Improper rotation

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Example polyhedra with rotoreflection symmetry
Group S4 S6 S8 S10 S12
Example 2-antiprism rotoreflection.png
beveled digonal antiprism
3-antiprism rotoreflection.png
triangular antiprism
Rotoreflection example square antiprism.png
square antiprism
Rotoreflection example antiprism.png
pentagonal antiprism
6-antiprism rotorereflection.png
hexagonal antiprism
Antiprisms with directed edges have rotoreflection symmetry.
p-antiprisms for odd p contain inversion symmetry, Ci.

In geometry, an improper rotation,[1] also called rotation-reflection,[2] rotoreflection,[1] rotary reflection,[3] or rotoinversion[4] is, depending on context, a linear transformation or affine transformation which is the combination of a rotation about an axis and a reflection in a plane perpendicular to that axis.[5]

Three dimensions[edit]

Subgroups for Schoenflies groups S2 to S20



An improper rotation of an object thus produces a rotation of its mirror image. The axis is called the rotation-reflection axis.[6] This is called an n-fold improper rotation if the angle of rotation is 360°/n.[6] There are several different systems for naming individual improper rotations:

  • The Schoenflies notation uses the symbol Sn (German, Spiegel, for mirror) denotes the symmetry group generated by an n-fold improper rotation. For example, the symmetry operation S6 is the combination of a rotation of (360°/6)=60° and a mirror plane reflection. (This should not to be confused with the same notation for symmetric groups).[6]
  • In Hermann–Mauguin notation the symbol n is used for n-fold rotoinversion; i.e., rotation by an angle of rotation of 360°/n with inversion.
  • The Coxeter notation for S2n is [2n+,2+].
  • The Orbifold notation is n×, order 2n.



As an indirect isometry[edit]




Physical systems[edit]


See also[edit]


  2. ^ Miessler, Gary; Fischer, Paul; Tarr, Donald (2014), Inorganic Chemistry (5 ed.), Pearson, p. 78