Fixed Points Of Isometry Groups In Euclidean Space
A fixed point of an isometry group is a point that is a fixed point for every isometry in the group. For any isometry group in Euclidean space the set of fixed points is either empty or an affine space.
For an object, any unique center 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 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.
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, hence there are infinitely many of such points. On the other hand, in the cases of e.g. C3h 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.
Read more about Fixed Points Of Isometry Groups In Euclidean Space: 1D, 2D, 3D, Arbitrary Dimension
Famous quotes containing the words fixed, points, groups and/or space:
“The word “obligation” is very seldom in my thoughts, and consequently very seldom is it uttered by my tongue. For I am satisfied that whoever hath the word “obligation” continually in his mouth hath the love of tyranny steadily fixed in his heart.”
—Sarah Fielding (1710–1768)
“The three main medieval points of view regarding universals are designated by historians as realism, conceptualism, and nominalism. Essentially these same three doctrines reappear in twentieth-century surveys of the philosophy of mathematics under the new names logicism, intuitionism, and formalism.”
—Willard Van Orman Quine (b. 1908)
“Instead of seeing society as a collection of clearly defined “interest groups,” society must be reconceptualized as a complex network of groups of interacting individuals whose membership and communication patterns are seldom confined to one such group alone.”
—Diana Crane (b. 1933)
“When Paul Bunyan’s loggers roofed an Oregon bunkhouse with shakes, fog was so thick that they shingled forty feet into space before discovering they had passed the last rafter.”
—State of Oregon, U.S. public relief program (1935-1943)