In geometry, an improper rotation[1] (also called rotation-reflection,[2]rotoreflection,[1]rotary reflection,[3] or rotoinversion[4]) is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicular to that axis. Reflection and inversion are each a special case of improper rotation. Any improper rotation is an affine transformation and, in cases that keep the coordinate origin fixed, a linear transformation.[5]
It is used as a symmetry operation in the context of geometric symmetry, molecular symmetry and crystallography, where an object that is unchanged by a combination of rotation and reflection is said to have improper rotation symmetry.
Example polyhedra with rotoreflection symmetry
Group
S4
S6
S8
S10
S12
Subgroups
C2
C3, S2 = Ci
C4, C2
C5, S2 = Ci
C6, S4, C3, C2
Example
beveled digonal antiprism
triangular antiprism
square antiprism
pentagonal antiprism
hexagonal antiprism
Antiprisms with directed edges have rotoreflection symmetry. p-antiprisms for odd p contain inversion symmetry, Ci.
^ abMorawiec, Adam (2004), Orientations and Rotations: Computations in Crystallographic Textures, Springer, p. 7, ISBN 978-3-540-40734-8.
^Miessler, Gary; Fischer, Paul; Tarr, Donald (2014), Inorganic Chemistry (5 ed.), Pearson, p. 78
^Kinsey, L. Christine; Moore, Teresa E. (2002), Symmetry, Shape, and Surfaces: An Introduction to Mathematics Through Geometry, Springer, p. 267, ISBN 978-1-930190-09-2.
^Klein, Philpotts (2013). Earth Materials. Cambridge University Press. pp. 89–90. ISBN 978-0-521-14521-3.
^Salomon, David (1999), Computer Graphics and Geometric Modeling, Springer, p. 84, ISBN 978-0-387-98682-1.
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Euclidean space are stiff rotations, reflections, or combinations of a rotation and a reflection (also known as improperrotations). Reflections are transformations...
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