Point group ↩

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created: 2021-10-19 17:17:40
modified: 2022-01-13 19:55:07

A group of geometric symmetries (or isometries) that leave at least one point (origin) in place, so translation is not included.
(Using Schoenflies notation)

Isometries that leave the origin in place

: inversion ^{^c64d66}
* : proper rotation by , -fold rotation ^{^9ff180}
* , , : reflection, with the subscript indicating the orientation of the relfection plane's orientation ( = "vertical", includes the main rotation axis; = "horizontal", perpendicular to the main rotation axis; = "diagonal", between the generated planes)
: improper rotation by , -fold roto-reflection (a then a )

Finite point groups in 3D

: only inversion
: only a mirror plane (Spiegel)
: -fold rotation axis
: plus a horizontal mirror plane
: plus vertical mirror planes
: -fold roto-reflection axis
: plus perpendicular axes (dihedral axes)
: plus a horizontal mirror plane
: plus vertical mirror planes between the axes
: tetrahedral symmetry
: plus horizontal reflection
: plus diagonal reflection
: octahedral of cubic symmetry (without reflection)
: plus horizontal mirror plane
, : icosahedral symmetries (not possible in crystals)