Symmetrical
Collier's New Encyclopedia (1921)
in botany (of the parts of a flower), related to each other in number, the same in number, or one a multiple of the other, as in Saxifraga, which has five divisions of the calyx, five petals, and five stamens; or Epilobium, which has a four-parted calyx, four petals, and eight stamens. In mathematics, possessing the attribute of symmetry; having corresponding parts or relations. In geometry, two points are symmetrically disposed with respect to a straight line, when they are distant from it, so that a straight line joining them intersects the given line, and is at right angles to it. A curve is symmetrical with respect to a straight line, when for each point on one side of the line there is a corresponding point on the other side, and equally distant from it. The line is called an axis of symmetry. In conic sections, the axes are the only true axes of symmetry. Two plane figures are symmetrically situated with respect to a straight line, when each point of one has a corresponding point in the other on the opposite side of the axis, and equally distant from it. A line or surface is symmetrical with respect to a plane, when for each point on one side of the plane there is a second point on the other side, equally distant from it. The plane is called the plane of symmetry, and is, in conical sections, a principal plane. Symmetrical lines and surfaces in space cannot, in general, be made to coincide with each other. Spherical triangles are symmetrical when their sides and angles are equal each to each, but not similarly situated. In analysis, an expression is symmetrical with with respect respect to two letters, when the places of these letters may be changed without changing the expression. Thus the expression x² + ax + ab + b²x is symmetrical with respect to a and b; for, if we change the place of a and b, we have x² + b²x + ba + a²x, the same expression. An expression is symmetrical with respect to several letters, when any two of them may change places without affecting the expression; thus, the expression ab + ba² + a²c + ca + b²c + be is symmetrical with respect to the three letters, a, b, c.
What Has Changed in the Last Century
Every definition Collier's gave in 1921 — the axis of symmetry, the plane of symmetry, the symmetric expression in which a and b may swap places without altering the whole — is still taught in those words, and symmetric polynomials remain a standard topic in algebra. What has moved is the subject's weight: through group theory and Emmy Noether's 1918 theorem linking every continuous symmetry to a conservation law, symmetry stopped being a descriptive property and became the organising principle of modern physics, from crystallography's 230 space groups to the gauge symmetries behind the Standard Model. The botanical sense has faded fastest; where Collier's counted the five sepals, five petals and five stamens of Saxifraga, botanists now usually say isomerous, and reserve talk of floral symmetry for the actinomorphic-versus-zygomorphic distinction.
Written for this edition, 2026.