Home >> Encyclopedia Americana, Volume 26 >> Swordfish to Tanning >> Symmetry

Symmetry

symmetrical, respect, line and axis

SYMMETRY. From the Latin symmetria; proportion, symmetry; which, in turn was de rived from the Greek symmetric', meaning agreement in dimensions, proportionate. In its present-day intent the word symmetry in the language of the laity can be defined as har mony or balance in the proportions of parts as to the whole.

Two points are said to be symmetrical with respect to a straight line, when the straight line bisects at right angles the straight line joining the two points. Thus in Fig. 1 the two points P and P are symmet rical with respect to the line M N, if M N bisects P P at right angles. In such a case the line M N is termed the axis of symmetry. Two figures are said to be symmetrical with respect to an axis when every point in one figure has its symmetrical point in the other. Thus in Fig. 2 the figures A B C. A' B' C' are symmetrical with respect to the axis M N, if every point in the figure A B C has a sym metrical point in A' B C' with respect to the median line M N. When figures are sym metrical with respect to an axis, by revolving either about the axis and superimposing one over the other, they will be found equal and similar. Two points are said to be symmetri cal to a third point, when this third point bi sects the straight line joining the two points.

Thus in Fig. 3, P and P' are symmetrical with respect to A, if the straight line P P' is bi sected at A. And the point A is called the centre of symmetry. Two figures are said to be symmetrical with respect to a centre, when as in Fig. 4, the triangles A B C, A' B' Cs are symmetrical to the centre 0, and every point in the triangle A B C has a symmetrical point in A' B' C'. A figure is symmetrical with re spect to an axis when it can be divided by that axis into two figures symmetrical with respect to the axis. And a figure is said to be sym metrical with respect to a centre when every straight line drawn through that centre cuts the figure in two points symmetrical with re spect to this centre. In solid geometry when two planes intersect they are said to form a diedral angle. When three or more planes meet in a common point, they are said to form a polyedral angle at that point. Two polyedral angles are symmetrical, when the face and diedral angles of one are equal to the face and diedral angles of the other each to each, but arranged in ri.-verse order. As example the triedral angles SA BC and S' A' B' C' (in Fig 5)•are symmetrical when the face-angles