Radial symmetry

“ It is independent of the observer's mesh-system, but it depends on his gauge. It is obvious that the number expressing the radius of curvature of the world at a point must depend on the unit of length; so we cannot say that the curvatures at two points are absolutely equal, because they depend on the gauges assigned at the two points. Conversely the radius of curvature of the world provides a natural and absolute gauge at every point; and it will presumably introduce the greatest possible symmetry into our laws if the observer chooses this, or some definite fraction of it, as his gauge. ”

“ With the symmetry of solids, or symmetry with relation to an axial plane, no such simple movement as the foregoing suffices to produce or explain it, because symmetry about a plane implies four-dimensional movement. It is easy to see why this must be so. In order to achieve symmetry in any space—that is, in any given number of dimensions—there must be revolution in the next higher space: one more dimension is necessary. ”

“ In the Articulates there is but one cavity, and the parts are here again arranged on either side of the longitudinal axis, but in these animals the whole body is divided from end to end into transverse rings or joints movable upon each other. In the Radiates we lose sight of the bilateral symmetry so prevalent in the other three, except as a very subordinate element of structure; the plan of this lowest type is an organic sphere, in which all parts bear definite relations to a vertical axis. ”

“ Here there are three similar planes of symmetry intersecting in the triad axis, and perpendicular to them is a fourth plane of symmetry; at the intersection of the three vertical planes with the horizontal plane are three similar dyad axes; there is no centre of symmetry.
The general form is bounded by twelve scalene triangles and is a ditrigonal bipyramid. Like the general form of the last class, this has two sets of indices {hkl, pqr}, (hkl) for faces above the equatorial plane of symmetry and (pqr) for faces below: with hexagonal axes there would be only one set of indices. ”

“ The symmetry also requires that the intensity of the scattered light should vanish for the ray which would be propagated along the axis; for there is nothing to distinguish one direction transverse to the ray from another. We have now got what we want. Suppose, for distinctness of statement, that the primary ray is vertical, and that the plane of vibration is that of the meridian. The intensity of the light scattered by a small particle is constant, and a maximum for rays which lie in the vertical plane running east and west, while there is no scattered ray along the north and south line. ”

“ Axes of Symmetry.—Consider the vertical axis joining the opposite corners a3 and ā3 of an octahedron (fig. 3) and passing through its centre O: by rotating the crystal about this axis through a right angle (90°) it reaches a position such that the orientation of its faces is the same as before the rotation; the face ā1ā2ā3, for example, coming into the position of a1ā2a3. During a complete rotation of 360° (= 90° × 4) , the crystal occupies four such interchangeable positions. Such an axis of symmetry is known as a tetrad axis of symmetry. ”

“ Victor von Lang showed that the true elements of symmetry are planes of symmetry and axes of symmetry. A crystal possessing a plane of symmetry is symmetrical on both sides of that plane, both as regards the number of the faces and their precise angular disposition with respect to one another. ”

“ Now could it be shown that the two-dimensional symmetry observed in nature is the result of a three-dimensional movement, the right-and left-handed symmetry of solids would by analogy be the result of a four-dimensional movement. Such revolution (about a plane) would be easily achieved, natural and characteristic, in four space, just as the analogous movement (about a line) is easy, natural, and characteristic, in our space of three dimensions. ”

“ In it, even though the rays should pass through the optic axis itself, where there is no double refraction, the primitive symmetry of the ray is destroyed, and the plane of primitive polarization deviates either to the right or left of the observer, by an angle proportional to the thickness of the plate of quartz. This angular motion, or true rotation of the plane of polarization, which is called circular polarization, is clearly proved by the phenomena. The coloured rings produced by all crystals having but one optic axis are circular, and traversed by a black cross concentric with the rings ”

“ This fresh example, if it were necessary, might serve to shew the difference which exists between the retrogradations on primitive crystals, where the geometrical form is one in which all the pants are perfectly symmetrical, and those on primitive crystals, in which the parts are not so circumstanced. A symmetry is established between the new planes produced on the first, while the planes which occur in the second, are more or less distant from it. ”

“ Symmetry in Physical Law Symmetry seems to be absolutely fascinating to the human mind. We like to look at symmetrical things in nature, such as perfectly symmetrical spheres like planets and the sun, or symmetrical crystals like snowflakes, or flowers which are nearly symmetrical. However, it is not the symmetry of the objects in nature that I want to discuss here; it is rather the symmetry of the physical laws themselves. It is easy to understand how an object can be symmetrical, but how can a physical law have a symmetry? ”

“ It may be different in kind from some of the motions known to physicists, and it is conceivable that life may be the transmission to dead matter, the molecules of which have already a special kind of motion, of a form of motion sui generis.”
Now, in the realm of organic beings symmetrical structures—those with a right and left symmetry—are everywhere in evidence. Granted that four dimensions exist, the simplest turning produces the image form, and by a folding-over structures could be produced, duplicated right and left, just as is the case of symmetry in a plane. ”

“ Mass, electric charge, temperature, have the same symmetry, of a type called scalar, that of the sphere. A current of water and a rectilineal electric current have the symmetry of an arrow, of the type polar vector. The symmetry of an upright circular cylinder is of the type tensor. All of the physics of crystals can be expressed in a form in which the particular phenomena in question are not specified, but in which are examined only the geometrical and analytical relations between the types of quantities where certain ones are considered as causes and the other as effects. ”

“ In the first place there is a complete symmetry round the direction of the force. The disturbance, consisting of transverse vibrations, is propagated outwards in alt directions from the centre; and in consequence of the symmetry, the direction of vibration in any ray lies in the plane containing the ray and the axis; that is to say, the direction of vibration in the scattered or diffracted ray makes with the direction of vibration in the incident or primary ray the least possible angle. ”

“ In every symmetrical system every deformation that tends to destroy the symmetry is complemented by an equal and opposite deformation that tends to restore it. In each deformation positive and negative work is done. One condition, therefore, though not an absolutely sufficient one, that a maximum or minimum of work corresponds to the form of equi­lib­rium, is thus supplied by symmetry. Regularity is successive symmetry. ”

“ In the core proper below the teeth the length of path continually shortens as we pass from the middle of the pole towards the centre line of symmetry. On the other hand, as the lines gradually accumulate in the core, their density increases from zero midway under the poles until it reaches a maximum on the line of symmetry. The two effects partially counteract one another, and tend to equalize the difference of magnetic potential required over the paths of varying lengths ”

“ That Born felt compelled to assume spherical symmetry, lies in the fact that he didn't consider the mechanical momentum of the electron, which can only be assumed as zero at certain assumptions concerning the form and charge distribution, for example when one assumes spherical symmetry ”

“ The optical properties afford conclusive proof of the monoclinic nature of the symmetry. The plane of the optic axes is perpendicular to the possible symmetry plane, b = {010}, and the first median line 160makes an angle of 23° with the vertical axis c, emerging consequently nearly normal to the basal plane c = {001}, so that a section-plate parallel to the c-faces, or a tabular crystal or cleavage plate parallel to c, shows the optic axial rings and brushes well. ”

“ The symmetry, therefore, arising from the simultaneous contemplation of the object and its image will be improved; but it will still be imperfect, as the image will appear to be distant from the plane of the mirror, only by the space m x, while the distance of the object is M x. As the eye moves from e to E, the line e A x will move into E A B, and the object and its image will seem to be placed at the equal distances M B, m B from the plane of the mirror, and will therefore form a symmetrical combination. ”

“ In the case of sound, light, heat, or in fact any radial emanation of energy from a point, the law of the inverse square is a direct consequence of the fact that the action takes place in three dimensional space. If our universe of matter were two dimensional, the law would be that of the inverse first power. In the first case the constant amount of energy, at a distance r, is spread over a spherical surface of area 4πr2 ”

“ But these spheres are not to be rigid in the sense of Euclidean geometry; their radius is to increase (in the sense of Euclidean geometry) when they are moved away from S towards infinity, and this increase is to take place in exact accordance with the same law as applies to the increase of the radii of the disc-shadows L' on the plane.
After having gained a vivid mental image of the geometrical behaviour of our L' spheres, let us assume that in our space there are no rigid bodies at all in the sense of Euclidean geometry, but only bodies having the behaviour of our L' spheres. ”

“ According to Symmes's Theory, the earth, as well as all the celestial orbicular bodies existing in the universe, visible and invisible, which partake in any degree of a planetary nature, from the greatest to the smallest, from the sun, down to the most minute blazing meteor or falling star, are all constituted in a greater or less degree, of a collection of spheres, more or less solid, concentric with each other, and more or less open at their poles; each sphere being separated from its adjoining compeers by space replete with aerial fluids ”

“ The disciples of Symmes believe that each sphere has a cavity, or mid-plane space near the centre of the matter composing it, filled with a very light, subtile, elastic substance, partaking somewhat, perhaps, of the nature of hydrogen gas; which aerial fluid is composed of molecules greatly rarified in comparison with the gravity of the extended or exposed surfaces of the sphere. This mid-plane space tends to give the sphere a degree of lightness and buoyancy. ”

“ Many hollow cylinders and spherical figures, we know do exist on the surface of the earth; and notwithstanding their own gravity, which the different parts exert on each other, as well as the gravity of the earth, they retain their shape and position; and had the matter in the earth originally been thrown by a centrifugal force into the form of a hollow sphere, or had the first creating power originally given it that shape,—I can discover no good reason for a change; neither should I entertain any apprehensions of the particles of matter coalescing at the centre. ”

“ For this purpose we will first give our attention once more to the geometry of two-dimensional spherical surfaces. In the adjoining figure let K be the spherical surface, touched at S by a plane, E, which, for facility of presentation, is shown in the drawing as a bounded surface. Let L be a disc on the spherical surface. Now let us imagine that at the point N of the spherical surface, diametrically opposite to S, there is a luminous point, throwing a shadow L′ of the disc L upon the plane E. Every point on the sphere has its shadow on the plane. ”

“ The application which the Dr. makes of this structure of the earth is this: that the concave sides of the spheres are made up of magnetic matter; that they revolve about their diurnal axes in about twenty-four hours; that the outer sphere moves either a little faster or a little slower than the internal ball; that the magnetic pole, both of the external shell and included globe, are distant from the poles of rotation; and that the variation arises from a change of the relative distances of the external and internal poles in consequence of the difference of their revolutions. ”

“ Suppose the attraction of gravitation, inherent in matter, to be so much increased, that a hollow sphere would possess the same attractive power, as if it were a solid globe, would not all the results and consequences be exactly the same? [109] This being the case,—and I know no reason why we should conclude differently,—the whole force of the objection appears to fall to the ground. According to Newton's principle of gravity, the matter of the sphere would attract all particles of matter placed on the surface, as well upon the concave as convex, in nearly equal proportions ”