SYMMETRY (Lat. symmetric, from Gk. arAperpia, from eLpperpoc, symmetros, having a common measure, from syn, together + mctron, measure, from perpeiv, mctrein, to measure). A term used in geometry to ex press a characteristic property of two congruent or quasi-congruent figures which have a certain relation with respect to a point, line, or plane.
Two systems of points, A,, B,, C„ A„ B,, ...., are said to he symmetric with respect to an axis when all lines A, A2, B1 • • • • are bisected at right angles by that axis. Two figures are said to be symmetric with respect to an axis when their systems of points are sym metric with respect to that axis. A figure is said to be symmetric with respect to an axis when the axis divides it into two symmetric fig ures. Two systems of points A,, B,, C,, ...., and A,, B„ C,, ...., are said to be symmetric icith respect to a centre 0 when all lines A, A2, C, C„ , are bisected by 0.
Two figures are said to he symmetric with re spect to a centre when their systems of points are symmetric with respect to that centre. E.g. in the figure triangles B, C,, A, B, C, are sym metric with respect to 0.
Figures of three dimensions besides being sym metric with respect to all axis or a centre may be symmetric with respect to a plane. E.g. the sphere is symmetric with respect to its centre, with re spect to any diameter as axis, and with respect to any diametrical plane as a plane of sym metry. Symmetric polyhedral angles may be con sidered as quasi-congruent. and are such as have
their dihedral angles equal, and the plane angles of their faces also equal, hut arranged in reverse order.
Thus, in the following figure. V and V' are sym metric trihedral angles, the letters showing +he reverse arrangement. Opposite polyhedral angles are such that each is formed by producing the edges and faces of the other through the vertex, and are symmetric. The theory of symmetric figures is closely related to that of similarity (q.v.).
In algebra, an algebraic function is said to be symmetric with respect to certain letters when these letters can be interchanged without chang ing the form of the expression. E.g. 2,ry y' is symmetric as to x and y, because if a' and y are interchanged it becomes which is the same as the original expression.
A knowledge of symmetry and homogeneity (q.v.) is of great value in factoring. E.g. to factor = z — y'). The expression vanishes for X = y, hence —y is a factor by the remainder the orem (q.v.). But f(x,Thz) is symmetric with re spect to x.y.z, therefore y z— a^, are also factors. And f(x, y, z) being homogeneous of the fourth degree, it must contain another factor of the first degree: but such a homogeneous sym metric factor can be x+ z only. Whence the literal factors are x—y, y—z, z—x, x y z. On algebraic symmetry consult Beman and Smith, Elements of Algebra (Boston, 1900).