APPENDIX V.

SYMMETRY IN SPACE.

THERE are two kinds of symmetry for polyhedrons, symmetry of

form and symmetry of position.

To give an idea of these two kinds of symmetry, let us consider, first, a tetrahedron SABC, and upon its edges, prolonged above the vertex S, take distances SA' = SA, SB' = = SB, SC' = SC, and draw A'B', A'C', B'C'; the parts of the two tetrahedrons (edges, faces, diedral angles) are evidently equal each to each, but disposed in an inverse order. They are called symmetric.

C

B

The second tetrahedron may be detached from the first, and is still symmetric, whatever may be their relative position.

Two polyhedrons are said to be symmetric [and that independent of their position in space] when they can be decomposed into the same number of tetrahedrons symmetric each to each, and disposed in an inverse order.

Whence it follows that, 1°. A polyhedron can have but one symmetric with it. 2°. Two symmetric polyhedrons have their edges, faces, diedral and polyhedral angles equal each to each.

SYMMETRY OF POSITION.

This exists in three ways: 1°. With reference to a point, which is called a center of symmetry; 2°. With reference to a line, called an axis of symmetry; 3°. With reference to a plane, called the plane of symmetry. We shall treat, first, of

SYMMETRY RELATIVE TO AN AXIS.

Definition. Two points are symmetrical with respect to a line when the line which joins them is perpendicular to the first, and divided by it into two equal parts.

A polyhedron is symmetric, or two polyhedrons are symmetric with reference to a line, when this line passes through the middle point of all the lines [other than the edges or diagonals of the faces] which join the vertices of the polyhedron, two and two, and is perpendicular to them.

THEOREM 1. Two figures which are symmetric with reference to a line are identical.

This may be proved by revolving the perpendiculars about the axis ; the vertices will all describe similar arcs.

Corollaries. In a polyhedron symmetric with reference to an axis, 10. Every line meeting the axis at right angles, and terminating at the surface, is equally divided by the axis. 20. Every plane through the axis cuts the polyhedron into two equal parts. 3°. Every plane perpendicular to the axis determines a symmetric section with reference to the point of intersection of this plane with the axis, and this point is the center of symmetry of the section.

Schol. 1. The most simple of polyhedrons symmetrical with reference to an axis is the right prism, the base of which is symmetric with reference to a point.

When the base of the right prism is a rectangle it has for axes of symmetry the three lines which join the centers of the opposite faces.

If, moreover, the base is a square, there exist two other axes of symmetry which join the middle of the opposite edges.

When the base of the right prism is a rhombus, there are three axes of symmetry, one joining the centers of the two bases, and two others joining the middle points of the opposite edges.

Schol. 2. The axis of a regular pyramid is also an axis of symmetry when the number of lateral faces is even.

Schol. 3. Symmetry, with reference to an axis, is, properly speaking, merely symmetry of position, since, by the preceding theorem, the figures are equal and capable of superposition. But the same is not the case with symmetry with reference to a point, or symmetry with reference to a plane, which are at the same time symmetry of form and position. For this reason we have commenced with symmetry referred to a line.

SYMMETRY WITH REFERENCE TO A POINT OR PLANE.

Definitions. Two points are said to be symmetrical with reference to a point when the latter divides into two equal parts the line joining the two former; and, with reference to a plane, when this plane is perpendicular to the line which joins the two points and bisects it.

THEOREM 2. If three points are in a right line, their symmetric

points with reference to a point or plane arc in a right line. The student will easily prove this.

Corollaries. 10. Two lines of determinate length, and symmetric with respect to a point, are equal and parallel. 20. Two triangles symmetric with respect to a point are equal and their planes parallel. 3°. Two lines of determinate length, and symmetric with reference to a plane, are equal, make equal angles with this plane, and, being prolonged, meet it at the same point, unless they are parallel.

THEOREM 3. If four points are in the same plane, their symmetric points, with reference to a point, are also in a same plane.

Schol. When the four points are in different planes their symmetrics are also, and then the two systems of points determine two tetrahedrons, whose angles, diedral and trihedral, are symmetric, and, consequently, the tetrahedrons themselves symmetric.

THEOREM 4. When two polyhedrons have their vertices, two and two, symmetric with reference to a point or a plane [in which case the polyhedrons are said to be symmetric], 1°. These polyhedrons have their faces equal each to each, their diedral angles equal each to each, and their polyhedral angles symmetric. 20. These polyhedrons are symmetric in form.

The first part of this theorem results from the last corollaries, and the second by observing that the two polyhedrons are composed of the same number of tetrahedrons symmetric, two and two, and inversely disposed.

THEOREM 5. When the vertices of a polyhedron are situated symmetrically with reference to a point, 10. This polyhedron has necessarily an even number of edges, equal and parallel two and two; and it is the same with the faces; 2°. The plane angles and diedral angles are also equal each to each; the polyhedral angles are symmetric in pairs; 3°. Every line passing through the center of symmetry and terminating at the surface is divided at this point into two equal parts; 4°. Finally, every plane passing through the center divides the polyhedron symmetric ally.

This follows from the last corollaries and scholium.

Schol. 1. The most simple of polyhedrons with reference to a point is the parallelopipedon. It has for a center of symmetry its center of figure. As every diagonal plane passes through its center of figure, such a plane divides the parallelopipedon symmetrically.

Schol. 2. After the parallelopipedon, the most simple are prisms having for bases polygons symmetric with reference to a point. The center of symmetry is the middle of the line which joins the centers of the two bases.

General Scholium upon symmetry with reference to a point and a plane compared with absolute symmetry.

It follows, from theorems four and five, that two polyhedrons symmetric with reference to a point or to a plane are at the same time absolutely symmetric.

Reciprocally, two polyhedrons symmetric to each other (absolutely) can always be placed symmetrically with reference to a point in space, or with reference to a plane, this point or plane being a common vertex or face of the two polyhedrons.*

THEOREM 6. Two symmetric polyhedrons are equivalent.

It is only necessary, after the first definition, to demonstrate this for tetrahedrons. These may be shown to have the same base and height (see th. 5, 30, of this App.), and are, consequently, equal.

The two following propositions may be easily established:

1o. When there exist in a polyhedron two planes of symmetry perpendicular to each other, their common intersection is an axis of symmetry; 20. And if there exist three, the point common to these three planes is a center of symmetry.

OF DIAMETRAL PLANES.

When a plane passes through a polyhedron in such a manner that a system of parallel lines terminating at the surface are equally divided by the plane, it is called a diametral plane. N.B.-The parallels are not necessarily perpendicular to the plane.

THEOREM 7. When the vertices of a polyhedron, or of two polyhedrons, are situated in pairs upon parallel lines, and a certain plane passes through the middle points of these lines, 10. Each couple of homologous edges produced will meet at a point of the plane, unless they are parallel; 2°. Each couple of homologous planes determined by three vertices of the one polyhedron and three corresponding of the other, intersect each other in a line of the first-mentioned plane (unless they are parallel); 3°. Every line parallel to any of the lines joining the homologous vertices, and terminating on either side the plane at the polyhedral surface, is equally divided by this plane, which is, consequently, a diametral plane.

N.B.-When the lines joining the homologous vertices are equal and parallel, the figures determined by the vertices are equal and their planes parallel.

* An object and its reflected image present a familiar example of two figures symmetric to each other.

The human body is a figure composed of two parts symmetric, with reference to what is called a median plane.