Symmetry, Dissymmetry and Asymmetry through The Looking Glass into the Anti-World

The fundamentally spatial or situalogical concepts of symmetry, dissymmetry, and asymmetry surge up from their abstract origins in geometry into the fourth apace dimension, become dynamic characteristics of the world of physics and find polar-to-physics embodiment in mythologies. From the one pole of the mathematical physicists’ passion to find symmetrical forma of explanations, thought becomes progressively more asymmetrically oriented in biology, in everyday life, in art and in mythology; until finally the form of mythology and magic embody a qualitative opposition between left and right -a fundamental asymmetry of world view. But it would now seem that the latest discoveries in physics and astronomy might be quietly and questioningly directing physics and cosmology toward such an asymmetrical world view.

I. The ideas of symmetry, dissymmetry and asymmetry are all necessarily entwined In each other – the meaning of one la bound up with the understanding of the others. And they are mentally dynamic ideas because they involve either imagined or real or imaging operations – of comparison, rotation, transformation and reflection (either by mathematics or by mirrors) and curious questions about the reality of mirror images and the fourth spatial dimension.

II. To the intrinsic abstract spatial interest of these concepts, the discoveries of most recent physics (about the lack of con­servation of parity) have added urgency in throwing into doubt the formerly quietly presupposed assumptions about the symmetry of physical laws.

III. In this way physics has been given in inclination toward the forms of thinking operating In biology, everyday observation, art and mythology, which show a progressive tendency to accept and finally to regard as fundamental an asymm­etrical mode of thought. The resultant – somewhat mythological sounding – cosmological opposition between World and Anti-World leaves the question of the ultimate symmetry or asymmetry of the universe empirically open.


To begin with the most simple, almost imperceptible, beginnings symmetry becomes abstracted almost to the point of losing its meaning when considered in one dimension only, but this is illuminating. Thus: in one dimension, a line X, a segment A is symmetrical to a segment B if A can be superimposed on B, has the same length. If A and B have the same direction, then A can be superimposed on B by moving it along line X – hence without leaving the one dimension, this is called translational symmetry.

However, if they have opposite directions, there is no way in which they can be moved along the line to make them coincide. To do this A would have to be turned around – to be rotated in a two dimensional plane in order to have the same direction as B.

More significantly, the case is similar for a two dimensional figure in a plane. Thus figures C and D have simple symmetry because they can be made to coincide by moving than around in the plane until C lies on D – translational symmetry} whereas figures E and F are symmetrical in a more complex way. They could only to made to coincide by rotating E from left to right so that it would have the same direction as F. But to do this E would have to be lifted out of the plane of the paper and rotated in three dimensions. Such a rotation through the third dimension would give the same result as reflection in a mirror. So here mirror reflection and real rotation are valid alternatives. This is called reflectional symmetry in two dimensions or rotational symmetry in three dimensions. A figure composed of two such mirror image parts is said to be bilaterally symmetrical, this is the fundamental structure of almost all the more highly developed forms of animal life. To this we shall return.

For three dimensional forms the case is parallel but also so essentially different that one uses a different word “dissymmetry” to designate it. Consider the most familiar of three dimensional objects – a left hand and a right hand. They have a teasing simi­larity of form that might lead one to think he might superimpose one on the other if only he twisted them about in the right way. Try it. Of course, it is impossible physically to turn a right hand into a left hand except by reflecting it in a mirror. But mirror reflections are metaphysically strange because half of the bilaterally symmetrical form or of the pair of twins so created has a different ontological status from the other half – it is “only” an image – unreal. You try to move into its space, and you are stopped by a hard vertical wall, or you break through the wall and thus destroy the image and its space. The other method used in the case of the two dimensional form – real rotation in another dimension is here excluded by the absence of a fourth spatial dimension. This is the essential difference which is marked by the use of the word “dissymmetry”. Absence is perhaps too objective a word here. Perhaps it is only we who are essentially three dimensional creatures or use our mind in a three dimensional way. What would it be like to “turn over” a left hand into a right hand in the fourth dimension? To being with the end: the result would be like exchanging it with its Mirror re­flection. Yes, but one can imagine how figure E is in the process of being turned over: how is a left hand when it is partially turned over in the fourth dimension? Gradually turning a left-handed glove inside out provides a suggestive picture of the process and effects a similar change, but then you have a glove that is inside out. But it la gruesome to imagine this actually happening to a solid hand. It might help to imagine the two dimensional rotation as it would look to two dimensional creatures whose conception and vision were limited to that plane. Thus as E is slowly lifted up into the third dimension and rotated, its projection on the plane shrinks gradually inward toward the axis of rotation until it is merely a line coinciding with the axis of rotation and then it gradually emerges on the other side. In the diagram forms at successive times are indicated by 1,2, etc.

The Oxford mathematician pseudonymed Lewis Carroll solved this problem graphically in “Alice through the Looking Glass”. As Alice, longing to move into the looking glass world – the world of opposltes, a pre-cognition of the newly adumbrated Anti-World, pushed herself against the mirror, she flattened herself out against its plane, dissolved into a two dimensional creature, and emerged on the other aide into the dissymmetrical looking glass world (as left-handed, no doubt, although he didn’t mention this). This doesn’t sound as traumatic as turning herself inside out but has the same result. The use of “dissymmetrical” emphasises this present lack of possibility of a real physical process of rotating a left-handed object into a right-handed object corresponding to the mirror reflection. (There is a developing theory of physics which realizes this possibility mathematically and aims from this more complex really four dimensional space to deduce the laws and entities of physics. But that is “Work in Progress”).

The physical importance of these relations was focused to attention only recently as the result of experiments suggested by the question of two American Chinese physicists, Yang and Lee; Why assume that the physical world is indifferent to mirror (or mathematical) reflection? Why assume that parity – equality of right-handedness and left-handedness must be preserved in all physical occurrences? Then to physicists’ astonishment, it was answered that in some cases it was not so. In these experiments the behaviour of one of the subtlest of fundamental particles, the neutrino, was investigated, this particle travels with the velocity of light, has no mass and has an angular momentum or intrinsic spin around the direction of its movement. This spin can go in a clock­wise or anti-clockwise way around the direction of the particle. If the thumb points in the direction of movement, then a clockwise spin is right-handed, and a counterclockwise movement is left-handed. It was then presumed, in accordance with the tacit assumption that nature did not have any preference for left-handed or right-handed config­urations, that in any neutrino-creating operation equal numbers of left-handed and of right-handed neutrinos would appear. However, to the astonishment of physicists only left-handed neutrinos were created. Nature seemed to show a bias – there had been introduced into the dissymmetrical balance of the physical world an essentially asymmetrical element, this came as a shock, like looking into a mirror, recognising your own face, but seeing that it had one eye on the left side but two on the right side. Puzzlement. Disequilibrium also in the mental sphere. What to do???

What they did do is illuminating about how far the mathematical passion for symmetry penetrates physicists’ thinking. The search for a balance for this disquieting asymmetry led into the Anti-World.

But what and where is the Anti-World? In the course of high speed, complex synchrotron experiments on fundamental particles one creates and destroys these fundamental particles. But one of the guiding principles of physical thought is that the number of physical particles should be conserved. So it is assumed, hoped and looked for and thus far found that whenever a particle is created, a particle similar in all respects but opposite in electric charge, a sort of mirror image twin, is also created. This is called an anti-particle. It acts by subtraction. That is: if it meets a particle, they annihilate each other (with a dazzling liberated radiation – a kind of Liebestod). In a more complex situation, say an atom, it acts by subtracting one corresponding particle. These anti-particles are rare, flighty, unstable creatures in our world, only created in high powered laboratory experiments and nuclear explosions, but a vast predominance of them would make up a stable Anti-World.

Now it was found that while neutrinos had only left-handed spins, the symmetry was restored to the situation if one looked to anti-neutrinos, which have a right-handed spin. Like looking into a mirror for the here rare inhabitants of the Anti-World. Thus the observed asymmetrical behaviour of neutrinos was attributed to (excused by?) the fact that here now in this world there is in general a vast pre­ponderance of particles over anti-particles (of World over Anti-World). But since the laws of physics are symmetrical with respect to particles and anti-particles, there is no a priori reason why at other parts of the universe there should not be similar concentrations of anti-particles making up Anti-Worlds – worlds of anti-matter with laws reflecting those in our world but oppositely charged – mirror image worlds. Indeed for the sake of symmetry one more or less expects them to exist somewhere among the nebulae. If such a hypothetical Anti-World should ever meet a World, in a flash of perfect subtraction they would annihilate each other with the liberation of vast quantities of energy as radiation. (Such an annihilating meeting of a World and an Anti-World has been suggested as one possible explanation of the almost Impossibly enormous radiations coming from the newly discovered Quasars – quasi-stellar radio sources – which are perplexing astronomers and cosmologists). Thus having started out with geometrical considerations of symmetry and asymmetry, we end with the Apocalypse, but perhaps this is the natural direction of things.

In this way the question of the ultimate symmetry or asymmetry of the universe has been transformed from a presupposition to an open empirical question of cosmology. Thus physics has been somewhat shifted toward the forms of thinking embodied in biology, observation of people, painting and mythology. These progressively tend to take asymmetry as more significant and fundamental.

In the domain of biology most complex living forms are bilaterally symmetrical – the left side is a reflection of the right. But embedded within this symmetrical framework are essential asymmetries. For example, the heart lies on the left side of the body, and its intrinsic form is that of an asymmetrical left-handed screw. The intestines are asymm­etrically curled. And there are more permeating asymmetries within the very structure of living tissue – in the structure of the molecules that make up organic matter. The glucose molecules contained in animal bodies have the form of right-handed screws, while the constituent fructose molecules have the left-handed form. Should these be interchanged, if a left-handed molecule is introduced where the right-handed form is natural, the whole organism dies. Thus life is dependent on certain asymmetries. Indeed this seems poetically right – an excessive perfection of symmetry seems anti-organic, anti-lifelike, cold like perfect snow crystals.

In “Der Zauberberg” the hero meditates thus on snow crystals: “Each in itself – this was the uncanny, the anti-organic, the life-denying character of them all – each of them was absolutely symmetrical, icily regular in form. They were too regular, as substance adapted to life never was to this degree – the living principle shuddered at this perfect precision, found it deathly, the very marrow of death”.

The life-like qualities arising from such asymmetries embedded within a larger order of reflected symmetry may partly account for the mysterious power of Shang Dynasty bronzes. These strange animal or dragon heads have a slanting almost writing-like dissymmetry reflected across the face’s centre as in an invisible mirror, but containing surprising expressive asymmetries. When one is expecting dissymmetrical balance, the asymmetrical detail is invested with meaning. They act like clues, solidified and dropped in time – like enigmatic meaningful faces – meaning what?

Indeed one sometimes looks into people’s faces, tries to read or decipher them in this sort of scrutinising way. Within the generally symmetrical frame of the face the subtle asymmetries function as keys to character or clues to personality or to “what lies behind”. A perfectly bilaterally symmetrical face would be a monster of vacuity. The asymmetries embedded within the usually two-eyed, two nostrilled faces one meets give to them their life and meaning and expression. Professor David Martin in the British Journal of Aesthetics for December 1964 heightened and formulated these observations about the significance of asymmetries in faces thus: “Often faces have ‘spiritual asymmetry – one side of the face is more the window of the inner life, expressing introverted tendencies, whereas the other side is more the mirror of the social life and the soul’s active struggle with the outer world, the extroverted tendencies”.

Further, more all-pervading than these asymmetries with larger symmetries, there is in the process of human perception a basic asymmetry which invests any given space that is looked at with a dynamic asymmetry. This is important for painting because a canvas is at the outset a given enclosed space that is looked at. Quotation from Fanchon Fröhlich, British Journal of Aesthetics, October 1964: “The natural path of the eye, which is very simple, interacts with the movements and placings within the painting to produce an endless variety of dynamic effects, even as a very simple physical law in operation under different circumstances in the exterior world produces and also explains a multitude of effects. Thus even with the blank canvas there are given perceptual tensions. Any mark within this natural structure interacts with it – works either with or against it or interlaces with it in some complex way capable of being unravelled. One can see this very clearly in looking at a mirror image of a known painting wherein left and right are reversed so that the given structure of the enclosed space is inverted without, in the case of representational painting, changing the mimetic significance of what is depicted. In such a case although the same scene is presented, the whole dynamic and emotional or psychological signifi­cance of the painting is changed.

“The empirical basis for this idea of a perceptual structure of an enclosed space is a series of experiments conducted by the artist, William Hayter, and the psychologist, Max Wertheimer, at the New School for Social Research in New York (These experiments, among other things, are discussed in S.W. Hayter’s forthcoming monograph “Nature, Art and Motion” in a series on ‘Vision and Value’ published by Braziller in U.S.A.). In the course of them, trying to find whether there were any noticeable regularities in the way people see a presented space, Hayter required his subjects to sit in a darkened room and fix their attention on a point on a line drawn within an illuminated rectangle while the movements of the eyeball were photographically recorded. The lines were drawn as diagonals from the lower left section to the upper right section of the space and in a separate experiment from the upper left to the lower right. (It would also have been of interest to observe the reactions to such points on vertical lines in order to determine whether the left/right and up/down movements were independent, but such an experiment was not carried out). The results of
these experiments were surprising in their regularity. For it was found in all cases that the eye moved along the line in the direction from lower left toward upper right or from the upper left to the lower
right and never the contrary. —- Thus picture space as activated by the vision of any spectator has an intrinsic dynamic structure ready to begin a dialogue with any marks placed within it”. In this asymmetry has become all-permeating and an actively operative factor.

The pole opposite to mathematical physics at the beginning of this paper is presented by mythology. Therein left and right function as completely asymmetrical. Their difference is taken as qualitative, radical, profound – as corresponding to, or to put it more exactly but more mysteriously, as somehow embodying, making manifest within space – the difference between good and evil. Such a difference recurs in the mythologies, religions and magic practices of various cultures. An extreme example of the mythological significance of asymmetry is the Last Judgement. In the Gospel according to Saint Matthew it is said: “And He shall set the sheep on His right hand but the goats on the left. Then shall the King say unto them on His right hand. Come ye, blessed of my Father, inherit the Kingdom prepared for you from the foundation of the world – Then He shall say also unto them on the left hand, Depart from me, ye cursed, into everlasting fire, prepared for the devil and his angels”.

Here again as in the possibility of the annihilating meeting of World and Anti-World, the meeting of the asymmetrical opposites leads to the Apocalypse. Thus science in cosmology approaches mythology in cosmology. Perhaps there is some deep, internal connection.