Bilateral symmetry

“ But such arrangements are apt to be too stiffly regular for art, which, needing abundance of freedom and variety, usually contents itself with symmetry in one direction, namely, bilateral symmetry. Why symmetry in an horizontal direction should please rather than in a vertical or any other direction will be explained further on. ”

“ This kind of symmetry is suited for a more strenuous life than radial animals show; it is suited for pursuing food, for avoiding enemies, for chasing mates. And with the establishment of bilateral symmetry must be associated the establishment of head-brains, the beginning of which is to be found in some simple worm-types. ”

“ There are many curious phenomena seen in the behaviour of parts normally repeated in bilateral symmetry which may some day guide us towards an understanding of the mechanics of division. A part like a hand, which needs the other hand to complete its symmetry, cannot twin by mere division, yet by proliferation and special modifications on the radial side of the same limb, even a hand may be twinned. ”

“ Radially-symmetrical as is the type, and radially symmetrical as are those centrally-placed individuals which are equally crowded all round, we see that the peripheral individuals, dissimilarly circumstanced on their outer sides and on their sides next the group, have partially changed their radial symmetry into bilateral symmetry. It is no longer possible to make two corresponding halves by any vertical plane cutting down through the pileus and the stem ”

“ The reader may think he can draw more arrangements than the fifteen stated in the article to be the number that can exist. So he can! However, the porphyrin ring possesses four-fold radial symmetry and front-back bilateral symmetry which reduces the number of different arrangements eightfold. ”

“ 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. ”

“ But, unless the two bubbles be of precisely equal size (and therefore of equal curvature) it is obvious that the tangents to the spheres will not meet the plane of their circle of contact at equal angles, and therefore that the partition-wall must be a curved surface: it is only plane when it divides two equal and symmetrical cells. It is also obvious, from the symmetry of the figure, that the centres of the spheres, the centre of the partition, and the centres of the two spherical surfaces are all on one and the same straight line. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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? ”

“ The straight line is felt, to put the matter briefly, as symmetrical to itself, which is the case also with the plane. Curved lines are felt as deviations from straight lines, that is, as deviations from symmetry. [22] The presence of a sense for symmetry in people possessing only one eye from birth, is indeed a riddle. Of course, the sense of symmetry, although primarily acquired by means of the eyes, cannot be wholly limited to the visual organs. It must also be deeply rooted in other parts of the organism by ages of practice and can thus not be eliminated forthwith by the loss of one eye. ”

“ But where there is symmetry about a plane no simple physical motion, such as we [Pg 77] are accustomed to, suffices to explain it. In our space a symmetrical object must be built up by equal additions on each side of a central plane. ”

“ It is this, approximately, that occurs in the external appearance of man and of numerous animals. An object possesses an axis of symmetry of the order n, if it preserves the same appearance after a rotation on this axis of the nth part of a revolution. Thus a regular flower of four petals has an axis of symmetry of the order four, or a quarternary axis. Crystals like those of rock salt or of alum possess many planes of symmetry and many axes of symmetry of different orders. ”

“ For the opposition, in a symmetrical object, is of like things reflecting each other: it is not the balance of contrary natures (like that of day and night) , but of like natures or like forms; one side of a leaf being set like the reflection of the other in water.
Symmetry in Nature is, however, never formal nor accurate. She takes the greatest care to secure some difference between the corresponding things or parts of things; and an 140 approximation to accurate symmetry is only permitted in animals, because their motions secure perpetual difference between the balancing parts. ”

“ 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 organic life the universal type is of right- and left-handed symmetry, there is a plane on each side of which the parts correspond. Now we have seen that in four dimensions a plane takes the place of a line in three dimensions. In our space, rotation about an axis is the type of rotation, and the origin of bodies symmetrical about a line as the earth is symmetrical about an axis can easily be explained. ”

“ 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. ”

“ It will be seen, as soon as the notion of Symmetry in general is well apprehended, that it is or includes a peculiar Fundamental Idea, not capable of being resolved into any of the ideas hitherto examined. It may be said, perhaps, that the Idea of Symmetry is a modification or derivative of our ideas of space and number;—that a symmetrical shape is one which consists of parts exactly similar, repeated a certain number of times, and placed so as to correspond with each other. ”

“ In things in which perfect symmetry is, from their nature, impossible or improper, a balance must be at least in some measure expressed before they can be beheld with pleasure.... Symmetry is the opposition of equal quantities to each other. Proportion the connection of unequal quantities with each other. The property of a tree in sending out equal boughs on opposite sides is symmetrical. Its sending out shorter and smaller towards the top, proportional. In the human face its balance of opposite sides is symmetry, its division upwards, proportion. ”

“ Hence we conclude in general, that when an object is placed at any distance before two mirrors inclined at an angle, which is an even aliquot part of 360°, the symmetry of the picture is perfect, when the eye, considered as a mathematical point, is placed at E, and that the deviation from symmetry increases as the eye recedes from E towards ε. ”

“ 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. ”

“ Lines balance when in opposite attitudes. We get Balance in all inversions, whether single or double.
Fig. 129
Here similar lines are drawn in opposite attitudes and we get Measure and Shape-Balance. In the above case the axis of balance is vertical. The balance is, therefore, symmetrical. Symmetrical Balance is obtained by the single inversion of any line or lines on a vertical axis. Double inversion gives a Balance of Measures and Shapes on a center. We have no Symmetry in double inversions. All this has been explained.
Fig. 130
We have Measure and Shape-Balance on a center in this case. ”

“ 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. ”

“ By the emphasis which it lays upon the recurring elements, it cuts up the field into determinate units; all that lies between the beats is one interval, one individual. If there were no recurrent impressions, no corresponding points, the field of perception would remain a fluid continuum, without defined and recognizable divisions. The outlines of most things are symmetrical because we choose what symmetrical lines we find to be the boundaries of objects. Their symmetry is the condition of their unity, and their unity of their individuality and separate existence. ”

“ Experience, to be sure, can teach us to regard unsymmetrical objects as wholes, because their elements move and change together in nature; but this is a principle of individuation, a posteriori, founded on the association of recognized elements. These elements, to be recognized and seen to go together and form one thing, must first be somehow discriminated; and the symmetry, either of their parts, or of their position as wholes, may enable us to fix their boundaries and to observe their number. ”

“ All these considerations lead us to the definition of Symmetry. By Symmetry I mean opposite directions or inclinations, opposite and equal distances, opposite positions, and in those positions equal, corresponding, and opposed attractions on a vertical axis. Briefly, Symmetry is right and left balance on a vertical axis. This axis will be imagined when not drawn. In Symmetry we have a balance which is perfectly obvious and instinctively felt by everybody. ”

“ If the movement, though over a solid surface, is not constant in direction, but takes place as often on one side as on another, radial symmetry may be again looked for; and if the motions are still more variously directed—if they are not limited to approximately-plane surfaces, but extend to surfaces that are distributed all around with a regular irregularity—an approach of the radial towards the spherical symmetry is to be anticipated. ”

“ To take one of the conceptions we have already formed, the turning of a real thing into its mirror image would be an occurrence which it would be hard to explain, except on the assumption of a fourth dimension.
We know of no such turning. But there exist a multitude of forms which show a certain relation to a plane, a relation of symmetry, which indicates more than an accidental juxtaposition of parts. ”

“ Very recently this strange little particle has upset one of our most unquestioned concepts about symmetry. We have always believed that nature made no distinction between her right hand and her left hand ”