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'' mea ...

in the group. For any

isometry group In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the ...

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

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 relate ...

. For an object, any unique

centre Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentricity ...

and, more generally, any point with unique properties with respect to the object is a fixed point of its

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient ...

. 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 surface of the figure. The same definition extends to any ob ...

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 In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...

. If the set of fixed points of the symmetry group of an object is a

singleton Singleton may refer to: Sciences, technology Mathematics * Singleton (mathematics), a set with exactly one element * Singleton field, used in conformal field theory Computing * Singleton pattern, a design pattern that allows only one instance o ...

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 In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by . In physics and mathematics, continuous translational symmetry is the invariance of a system of equatio ...

, 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 In geometry, a two-dimensional point group or rosette group is a group of geometric symmetries (isometries) that keep at least one point fixed in a plane. Every such group is a subgroup of the orthogonal group O(2), including O(2) itself. Its ele ...

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 In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries tha ...

;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. In Euclidea ...

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. {{DEFAULTSORT:Fixed Points Of Isometry Groups In Euclidean Space Euclidean symmetries Group theory Fixed points (mathematics) Geometric centers