Symmetry

“ 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. ”

“ 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 can be argued that the occurrence of symmetry in two dimensions involves the existence of a three-dimensional process, as when a stone falls into water and makes rings of ripples, or as when a mass of soft material rotates about an axis. It can be argued that symmetry in any number of dimensions is the evidence of an action in a higher dimensionality. Thus considering living beings, there is an evidence both in their structure, and their different mode of activity, of a something coming in from without into the inorganic world. ”

“ 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. ”

“ The symmetry of the living things is like the symmetry of the concentric waves in a pool caused by a splash. Perfect circles are made only in the imaginary case of mathematical uniformity, but the system maintains an approximate symmetry though liable to manifold deformation. ”

“ It will of course be understood that by the term Symmetry I here intend, not that more indefinite attribute of form which belongs to the domain of the fine arts, as when we speak of the ‘symmetry’ of an edifice 68 or of a sculptured figure, but a certain definite relation or property, no less rigorous and precise than other relations of number and position, which is thus one of the sure guides of the scientific faculty, and one of the bases of our exact science. ”

“ 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. ”

“ For there are seven such systems of crystal symmetry, each distinguished by its own specific elements of symmetry.
It is only within recent years that we have come to appreciate what are the real elements of symmetry. ”

“ 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 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. ”

“ The whole problem of the solid, up to a certain point, is contained in our plane diagram of Fig. 138. And when our particular solid is a solid of revolution, then it is obvious that a study of its plane of symmetry (that is to say any plane passing through its axis of rotation) gives us the solution of the whole problem. The right cone is a case in point, for here the in­ves­ti­ga­tion of its modes of symmetrical subdivision is completely met by an examination of the isosceles triangle which constitutes its plane 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. ”

“ 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. ”

“ Again since the one face, constant in symmetry, appears sometimes fair and sometimes not, can we doubt that beauty is something more than symmetry, that symmetry itself owes its beauty to a remoter principle? ”

“ Stand before a mirror; hold your arms in precisely the same position at each side, your head upright, your body straight; divide your hair exactly in the middle and get it as nearly as you can into exactly the same shape over each ear; and you will see the effect of accurate symmetry: you will see, no less, how all grace and power in the human form result from the interference of motion and life with symmetry, and from the reconciliation of its balance with its changefulness. Your position, as seen in the mirror, is the highest type of symmetry as understood by modern architects. ”

“ The symmetry is entirely due to the instrument. And if a competent enquirer looks into the instrument and examines its construction, he will be able to lay down with absolute certainty the laws of that symmetry which every pattern as seen through the instrument must obey. ”

“ 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? ”

“ 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. ”

“ Finally, symmetry and order are beautiful things, and at the same time, are useful things, because they economize space, because objects symmetrically disposed are easier to find when one wants them; but that is not what makes for us the beauty of symmetry, for we immediately seize this kind of beauty, and it is often late enough before we recognize the utility that is found in it. ”

“ The principle of symmetry furnishes an example of an analogous evolution. To begin with, observation of Nature was able to suggest the idea of symmetry; though such observations reveal only imperfectly any regular dispositions in the aspects of animals and plants. ”

“ If the principle of symmetry could be asserted concerning the human body or concerning the solar system, that symmetry might answer many questions; it might be said, "This or that is so, because there is no reason why it should be otherwise." But there is an absence of symmetry from many parts of nature, and, when no geometrical or other cause can be assigned, you need the hypothesis of an independent will in order to render the irregular formation in any degree intelligible. A supreme will throws light upon the darkness ”

“ Symmetry is a proper agreement between the members of the work itself, and relation between the different parts and the whole general scheme, in accordance with a certain part selected as standard. Thus in the human body there is a kind of symmetrical harmony between forearm, foot, palm, finger, and other small parts; and so it is with perfect buildings. In the case of temples, symmetry may be calculated from the thickness of a column, from a triglyph, or even from a module; in the ballista, from the hole or from what the Greeks call the περἱτρητος ”

“ There must be no axis of symmetry higher than a twofold and three prominent directions (i.e., parallel to important faces) at right angles to each other, the highest grade of the three classes having three twofold axes ”

“ 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. ”

“ Cubic Crystals growing from Solution.
41We have thus attained at length to a truly scientific classification of crystal forms, by using axes and planes of symmetry as criteria. There is no occasion whatever to imagine suppression of faces in the classes of lower than the holohedral or highest symmetry of any system. In these classes it is simply the fact that less than the full number of elements of symmetry possible to the system are present and characterise the class, which still conforms, however, to the minimum symmetry absolutely essential to the system. ”

“ Over and above this the ends of support, and the utilities of life in general, serve to give bilateral symmetry a high practical value. Most of the products of the useful arts, from architecture down to the art of constructing common utensils, possess this bilateral symmetry. This prevalence of the relation, in objects of daily perception, would serve to fix a habit of looking for symmetry as the normal form of things. In other words, bilateral symmetry would tend to become, to speak after Kant, a sort of a priori form of æsthetic intuition. ”

“ If I should increase the force of the symmetry, by filling in the center with black, it would be necessary either to move the symmetry nearer to the center or to move the opposite attractions away from it. An unstable attitude in the symmetry would have to be counteracted, in some way, on the other side.
Intricate shapes from which the eye cannot easily or quickly escape often hold the eye with a force which must be added to that of their tone-contrasts. [Pg 180]
Fig. 231
In this case the shape on the right requires a pretty strong dark spot to balance its contrasts and its intricacy. ”

“ 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. ”

“ The classification of these different kinds of symmetry has been most attended to in Crystallography, in which science it is the highest and most general principle by which the classes of forms are governed. ”

“ 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 ”

“ This illustrious physicist pointed out the advantage of introducing into the study of physical phenomena the considerations on symmetry familiar to crystallographers; for a phenomenon to take place, it is necessary that a certain dissymmetry should previously exist in the medium in which this phenomenon occurs. A body, for instance, may be animated with a certain linear velocity or a speed of rotation ”

“ Having once established a theory of the æsthetic significance of real symmetry, we should next have to examine asymmetrical, beautiful objects with reference to the relation of their parts to a middle line; to isolate the elements which suggest motor impulses; to find out how far it is possible to establish a system of substitution of these psychological factors and how far such substitution takes place in works of art—i.e., to what extent a substitutional symmetry or balance is found in pleasing [469] arrangements. ”

“ If an organism advances through a homogeneous medium with one end always foremost, that end, being exposed to forces unlike those to which the other end is exposed, may be expected to become unlike it; and supposing this to be the only constant contrast of conditions, we may expect an equal distribution of the parts round the axis of movement—a radial symmetry. ”

“ In every phenomenon the elements of symmetry compatible with its existence may be determined. Certain elements can coexist with certain phenomena, but they are not necessary to them. That which is necessary is that certain ones among these elements shall not exist. It is dissymmetry that creates the phenomenon. When several phenomena are superposed in the same system, the dissymmetries are added together. "Works of Pierre Curie," page 127. ”