]], ['C2_fk', [[y, x, -z + 1]]], ['CC2_f', [[x, y, -z + 1]]], ['±C2_fe', [[-x, -y, -z + 1], [-x, y, z]]], ['CD4_fe', [[-x, y, -z + 1], [x, y, -z + 1]]], ['CD4_fk', [[y, x, -z + 1], [x, y, -z + 1]]], ['±C2_fk', [[-y, -x, -z + 1], [y, x, z]]], ['±C2_fc', [[x, y, -z + 1], [-x, -y, z]]], ['D4_fk', [[y, x, -z + 1], [-x, -y, z]]], ['D4_fe', [[-x, y, -z + 1], [-x, -y, z]]], ['CC4_fc', [[y, -x, -z + 1]]], ['±C4_fc', [[-y, x, z], [-x, -y, -z + 1]]], ['DD8_fk', [[y, x, -z + 1], [y, -x, -z + 1]]], ['D8_f', [[y, x, -z + 1], [-y, x, z]]], ['±D4_fk', [[-y, -x, z], [y, x, z], [x, y, -z + 1]]], ['±D4_fe', [[x, y, -z + 1], [-x, y, z], [-x, -y, z]]], ['DD8_fe', [[-y, -x, z], [y, -x, -z + 1]]], ['±D8_f', [[y, x, -z + 1], [-y, x, z], [-x, -y, -z + 1]]], ['C2_ek', [[x, -y + 1, -z + 1]]], ['±C1_e', [[-x, -y + 1, -z + 1]]], ['±C2_ef', [[-x, -y + 1, -z + 1], [x, y, -z + 1]]], ['±C2_ec', [[-x, -z + 1, -y + 1], [x, z, y]]], ['±C2_ek', [[-x, y, z], [x, -y + 1, -z + 1]]], ['D4_ec', [[-x, z, y], [x, -y + 1, -z + 1]]], ['CD4_eko', [[x, -y + 1, z], [x, -y + 1, -z + 1]]], ['C4_ek', [[x, z, -y + 1]]], ['CD4_ekd', [[x, -z + 1, -y + 1], [x, -y + 1, -z + 1]]], ['D4_ef', [[-x, y, -z + 1], [x, -y + 1, -z + 1]]], ['CC4_ek', [[-x, -z + 1, y]]], ['±C4_ek', [[x, z, -y + 1], [-x, -y + 1, -z + 1]]], ['DD8_ec', [[-x, z, y], [-x, -z + 1, y]]], ['D8_e', [[-x, z, y], [x, z, -y + 1]]], ['±D4_ec', [[x, -z + 1, -y + 1], [x, z, y], [-x, y, z]]], ['±D4_ef', [[-x, y, z], [x, y, -z + 1], [x, -y + 1, -z + 1]]], ['CD8_ek', [[x, -z + 1, -y + 1], [x, z, -y + 1]]], ['DD8_ef', [[x, -z + 1, -y + 1], [-x, -z + 1, y]]], ['±D8_e', [[-x, z, y], [x, z, -y + 1], [-x, -y + 1, -z + 1]]], ['±C1_k', [[-x + 1, -y + 1, -z + 1]]], ['D4_ke', [[-x + 1, -y + 1, z], [-x + 1, y, -z + 1]]], ['CC4_ke', [[y, -x + 1, -z + 1]]], ['±C2_kf', [[-x + 1, -y + 1, -z + 1], [y, x, z]]], ['±C2_ke', [[x, -y + 1, z], [-x + 1, y, -z + 1]]], ['D4_kf', [[-y + 1, -x + 1, -z + 1], [-x + 1, -y + 1, z]]], ['D6_k', [[-y + 1, -x + 1, -z + 1], [y, z, x]]], ['±C3_k', [[-y + 1, -z + 1, -x + 1]]], ['±D4_ke', [[-x + 1, -y + 1, z], [-x + 1, y, -z + 1], [-x + 1, y, z]]], ['±C4_ke', [[-y + 1, x, z], [-x + 1, -y + 1, -z + 1]]], ['±D4_kf', [[x, y, -z + 1], [y, x, z], [-x + 1, -y + 1, z]]], ['DD8_kf', [[x, -y + 1, z], [y, -x + 1, -z + 1]]], ['DD8_ke', [[-x + 1, y, -z + 1], [y, -x + 1, -z + 1]]], ['D8_k', [[-x + 1, y, -z + 1], [-y + 1, x, z]]], ['±D6_k', [[y, x, z], [-y + 1, -z + 1, -x + 1]]], ['T12_k', [[-z + 1, -x + 1, y], [x, -y + 1, -z + 1]]], ['±D8_k', [[-x + 1, y, -z + 1], [-y + 1, x, z], [-x + 1, -y + 1, -z + 1]]], ['±T12_k', [[x, -y + 1, -z + 1], [y, z, -x + 1]]], ['TO24_k', [[y, x, z], [-x + 1, -z + 1, y]]], ['O24_k', [[-y + 1, -x + 1, -z + 1], [x, z, -y + 1]]], ['±O24_k', [[y, x, z], [x, z, -y + 1]]] ]
Objects
Below is a list of what are, I believe, minimal objects with the corresponding symmetry group. The objects are given as lists of points.
Click on "Expand" to the right to view the points.
[ ['C1', [[0, -1, -1], [0, 0, 0], [1, 0, 1], [1, 1, 1]]], ['C2_cf', [[-1, 0, -1], [0, 0, -1], [0, 0, 1], [0, 1, 0], [1, 0, 1]]], ['C2_ce', [[0, -1, -1], [0, 0, 0], [1, 0, 1]]], ['CC2_/', [[-1, -1, -1], [0, 0, 0], [1, 1, 0]]], ['±C1_c', [[-1, -1, -2], [-1, 0, -1], [0, 0, 0], [1, 0, 1], [1, 1, 2]]], ['CC2_c', [[0, -1, 0], [0, 0, 0], [0, 1, 1]]], ['C3', [[-1, 0, 1], [0, 0, 1], [0, 1, -1], [0, 1, 0], [1, -1, 0], [1, 0, 0]]], ['D4_cf', [[-1, -1, 1], [-1, 0, 0], [-1, 1, -1], [0, -2, 0], [0, 2, 0], [1, -1, -1], [1, 0, 0], [1, 1, 1]]], ['C4_cf', [[-2, -1, 1], [-1, 0, 0], [-1, 2, 1], [0, -1, 0], [0, 1, 0], [1, -2, 1], [1, 0, 0], [2, 1, 1]]], ['CC4_cf', [[-2, -1, 1], [-1, 0, 0], [-1, 2, -1], [0, -1, 0], [0, 1, 0], [1, -2, -1], [1, 0, 0], [2, 1, 1]]], ['CD4_cfd', [[-1, -1, 1], [0, 0, 0], [1, 1, 1]]], ['CD4_ce', [[0, 0, 0], [0, 1, 0], [1, 0, 0]]], ['±C2_ce', [[-1, -1, -1], [-1, -1, 0], [0, 0, 0], [1, 1, 0], [1, 1, 1]]], ['±C2_cf', [[-1, 0, -1], [-1, 0, 0], [0, 0, 0], [1, 0, 0], [1, 0, 1]]], ['CD4_cfo', [[-1, 1, 0], [0, 0, 0], [1, 1, 0]]], ['D4_ce', [[-2, -1, 1], [-1, -2, -1], [-1, -1, 0], [0, 0, 0], [1, 1, 0], [1, 2, -1], [2, 1, 1]]], ['CD6', [[0, 0, 0], [0, 0, 1], [0, 1, 0], [1, 0, 0]]], ['D6_c', [[-1, 0, 1], [0, 0, 0], [0, 1, -1], [1, -1, 0]]], ['±C3_c', [[-2, 0, -1], [-1, -2, 0], [-1, -1, -1], [0, -1, -2], [0, 0, 0], [0, 1, 2], [1, 1, 1], [1, 2, 0], [2, 0, 1]]], ['±D4_cf', [[-1, -1, 0], [-1, 0, 0], [-1, 1, 0], [0, 0, 0], [1, -1, 0], [1, 0, 0], [1, 1, 0]]], ['±C4_cf', [[-2, -1, 0], [-1, 0, 0], [-1, 2, 0], [0, -1, 0], [0, 1, 0], [1, -2, 0], [1, 0, 0], [2, 1, 0]]], ['±D4_ce', [[-1, -1, 0], [0, 0, 0], [1, 1, 0]]], ['DD8_ce', [[-1, 0, 1], [0, -1, -1], [0, 0, 0], [0, 1, -1], [1, 0, 1]]], ['DD8_cf', [[-1, -1, 1], [-1, 1, -1], [0, 0, -1], [0, 0, 0], [0, 0, 1], [1, -1, -1], [1, 1, 1]]], ['CD8_cf', [[-1, 0, 1], [0, -1, 1], [0, 0, 0], [0, 1, 1], [1, 0, 1]]], ['D8_c', [[-2, -1, 1], [-2, 1, -1], [-1, -2, -1], [-1, 0, 0], [-1, 2, 1], [0, -1, 0], [0, 1, 0], [1, -2, 1], [1, 0, 0], [1, 2, -1], [2, -1, -1], [2, 1, 1]]], ['±D6_c', [[-1, -1, -1], [0, 0, 0], [1, 1, 1]]], ['T12_c', [[-2, 0, -1], [-2, 0, 1], [-1, -2, 0], [-1, -1, 1], [-1, 1, -1], [-1, 2, 0], [0, -1, -2], [0, -1, 2], [0, 0, 0], [0, 1, -2], [0, 1, 2], [1, -2, 0], [1, -1, -1], [1, 1, 1], [1, 2, 0], [2, 0, -1], [2, 0, 1]]], ['±D8_c', [[0, 0, -1], [0, 0, 0], [0, 0, 1]]], ['±T12_c', [[-2, -1, 0], [-2, 1, 0], [-1, 0, -2], [-1, 0, 0], [-1, 0, 2], [0, -2, -1], [0, -2, 1], [0, -1, 0], [0, 0, -1], [0, 0, 1], [0, 1, 0], [0, 2, -1], [0, 2, 1], [1, 0, -2], [1, 0, 0], [1, 0, 2], [2, -1, 0], [2, 1, 0]]], ['TO24_c', [[-1, -1, 1], [-1, 1, -1], [0, 0, 0], [1, -1, -1], [1, 1, 1]]], ['O24_c', [[-3, -2, -1], [-3, -1, 2], [-3, 1, -2], [-3, 2, 1], [-2, -3, 1], [-2, -1, -3], [-2, -1, 0], [-2, 0, -1], [-2, 0, 1], [-2, 1, 0], [-2, 1, 3], [-2, 3, -1], [-1, -3, -2], [-1, -2, 0], [-1, -2, 3], [-1, 0, -2], [-1, 0, 2], [-1, 2, -3], [-1, 2, 0], [-1, 3, 2], [0, -2, -1], [0, -2, 1], [0, -1, -2], [0, -1, 2], [0, 1, -2], [0, 1, 2], [0, 2, -1], [0, 2, 1], [1, -3, 2], [1, -2, -3], [1, -2, 0], [1, 0, -2], [1, 0, 2], [1, 2, 0], [1, 2, 3], [1, 3, -2], [2, -3, -1], [2, -1, 0], [2, -1, 3], [2, 0, -1], [2, 0, 1], [2, 1, -3], [2, 1, 0], [2, 3, 1], [3, -2, 1], [3, -1, -2], [3, 1, 2], [3, 2, -1]]], ['±O24_c', [[0, 0, 0]]], ['C2_fe', [[-1, 1, 1], [0, 0, 0], [0, 0, 1], [1, 1, 0]]], ['±C1_f', [[-1, -1, 1], [-1, 0, 1], [0, 0, 0], [0, 0, 1], [1, 0, 0], [1, 1, 0]]], ['C2_fk', [[0, 0, 0], [0, 0, 1], [0, 1, 1], [1, 0, 0]]], ['CC2_f', [[-1, 0, 0], [-1, 0, 1], [0, 0, 0], [0, 0, 1], [1, 1, 0], [1, 1, 1]]], ['±C2_fe', [[0, -1, 0], [0, 0, 0], [0, 0, 1], [0, 1, 1]]], ['CD4_fe', [[0, 0, 0], [0, 0, 1], [0, 1, -1], [0, 1, 2]]], ['CD4_fk', [[0, 0, 0], [0, 0, 1], [1, 1, -1], [1, 1, 2]]], ['±C2_fk', [[-1, -1, 1], [0, 0, 0], [0, 0, 1], [1, 1, 0]]], ['±C2_fc', [[-1, -1, 0], [-1, -1, 1], [-1, 0, 0], [-1, 0, 1], [0, 0, 0], [0, 0, 1], [1, 0, 0], [1, 0, 1], [1, 1, 0], [1, 1, 1]]], ['D4_fk', [[-2, -1, 0], [-1, -2, 1], [-1, 0, 0], [0, -1, 1], [0, 1, 1], [1, 0, 0], [1, 2, 1], [2, 1, 0]]], ['D4_fe', [[-1, -1, 1], [-1, 0, 0], [-1, 0, 1], [-1, 1, 0], [0, 0, 0], [0, 0, 1], [1, -1, 0], [1, 0, 0], [1, 0, 1], [1, 1, 1]]], ['CC4_fc', [[-1, -1, 2], [-1, 0, 0], [-1, 1, -1], [0, -1, 1], [0, 1, 1], [1, -1, -1], [1, 0, 0], [1, 1, 2]]], ['±C4_fc', [[-2, -1, 0], [-2, -1, 1], [-1, 0, 0], [-1, 0, 1], [-1, 2, 0], [-1, 2, 1], [0, -1, 0], [0, -1, 1], [0, 1, 0], [0, 1, 1], [1, -2, 0], [1, -2, 1], [1, 0, 0], [1, 0, 1], [2, 1, 0], [2, 1, 1]]], ['DD8_fk', [[-1, 0, 0], [0, -1, 1], [0, 1, 1], [1, 0, 0]]], ['D8_f', [[-2, -1, 1], [-2, 1, 0], [-1, -2, 0], [-1, 0, 0], [-1, 0, 1], [-1, 2, 1], [0, -1, 0], [0, -1, 1], [0, 1, 0], [0, 1, 1], [1, -2, 1], [1, 0, 0], [1, 0, 1], [1, 2, 0], [2, -1, 0], [2, 1, 1]]], ['±D4_fk', [[-1, -1, 0], [-1, -1, 1], [0, 0, 0], [0, 0, 1], [1, 1, 0], [1, 1, 1]]], ['±D4_fe', [[-1, 0, 0], [-1, 0, 1], [0, 0, 0], [0, 0, 1], [1, 0, 0], [1, 0, 1]]], ['DD8_fe', [[-1, -1, 0], [-1, 1, 1], [0, 0, 0], [0, 0, 1], [1, -1, 1], [1, 1, 0]]], ['±D8_f', [[0, 0, 0], [0, 0, 1]]], ['C2_ek', [[-1, 0, -1], [-1, 1, 2], [0, 0, 0], [0, 1, 1]]], ['±C1_e', [[-1, 2, 1], [0, 0, 0], [0, 1, 1], [1, -1, 0]]], ['±C2_ef', [[-1, 1, 0], [-1, 1, 1], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, 0, 0], [1, 0, 1]]], ['±C2_ec', [[-1, 1, 1], [0, 0, 0], [0, 1, 1], [1, 0, 0]]], ['±C2_ek', [[0, -1, 0], [0, 0, 0], [0, 1, 1], [0, 2, 1]]], ['D4_ec', [[-1, 0, -1], [-1, 1, 2], [0, 0, 0], [0, 1, 1], [1, -1, 0], [1, 2, 1]]], ['CD4_eko', [[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, -1, 0], [1, -1, 1], [1, 2, 0], [1, 2, 1]]], ['C4_ek', [[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, -1, 0], [1, 0, 2], [1, 1, -1], [1, 2, 1]]], ['CD4_ekd', [[0, 0, 0], [0, 1, 1], [1, -1, -1], [1, 2, 2]]], ['D4_ef', [[-1, -1, 1], [-1, 2, 0], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, -1, 0], [1, 2, 1]]], ['CC4_ek', [[-1, 0, 2], [-1, 1, -1], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, -1, 0], [1, 2, 1]]], ['±C4_ek', [[0, -1, 0], [0, 0, 0], [0, 0, 1], [0, 0, 2], [0, 1, -1], [0, 1, 0], [0, 1, 1], [0, 2, 1]]], ['DD8_ec', [[-1, 0, -1], [-1, 0, 2], [-1, 1, -1], [-1, 1, 2], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, -1, 0], [1, -1, 1], [1, 2, 0], [1, 2, 1]]], ['D8_e', [[-1, -1, 1], [-1, 0, -1], [-1, 1, 2], [-1, 2, 0], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, -1, 0], [1, 0, 2], [1, 1, -1], [1, 2, 1]]], ['±D4_ec', [[0, 0, 0], [0, 1, 1]]], ['±D4_ef', [[0, -1, 0], [0, -1, 1], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [0, 2, 0], [0, 2, 1]]], ['CD8_ek', [[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, -1, 0], [1, -1, 1], [1, 0, -1], [1, 0, 2], [1, 1, -1], [1, 1, 2], [1, 2, 0], [1, 2, 1]]], ['DD8_ef', [[-1, 0, 1], [-1, 1, 0], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, 0, 0], [1, 1, 1]]], ['±D8_e', [[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1]]], ['±C1_k', [[-1, 0, 0], [0, 1, 0], [1, 0, 1], [2, 1, 1]]], ['D4_ke', [[-1, -1, 0], [-1, 0, 0], [-1, 1, 1], [-1, 2, 1], [0, -2, 0], [0, 3, 1], [1, -2, 1], [1, 3, 0], [2, -1, 1], [2, 0, 1], [2, 1, 0], [2, 2, 0]]], ['CC4_ke', [[-1, 0, -1], [0, 0, 0], [0, 1, 1], [0, 2, 2], [1, -1, 2], [1, 0, 1], [1, 1, 0], [2, 1, -1]]], ['±C2_kf', [[0, 0, -1], [0, 0, 0], [1, 1, 1], [1, 1, 2]]], ['±C2_ke', [[-1, 0, 0], [-1, 1, 0], [0, 0, 0], [0, 1, 0], [1, 0, 1], [1, 1, 1], [2, 0, 1], [2, 1, 1]]], ['D4_kf', [[-1, 0, 0], [0, -1, 1], [0, 0, 0], [0, 0, 1], [1, 1, 0], [1, 1, 1], [1, 2, 1], [2, 1, 0]]], ['D6_k', [[-1, 0, 1], [0, 0, 0], [0, 1, -1], [0, 1, 2], [1, -1, 0], [1, 1, 1], [1, 2, 0], [2, 0, 1]]], ['±C3_k', [[-1, 0, 1], [0, 0, 0], [0, 1, -1], [0, 2, 1], [1, -1, 0], [1, 0, 2], [1, 1, 1], [2, 1, 0]]], ['±D4_ke', [[-1, -1, 0], [-1, -1, 1], [-1, 0, 0], [-1, 0, 1], [-1, 1, 0], [-1, 1, 1], [-1, 2, 0], [-1, 2, 1], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, 0, 0], [1, 0, 1], [1, 1, 0], [1, 1, 1], [2, -1, 0], [2, -1, 1], [2, 0, 0], [2, 0, 1], [2, 1, 0], [2, 1, 1], [2, 2, 0], [2, 2, 1]]], ['±C4_ke', [[-1, 0, 0], [-1, 0, 1], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [0, 2, 0], [0, 2, 1], [1, -1, 0], [1, -1, 1], [1, 0, 0], [1, 0, 1], [1, 1, 0], [1, 1, 1], [2, 1, 0], [2, 1, 1]]], ['±D4_kf', [[0, 0, 0], [0, 0, 1], [1, 1, 0], [1, 1, 1]]], ['DD8_kf', [[-1, 0, 0], [-1, 1, 0], [0, -1, 1], [0, 2, 1], [1, -1, 1], [1, 2, 1], [2, 0, 0], [2, 1, 0]]], ['DD8_ke', [[0, 0, -1], [0, 0, 0], [0, 1, 1], [0, 1, 2], [1, 0, 1], [1, 0, 2], [1, 1, -1], [1, 1, 0]]], ['D8_k', [[-1, 0, -1], [-1, 1, 2], [0, -1, 2], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [0, 2, -1], [1, -1, -1], [1, 0, 0], [1, 0, 1], [1, 1, 0], [1, 1, 1], [1, 2, 2], [2, 0, 2], [2, 1, -1]]], ['±D6_k', [[0, 0, 0], [1, 1, 1]]], ['T12_k', [[-2, 0, 2], [-2, 1, -1], [-1, -2, 1], [-1, 0, 1], [-1, 1, 0], [-1, 3, 0], [0, -1, 1], [0, -1, 3], [0, 0, 2], [0, 1, -1], [0, 2, -2], [0, 2, 0], [1, -1, -2], [1, -1, 0], [1, 0, -1], [1, 1, 2], [1, 2, 1], [1, 2, 3], [2, -2, 0], [2, 0, 0], [2, 1, 1], [2, 3, 1], [3, 0, -1], [3, 1, 2]]], ['±D8_k', [[0, 0, -1], [0, 0, 0], [0, 0, 1], [0, 0, 2], [0, 1, -1], [0, 1, 0], [0, 1, 1], [0, 1, 2], [1, 0, -1], [1, 0, 0], [1, 0, 1], [1, 0, 2], [1, 1, -1], [1, 1, 0], [1, 1, 1], [1, 1, 2]]], ['±T12_k', [[-2, -1, 0], [-2, -1, 1], [-2, 2, 0], [-2, 2, 1], [-1, -1, -1], [-1, -1, 2], [-1, 0, -2], [-1, 0, 3], [-1, 1, -2], [-1, 1, 3], [-1, 2, -1], [-1, 2, 2], [0, -2, -1], [0, -2, 2], [0, 3, -1], [0, 3, 2], [1, -2, -1], [1, -2, 2], [1, 3, -1], [1, 3, 2], [2, -1, -1], [2, -1, 2], [2, 0, -2], [2, 0, 3], [2, 1, -2], [2, 1, 3], [2, 2, -1], [2, 2, 2], [3, -1, 0], [3, -1, 1], [3, 2, 0], [3, 2, 1]]], ['TO24_k', [[0, 0, 0], [0, 1, 1], [1, 0, 1], [1, 1, 0]]], ['O24_k', [[-2, -1, 1], [-2, 0, -1], [-2, 1, 2], [-2, 2, 0], [-1, -2, 0], [-1, -1, -1], [-1, -1, 2], [-1, 0, 3], [-1, 1, -2], [-1, 2, -1], [-1, 2, 2], [-1, 3, 1], [0, -2, 2], [0, -1, -2], [0, 2, 3], [0, 3, -1], [1, -2, -1], [1, -1, 3], [1, 2, -2], [1, 3, 2], [2, -2, 1], [2, -1, -1], [2, -1, 2], [2, 0, -2], [2, 1, 3], [2, 2, -1], [2, 2, 2], [2, 3, 0], [3, -1, 0], [3, 0, 2], [3, 1, -1], [3, 2, 1]]], ['±O24_k', [[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, 0, 0], [1, 0, 1], [1, 1, 0], [1, 1, 1]]] ]
Kinetic symmetries
In addition to the 92 above static symmetries there are also 245 kinetic symmetries with mod unequal to their period (220 with period/mod=2, 7 with period/mod=3, 14 with period/mod=4, and 4 with period/mod=6).
"Composite symmetry" refers to the resulting symmetry of the pattern created from each of the oscillator's phases.
| Static symmetry | Composite symmetry | period/mod | Description | Gutteroids |
|---|---|---|---|---|
| C1 | C2_cf | 2 | ||
| C1 | C2_ce | 2 | ||
| C1 | CC2_/ | 2 | ||
| C1 | ±C1_c | 2 | ||
| C1 | CC2_c | 2 | ||
| C2_cf | D4_cf | 2 | ||
| C2_cf | C4_cf | 2 | ||
| C2_cf | CC4_cf | 2 | ||
| C2_cf | CD4_cfd | 2 | ||
| CC2_/ | CD4_cfd | 2 | ||
| C2_ce | CD4_ce | 2 | ||
| CC2_/ | CD4_ce | 2 | ||
| CC2_c | CD4_ce | 2 | ||
| C2_ce | ±C2_ce | 2 | ||
| CC2_/ | ±C2_ce | 2 | ||
| ±C1_c | ±C2_ce | 2 | ||
| C2_cf | ±C2_cf | 2 | ||
| ±C1_c | ±C2_cf | 2 | ||
| CC2_c | ±C2_cf | 2 | ||
| C2_cf | CD4_cfo | 2 | ||
| CC2_c | CD4_cfo | 2 | ||
| C2_cf | D4_ce | 2 | ||
| C2_ce | D4_ce | 2 | ||
| C3 | CD6 | 2 | ||
| C3 | D6_c | 2 | ||
| C3 | ±C3_c | 2 | ||
| D4_cf | ±D4_cf | 2 | ||
| ±C2_cf | ±D4_cf | 2 | ||
| CD4_cfo | ±D4_cf | 2 | ||
| C4_cf | ±C4_cf | 2 | ||
| CC4_cf | ±C4_cf | 2 | ||
| ±C2_cf | ±C4_cf | 2 | ||
| CD4_cfd | ±D4_ce | 2 | ||
| CD4_ce | ±D4_ce | 2 | ||
| ±C2_ce | ±D4_ce | 2 | ||
| ±C2_cf | ±D4_ce | 2 | ||
| D4_ce | ±D4_ce | 2 | ||
| CC4_cf | DD8_ce | 2 | ||
| CD4_cfo | DD8_ce | 2 | ||
| D4_ce | DD8_ce | 2 | ||
| D4_cf | DD8_cf | 2 | ||
| CC4_cf | DD8_cf | 2 | ||
| CD4_cfd | DD8_cf | 2 | ||
| C4_cf | CD8_cf | 2 | ||
| CD4_cfd | CD8_cf | 2 | ||
| CD4_cfo | CD8_cf | 2 | ||
| D4_cf | D8_c | 2 | ||
| C4_cf | D8_c | 2 | ||
| D4_ce | D8_c | 2 | ||
| CD6 | ±D6_c | 2 | ||
| D6_c | ±D6_c | 2 | ||
| ±C3_c | ±D6_c | 2 | ||
| ±D4_cf | ±D8_c | 2 | ||
| ±C4_cf | ±D8_c | 2 | ||
| ±D4_ce | ±D8_c | 2 | ||
| DD8_ce | ±D8_c | 2 | ||
| DD8_cf | ±D8_c | 2 | ||
| CD8_cf | ±D8_c | 2 | ||
| D8_c | ±D8_c | 2 | ||
| T12_c | ±T12_c | 2 | ||
| T12_c | TO24_c | 2 | ||
| T12_c | O24_c | 2 | ||
| ±T12_c | ±O24_c | 2 | ||
| TO24_c | ±O24_c | 2 | ||
| O24_c | ±O24_c | 2 | ||
| C1 | C2_fe | 2 | ||
| C1 | ±C1_f | 2 | ||
| C1 | C2_fk | 2 | ||
| C1 | CC2_f | 2 | ||
| C2_fe | ±C2_fe | 2 | ||
| CC2_c | ±C2_fe | 2 | ||
| ±C1_f | ±C2_fe | 2 | ||
| C2_fe | CD4_fe | 2 | ||
| CC2_c | CD4_fe | 2 | ||
| CC2_f | CD4_fe | 2 | ||
| CC2_/ | CD4_fk | 2 | ||
| C2_fk | CD4_fk | 2 | ||
| CC2_f | CD4_fk | 2 | ||
| CC2_/ | ±C2_fk | 2 | ||
| ±C1_f | ±C2_fk | 2 | ||
| C2_fk | ±C2_fk | 2 | ||
| C2_cf | ±C2_fc | 2 | ||
| ±C1_f | ±C2_fc | 2 | ||
| CC2_f | ±C2_fc | 2 | ||
| C2_cf | D4_fk | 2 | ||
| C2_fk | D4_fk | 2 | ||
| C2_cf | D4_fe | 2 | ||
| C2_fe | D4_fe | 2 | ||
| C2_cf | CC4_fc | 2 | ||
| ±C2_fc | ±C4_fc | 2 | ||
| C4_cf | ±C4_fc | 2 | ||
| CC4_fc | ±C4_fc | 2 | ||
| D4_fk | DD8_fk | 2 | ||
| CD4_cfo | DD8_fk | 2 | ||
| CC4_fc | DD8_fk | 2 | ||
| D4_fk | D8_f | 2 | ||
| C4_cf | D8_f | 2 | ||
| D4_fe | D8_f | 2 | ||
| CD4_fk | ±D4_fk | 2 | ||
| ±C2_fk | ±D4_fk | 2 | ||
| ±C2_fc | ±D4_fk | 2 | ||
| D4_fk | ±D4_fk | 2 | ||
| CD4_cfd | ±D4_fk | 2 | ||
| ±C2_fe | ±D4_fe | 2 | ||
| CD4_fe | ±D4_fe | 2 | ||
| ±C2_fc | ±D4_fe | 2 | ||
| CD4_cfo | ±D4_fe | 2 | ||
| D4_fe | ±D4_fe | 2 | ||
| CD4_cfd | DD8_fe | 2 | ||
| D4_fe | DD8_fe | 2 | ||
| CC4_fc | DD8_fe | 2 | ||
| ±C4_fc | ±D8_f | 2 | ||
| DD8_fk | ±D8_f | 2 | ||
| D8_f | ±D8_f | 2 | ||
| ±D4_fk | ±D8_f | 2 | ||
| ±D4_fe | ±D8_f | 2 | ||
| CD8_cf | ±D8_f | 2 | ||
| DD8_fe | ±D8_f | 2 | ||
| C1 | C2_ek | 2 | ||
| C1 | ±C1_e | 2 | ||
| C2_fe | ±C2_ef | 2 | ||
| CC2_f | ±C2_ef | 2 | ||
| ±C1_e | ±C2_ef | 2 | ||
| CC2_/ | ±C2_ec | 2 | ||
| ±C1_e | ±C2_ec | 2 | ||
| C2_ce | ±C2_ec | 2 | ||
| C2_ek | ±C2_ek | 2 | ||
| ±C1_e | ±C2_ek | 2 | ||
| CC2_c | ±C2_ek | 2 | ||
| C2_ek | D4_ec | 2 | ||
| C2_ce | D4_ec | 2 | ||
| C2_ek | CD4_eko | 2 | ||
| CC2_f | CD4_eko | 2 | ||
| C2_ek | C4_ek | 2 | ||
| CC2_/ | CD4_ekd | 2 | ||
| C2_ek | CD4_ekd | 2 | ||
| C2_ek | D4_ef | 2 | ||
| C2_fe | D4_ef | 2 | ||
| C2_ek | CC4_ek | 2 | ||
| ±C2_ek | ±C4_ek | 2 | ||
| C4_ek | ±C4_ek | 2 | ||
| CC4_ek | ±C4_ek | 2 | ||
| D4_ec | DD8_ec | 2 | ||
| CD4_eko | DD8_ec | 2 | ||
| CC4_ek | DD8_ec | 2 | ||
| D4_ec | D8_e | 2 | ||
| C4_ek | D8_e | 2 | ||
| D4_ef | D8_e | 2 | ||
| CD4_ce | ±D4_ec | 2 | ||
| ±C2_ec | ±D4_ec | 2 | ||
| ±C2_ek | ±D4_ec | 2 | ||
| D4_ec | ±D4_ec | 2 | ||
| CD4_ekd | ±D4_ec | 2 | ||
| ±C2_ef | ±D4_ef | 2 | ||
| CD4_fe | ±D4_ef | 2 | ||
| ±C2_ek | ±D4_ef | 2 | ||
| CD4_eko | ±D4_ef | 2 | ||
| D4_ef | ±D4_ef | 2 | ||
| CD4_eko | CD8_ek | 2 | ||
| C4_ek | CD8_ek | 2 | ||
| CD4_ekd | CD8_ek | 2 | ||
| CD4_ekd | DD8_ef | 2 | ||
| D4_ef | DD8_ef | 2 | ||
| CC4_ek | DD8_ef | 2 | ||
| ±C4_ek | ±D8_e | 2 | ||
| DD8_ec | ±D8_e | 2 | ||
| D8_e | ±D8_e | 2 | ||
| ±D4_ec | ±D8_e | 2 | ||
| ±D4_ef | ±D8_e | 2 | ||
| CD8_ek | ±D8_e | 2 | ||
| DD8_ef | ±D8_e | 2 | ||
| C1 | ±C1_k | 2 | ||
| C2_ek | D4_ke | 2 | ||
| C2_ek | CC4_ke | 2 | ||
| C2_fk | ±C2_kf | 2 | ||
| CC2_/ | ±C2_kf | 2 | ||
| ±C1_k | ±C2_kf | 2 | ||
| C2_ek | ±C2_ke | 2 | ||
| ±C1_k | ±C2_ke | 2 | ||
| CC2_f | ±C2_ke | 2 | ||
| C2_ek | D4_kf | 2 | ||
| C2_fk | D4_kf | 2 | ||
| C3 | D6_k | 2 | ||
| C3 | ±C3_k | 2 | ||
| D4_ke | ±D4_ke | 2 | ||
| ±C2_ke | ±D4_ke | 2 | ||
| CD4_eko | ±D4_ke | 2 | ||
| C4_ek | ±C4_ke | 2 | ||
| CC4_ke | ±C4_ke | 2 | ||
| ±C2_ke | ±C4_ke | 2 | ||
| CD4_ekd | ±D4_kf | 2 | ||
| CD4_fk | ±D4_kf | 2 | ||
| ±C2_kf | ±D4_kf | 2 | ||
| ±C2_ke | ±D4_kf | 2 | ||
| D4_kf | ±D4_kf | 2 | ||
| CC4_ke | DD8_kf | 2 | ||
| CD4_eko | DD8_kf | 2 | ||
| D4_kf | DD8_kf | 2 | ||
| D4_ke | DD8_ke | 2 | ||
| CC4_ke | DD8_ke | 2 | ||
| CD4_ekd | DD8_ke | 2 | ||
| D4_ke | D8_k | 2 | ||
| C4_ek | D8_k | 2 | ||
| D4_kf | D8_k | 2 | ||
| CD6 | ±D6_k | 2 | ||
| D6_k | ±D6_k | 2 | ||
| ±C3_k | ±D6_k | 2 | ||
| ±D4_ke | ±D8_k | 2 | ||
| ±C4_ke | ±D8_k | 2 | ||
| ±D4_kf | ±D8_k | 2 | ||
| DD8_kf | ±D8_k | 2 | ||
| DD8_ke | ±D8_k | 2 | ||
| CD8_ek | ±D8_k | 2 | ||
| D8_k | ±D8_k | 2 | ||
| T12_k | ±T12_k | 2 | ||
| T12_k | TO24_k | 2 | ||
| T12_k | O24_k | 2 | ||
| ±T12_k | ±O24_k | 2 | ||
| TO24_k | ±O24_k | 2 | ||
| O24_k | ±O24_k | 2 | ||
| C1 | C3 | 3 | ||
| ±C1_c | ±C3_c | 3 | ||
| D4_cf | T12_c | 3 | ||
| ±D4_cf | ±T12_c | 3 | ||
| ±C1_k | ±C3_k | 3 | ||
| D4_ke | T12_k | 3 | ||
| ±D4_ke | ±T12_k | 3 | ||
| C1 | C4_cf | 4 | ||
| C1 | CC4_cf | 4 | ||
| ±C1_c | ±C4_cf | 4 | ||
| CC2_c | ±C4_cf | 4 | ||
| C1 | CC4_fc | 4 | ||
| ±C1_f | ±C4_fc | 4 | ||
| CC2_f | ±C4_fc | 4 | ||
| C1 | C4_ek | 4 | ||
| C1 | CC4_ek | 4 | ||
| ±C1_e | ±C4_ek | 4 | ||
| CC2_c | ±C4_ek | 4 | ||
| C1 | CC4_ke | 4 | ||
| ±C1_k | ±C4_ke | 4 | ||
| CC2_f | ±C4_ke | 4 | ||
| C1 | ±C3_c | 6 | ||
| D4_cf | ±T12_c | 6 | ||
| C1 | ±C3_k | 6 | ||
| D4_ke | ±T12_k | 6 |
Spaceship symmetries
With period equal to mod
There are 22 spaceship symmetries with period equal to the mod.
| Static equivalent | Direction | Description |
|---|---|---|
| C1 | any | no symmetry |
| C2_cf | orthogonal | chiral digonal pyramidal symmetry, a.k.a., 180 degree rotation (line goes through cells and faces) |
| C2_ce | diagonal | chiral digonal pyramidal symmetry, a.k.a., 180 degree rotation (line goes through cells and edges) |
| CC2_/ | any in diagonal plane | reflection (plane goes through cells, faces, edges, and vertices) |
| CC2_c | any in orthogonal plane | reflection (plane goes through cells, edges, and faces) |
| C3 | paradiagonal | chiral triangular pyramidal symmetry, a.k.a., 120 degree rotation (line goes through vertices and cells) |
| C4_cf | orthogonal | chiral square pyramidal symmetry, a.k.a., 90 degree rotation (line goes through cells and faces) |
| CD4_cfd | orthogonal | digonal pyramidal symmetry (line goes through cells and faces), reflection planes have diagonal orientation |
| CD4_ce | diagonal | digonal pyramidal symmetry (line goes through cells and edges) |
| CD4_cfo | orthogonal | digonal pyramidal symmetry (line goes through cells and faces), reflection planes have orthogonal orientation |
| CD6 | paradiagonal | triangular pyramidal symmetry (line goes through vertices and cells) |
| CD8_cf | orthogonal | square pyramidal symmetry (line goes through cells and faces) |
| C2_fe | orthogonal | chiral digonal pyramidal symmetry, a.k.a., 180 degree rotation (line goes through faces and edges) |
| C2_fk | diagonal | chiral digonal pyramidal symmetry, a.k.a., 180 degree rotation (line goes through faces and vertices) |
| CC2_f | any in orthogonal plane | reflection (plane goes through faces, edges, and vertices) |
| CD4_fe | orthogonal | digonal pyramidal symmetry (line goes through faces and edges) |
| CD4_fk | diagonal | digonal pyramidal symmetry (line goes through faces and vertices) |
| C2_ek | orthogonal | chiral digonal pyramidal symmetry, a.k.a., 180 degree rotation (line goes through edges and vertices) |
| CD4_eko | orthogonal | digonal pyramidal symmetry (line goes through edges and vertices), reflection planes have orthogonal orientation |
| C4_ek | orthogonal | chiral square pyramidal symmetry, a.k.a., 90 degree rotation (line goes through edges and vertices) |
| CD4_ekd | orthogonal | digonal pyramidal symmetry (line goes through edges and vertices), reflection planes have diagonal orientation |
| CD8_ek | orthogonal | square pyramidal symmetry (line goes through edges and vertices) |
With period different from mod
There are 43 spaceship symmetries with period different from mod.
| Static symmetry | Composite symmetry | period/mod | direction | Description |
|---|---|---|---|---|
| C1 | C2_cf | 2 | orthogonal | Pattern is asymmetric 180 degree skew-rotation (line goes through cells and faces) |
| C1 | C2_fe | 2 | orthogonal | Pattern is asymmetric 180 degree skew-rotation (line goes through faces and edges) |
| C1 | C2_ek | 2 | orthogonal | Pattern is asymmetric 180 degree skew-rotation (line goes through edges and vertices) |
| C1 | C2_ce | 2 | diagonal | Pattern is asymmetric 180 degree skew-rotation (line goes through cells and edges) |
| C1 | C2_fk | 2 | diagonal | Pattern is asymmetric 180 degree skew-rotation (line goes through faces and vertices) |
| C1 | N/A | 2 | diagonal | Pattern is asymmetric 180 degree skew-rotation (line goes through edges) |
| C1 | N/A | 2 | diagonal | Pattern is asymmetric 180 degree skew-rotation (line goes through faces) |
| C1 | CC2_c | 2 | any in orthogonal plane | Pattern is asymmetric skew-reflection (plane goes through cells, edges, and faces) |
| C1 | CC2_f | 2 | any in orthogonal plane | Pattern is asymmetric skew-reflection (plane goes through faces, edges, and vertices) |
| C1 | CC2_/ | 2 | any in diagonal plane | Pattern is asymmetric skew-reflection (plane goes through cells, faces, edges, and vertices) |
| C1 | N/A | 2 | any in diagonal plane | Pattern is asymmetric skew-reflection (plane goes through faces and edges) |
| C2_cf | CD4_cfd | 2 | orthogonal | Pattern is symmetric under 180 degree rotation (line goes through cells and faces) skew-reflections through diagonal planes |
| C2_ek | CD4_ekd | 2 | orthogonal | Pattern is symmetric under 180 degree rotation (line goes through edges and vertices) skew-reflections through diagonal planes |
| C2_cf | CD4_cfo | 2 | orthogonal | Pattern is symmetric under 180 degree rotation (line goes through cells and faces) skew-reflections through orthogonal planes |
| C2_ek | CD4_eko | 2 | orthogonal | Pattern is symmetric under 180 degree rotation (line goes through edges and vertices) skew-reflections through orthogonal planes |
| C2_cf | C4_cf | 2 | orthogonal | Pattern is symmetric under 180 degree rotation (line goes through cells and faces) 90 degree skew-rotation |
| C2_ek | C4_ek | 2 | orthogonal | Pattern is symmetric under 180 degree rotation (line goes through edges and vertices) 90 degree skew-rotation |
| C2_ce | CD4_ce | 2 | diagonal | Pattern is symmetric under 180 degree rotation (line goes through cells and edges) skew-reflections |
| C2_fk | CD4_fk | 2 | diagonal | Pattern is symmetric under 180 degree rotation (line goes through faces and vertices) skew-reflections |
| C2_fe | CD4_fe | 2 | orthogonal | Pattern is symmetric under 180 degree rotation (line goes through faces and edges) skew-reflections |
| CC2_/ | CD4_cfd | 2 | orthogonal | Pattern is symmetric under reflection (plane goes through cells, faces, edges, and vertices) 180 degree skew-rotation (line goes through cells and faces) |
| CC2_/ | CD4_ekd | 2 | orthogonal | Pattern is symmetric under reflection (plane goes through cells, faces, edges, and vertices) 180 degree skew-rotation (line goes through edges and vertices) |
| CC2_/ | CD4_ce | 2 | diagonal | Pattern is symmetric under reflection (plane goes through cells, faces, edges, and vertices) 180 degree skew-rotation (line goes through cells and edges) |
| CC2_/ | CD4_fk | 2 | diagonal | Pattern is symmetric under reflection (plane goes through cells, faces, edges, and vertices) 180 degree skew-rotation (line goes through faces and vertices) |
| CC2_c | CD4_ce | 2 | diagonal | Pattern is symmetric under reflection (plane goes through cells, edges, and faces) 180 degree skew-rotation (line goes through cells and edges) |
| CC2_c | N/A | 2 | diagonal | Pattern is symmetric under reflection (plane goes through cells, edges, and faces) 180 degree skew-rotation (line goes through edges) |
| CC2_f | CD4_fk | 2 | diagonal | Pattern is symmetric under reflection (plane goes through faces, edges, and vertices) 180 degree skew-rotation (line goes through faces and vertices) |
| CC2_f | N/A | 2 | diagonal | Pattern is symmetric under reflection (plane goes through faces, edges, and vertices) 180 degree skew-rotation (line goes through faces) |
| CC2_c | CD4_cfo | 2 | orthogonal | Pattern is symmetric under reflection (plane goes through cells, edges, and faces) 180 degree skew-rotation (line goes through cells and faces) |
| CC2_f | CD4_eko | 2 | orthogonal | Pattern is symmetric under reflection (plane goes through faces, edges, and vertices) 180 degree skew-rotation (line goes through edges and vertices) |
| CC2_c | CD4_fe | 2 | orthogonal | Pattern is symmetric under reflection (plane goes through cells, edges, and faces) 180 degree skew-rotation (line goes through faces and edges) |
| CC2_f | CD4_fe | 2 | orthogonal | Pattern is symmetric under reflection (plane goes through faces, edges, and vertices) 180 degree skew-rotation (line goes through faces and edges) |
| C3 | CD6 | 2 | paradiagonal | Pattern is symmetric under 120 degree rotation (line goes through cells and vertices) skew-reflections |
| CD4_cfd | CD8_cf | 2 | orthogonal | Pattern has digonal pyramidal symmetry (line goes through cells and faces), reflection planes have diagonal orientation 90 degree skew-rotation |
| CD4_ekd | CD8_ek | 2 | orthogonal | Pattern has digonal pyramidal symmetry (line goes through edges and vertices), reflection planes have diagonal orientation 90 degree skew-rotation |
| CD4_cfo | CD8_cf | 2 | orthogonal | Pattern has digonal pyramidal symmetry (line goes through cells and faces), reflection planes have orthogonal orientation 90 degree skew-rotation |
| CD4_eko | CD8_ek | 2 | orthogonal | Pattern has digonal pyramidal symmetry (line goes through edges and vertices), reflection planes have orthogonal orientation 90 degree skew-rotation |
| C4_cf | CD8_cf | 2 | orthogonal | Pattern is symmetric under 90 degree rotation (line goes through cells and faces) skew-reflections |
| C4_ek | CD8_ek | 2 | orthogonal | Pattern is symmetric under 90 degree rotation (line goes through edges and vertices) skew-reflections |
| C1 | C3 | 3 | paradiagonal | Pattern is asymmetric 120 degree skew-rotation (line goes through cells and vertices) |
| C1 | N/A | 3 | paradiagonal | Pattern is asymmetric 120 degree skew-rotation (line goes paradiagonally through edges and faces) |
| C1 | C4_cf | 4 | orthogonal | Pattern is asymmetric 90 degree skew-rotation (line goes through cells and faces) |
| C1 | C4_ek | 4 | orthogonal | Pattern is asymmetric 90 degree skew-rotation (line goes through edges and vertices) |