Symmetry Operation
Symmetry operations include the Improper Rotation, Inversion Operation, Mirror Plane, andRotation. Together, these operations create 32 crystal classes corresponding to the 32 Point Groups.
The Inversion Operation takes
and is denoted
. When used in conjunction with a Rotation, it becomes an Improper Rotation. AnImproper Rotation by 
is denoted 
(or 
). For periodic crystals, the CrystallographyRestriction allows only the Improper Rotations 
, 
, 
, 
, and 
. The Mirror Plane symmetry operation takes
etc., which is equivalent to
. Invariance under reflection can be denoted 
or 
. TheRotation symmetry operation for is denoted 
(or 
). For periodic crystals, CrystallographyRestriction allows only 1, 2, 3, 4, and 6.Symmetry operations can be indicated with symbols such as , , 
, , , and .
- 1. indicates Rotation about an -fold symmetry axis.
- 2. indicates Improper Rotation about an -fold symmetry axis.
- 3. (or
) indicates invariance under Translation. - 4. indicates a center of symmetry under Inversion.
- 5. indicates invariance under vertical Reflections.
- 6. indicates invariance under horizontal Reflections.
- Hyperbolic Partial Differential Equation
- Hyperbolic Plane
- Hyperbolic Point
- Hyperbolic Polyhedron
- Hyperbolic Rotation
- Hyperbolic Rotation (Crossed)
- Hyperbolic Secant
- Hyperbolic Sine
- Hyperbolic Space
- Hyperbolic Spiral
- Hyperbolic Spiral Inverse Curve
- Hyperbolic Spiral Roulette
- Hyperbolic Substitution
- Hyperbolic Tangent
- Hyperbolic Tetrahedron
- Hyperbolic Umbilic Catastrophe
- Hyperboloid
- Hyperboloid Embedding
- Hypercomplex Number
- Hypercube
- Hyperdeterminant
- Hyperellipse
- Hyperelliptic Function
- Hyperelliptic Integral
- Hyperfactorial