Polar point group

In geometry, a polar point group is a point group in which every symmetry operation leaves more than one point unmoved. Therefore, a point group with more than one axis of rotation or a mirror plane perpendicular to the axis of rotation cannot be polar.

A straight line joining unmoved points defines a unique axis of rotation, unless symmetry operations do not allow any rotation at all, such as mirror symmetry, in which case, the polar direction must be parallel to any mirror planes.

The space groups associated with a polar point group do not have their origins uniquely determined by symmetry elements.

Of the 32 crystallographic point groups, 10 are polar:

Polar crystallographic point groups

Crystal system	Polar point groups
	Schönflies		Hermann–Mauguin		Orbifold		Coxeter
Triclinic	C1		1		11		[ ]+
Monoclinic	C2	Cs	2	m	22	*	[2]+	[ ]
Orthorombic	C2v		mm2		*22		[2]
Tetragonal	C4	C4v	4	4mm	44	*44	[4]+	[4]
Trigonal	C3	C3v	3	3m	33	*33	[3]+	[3]
Hexagonal	C6	C6v	6	6mm	66	*66	[6]+	[6]
Cubic	(none)

When materials having a polar point group crystal structure are heated or cooled, they may temporarily generate a voltage called pyroelectricity.