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

Fixed point (mathematics)

In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...

 for every isometry

Isometry

In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

 in the group. For any isometry group

Isometry group

In mathematics, the isometry group of a metric space is the set of all isometries from the metric space onto itself, with the function composition as group operation...

 in Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

 the set of fixed points is either empty or an affine space

Affine space

In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...

.

For an object, any unique center

Centre (geometry)

In geometry, the centre of an object is a point in some sense in the middle of the object. If geometry is regarded as the study of isometry groups then the centre is a fixed point of the isometries.-Circles:...

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

Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...

.

In particular this applies for the centroid

Centroid

In geometry, the centroid, geometric center, or barycenter of a plane figure or two-dimensional shape X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...

 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 center of mass

Center of mass

In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...

.

If the set of fixed points of the symmetry group of an object is a singleton then the object has a specific center of symmetry. The centroid and center of mass, if defined, are this point. Another meaning of "center 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

Translational symmetry

In geometry, a translation "slides" an object by a a: Ta = p + a.In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation...

, 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 center 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 center 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 center 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. The point groups in two dimensions

Point groups in two dimensions

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

 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

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, the group of all isometries that leave the origin fixed, or correspondingly, the group...

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

Parallelepiped

In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, its definition encompasses all four concepts...

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