A fixed point of an isometry group is a point that is a fixed point for every

isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...

in the group. For any isometry group in

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

the set of fixed points is either empty or an

affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...

. For an object, any unique centre and, more generally, any point with unique properties with respect to the object is a fixed point of its symmetry group. In particular this applies for the

centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the figure. The same definition extends to any object in n-d ...

of a figure, if it exists. In the case of a physical body, if for the symmetry not only the shape but also the density is taken into account, it applies to the centre of mass. If the set of fixed points of the symmetry group of an object is a singleton then the object has a specific centre of symmetry. The centroid and centre of mass, if defined, are this point. Another meaning of "centre of symmetry" is a point with respect to which inversion symmetry applies. Such a point needs not be unique; if it is not, there is translational symmetry, hence there are infinitely many of such points. On the other hand, in the cases of e.g. ''C_3h'' and ''D₂'' symmetry there is a centre of symmetry in the first sense, but no inversion. If the symmetry group of an object has no fixed points then the object is infinite and its centroid and centre of mass are undefined. If the set of fixed points of the symmetry group of an object is a line or plane then the centroid and centre of mass of the object, if defined, and any other point that has unique properties with respect to the object, are on this line or plane.

1D

;Line: :Only the trivial isometry group leaves the whole line fixed. ;Point: :The groups generated by a reflection leave a point fixed.

2D

;Plane: :Only the trivial isometry group ''C₁'' leaves the whole plane fixed. ;Line: :''C''_s with respect to any line leaves that line fixed. ;Point: :The point groups in two dimensions with respect to any point leave that point fixed.

3D

;Space: :Only the trivial isometry group ''C₁'' leaves the whole space fixed. ;Plane: :''C_s'' with respect to a plane leaves that plane fixed. ;Line: :Isometry groups leaving a line fixed are isometries which in every plane perpendicular to that line have common 2D point groups in two dimensions with respect to the point of intersection of the line and the planes. :*''C_n'' ( ''n'' > 1 ) and ''C_nv'' ( ''n'' > 1 ) :*cylindrical symmetry without reflection symmetry in a plane perpendicular to the axis :*cases in which the symmetry group is an infinite subset of that of cylindrical symmetry ;Point: :All other point groups in three dimensions ;No fixed points: :The isometry group contains translations or a screw operation.

Arbitrary dimension

;Point: :One example of an isometry group, applying in every dimension, is that generated by inversion in a point. An n-dimensional

parallelepiped In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. Three equiva ...

is an example of an object invariant under such an inversion.

References

* Slavik V. Jablan, ''Symmetry, Ornament and Modularity'', Volume 30 of K & E Series on Knots and Everything, World Scientific, 2002. {{isbn, 9812380809 Euclidean symmetries Group theory Fixed points (mathematics) Geometric centers