Animal, vegetable or mineral
A bird was footering about on the gravel of the roof below, and I was trying to make out what it was. Charcoal head shading to mid-grey body, white flashes on the wings, russet rump. Much of the time it was only visible in movement, when the parallel white lines showed up. And I wondered, knowing that the best method of camouflage breaks up the lines of the structure, why every bird and beast from the most ostentatious to the most sleekit has bilateral symmetry. And plants extend round a line, and minerals round a point. Mineral, vegetable, animal, growing from point, line, plane. Fixed, flexible, mobile: the stricter the symmetry, the freer the creature. Is there an order of symmetry above bilateral? or below spherical? And humans, with our moral codes and formality in art, are we looking for freedom in a higher symmetry? Could it be formulated for verse? That's my real question, because as usual I'm only interested in theory as raw material.
To be more precise, minerals in this world aren't spherical, unless you count the oblate spheroid itself. But there's the stars, and the planets. Other mineral states derive from them. Gravity, collusions and collisions make the fractures and structures of rocks and crystals here.
For the radial symmetry of plants I'm thinking of trees and their roots, of thistles, roses, clumps of gorse and grass. I know they can start with punctate seeds and end with superficial leaves, but the governing axis of symmetry is the line. Animals have symmetry about a plane, which is only pliant, into the reflex dimension, and when the muscles relax they straighten up (watch how fish swim); the line, though, can twist and loop into another two dimensions, so that even when they are bilateral, plants are never so straightforward: a leaf is mapped onto its mirror image, the pair rotates up a tendril helically, boughs bend and twigs twist out to the light. Ach I should be writing this in winter. Theories work better then. As it is, spiricles are unfurling from under crumbs of earth, shaking out into crossed pairs of smooth and serrated leaves. There are hundreds of different options: spiral fern shoots, sycamore hands, and maple, where leaves are certainly bilateral but also, like side aisles, echoes of the centre, diminishing fractally into their own serrations. Corinthian sprouts from doric severity.
As to animal symmetry, I know about amoeba and protozoa, but take this heuristic: if it's too wee to stamp on, it doesn't count. Sea anemones may plant themselves on rocks, and starfish revert from bilateral to apparent radial symmetry, but they're not the only creatures that don't know what they're about. It's curious that animals, which are not particularly symmetrical inside, are quite symmetrical outside: animal limbs are congruent, and the colouring, to return to the start (it was a black redstart), is just about absolutely symmetrical. Why? There are compelling functional explanations of the symmetry of bodies that want to follow their nose - such as avoidance of walking in circles. But why bilateral symmetry of decoration? Pierre Aubry, after some head-scratching, came up with a counter example. The sole, which is born a flat, vertical fish, develops into a horizontal flatfish. One of its eyes migrates round the corner so it can have a better view of the ocean squashing it some more. This means that one of its sides has two eyes, the other none; one side is one colour, the other another. I checked this up at the fishmonger's, and in a way he's right: the sole's mouth is along one edge, like the zip on a toilet bag. Nevertheless, it's more than half-way towards achieving a new axis of symmetry of the traditional sort, along a line that runs at right angles to the shortest route between its eyes. Why?
Another species of flat fish: imagine a dozen nacreous hatchets with the features of Tam Dalziel (sorry, Tam). If you can't, then get to Basel Zoo and look at the creatures tacking round over the label Selene vomer. They're called St Pierre or John Dory when they hit the fishmonger's slab, by which time they look more like Douglas Hurd. (wrong again: John Dory is Zeus faber. But anyway...) These creatures are symmetrical and identical: specimens of their species. This lump of clay has no symmetry. This human face has surpassed symmetry. A specimen abstracted from the species is symmetrical, but the unrepeatable individual steals from symmetry. Rubato. My symmetry says I am, and my mark on it says I'm only here the once. How does it go? asymmetry, symmetry, dissymetry, then either dissolution or higher order symmetry? Maybe.

Groups of symmetry

Space/Time Wave/Particle Rhythm
Point Mineral Logic
Line Vegetable Rhyme
Plane Animal Metaphor
Body Human Naming

Each successive group incorporates the previous one - geometrically in column 1, physically in column 2, and literally in column 3: metaphor can entertain logic, for example, whereas logic, with the law of non-contradiction, cannot accommodate metaphor.
I have extended the "animal, vegetable and mineral" categories by one at either end: subatomic, which I don't understand, and human, which I haven't really grasped either. In the nineteenth century there would have been a "superhuman" category, courtesy of Solovyov or Nietzsche, but that hubris has nearly lost us our human specificity. "God is dead" led to "I is dead". I don't think poetry can touch the godhead. It's only human. But human it is.
It would seem that Big-Bang archaeologists are projecting a golden age of supersymmetry in ten dimensions or so. People like John Ellis try to reassemble the Ur-version, rather as classicists infer Orphic cosmology from slighting references in Aristotle, or the way theologians chase through Coptic and Armenian translations for something nearer to the original "In the beginning" - but the nearest they'll get is Greek. Echoes is all we ever get, the air shaken by speech, the seismograph. There's never a sign but in translation.
Still, if time began, it's always beginning somewhere, quite apart from nostalgic algebra.

I gave my daughter a box of crayons.
First she chewed it
then she rattled it.
This went on for quite some time.
When she did put wax to paper
she just did rhythm,
cardiograms.

Note that none of the above groups could have been extrapolated from its predecessor. There's something miraculous about vegetable from mineral, birds budding from branches. A miracle, true, is that which no rational being can believe in. But then rational beings didn't deduce themselves from cognates in the trees. All they can do in this area is explain themselves away. Showing how something has evolved over time explains it in the etymological sense (it unfolds the thing origamically). It does not explain things in the common sense of saying why they're there. So why are we here? Wait a minute.

Words
What symmetry do we have in letters? numbers? words? This English alphabet (capital letters) has 6 letters without an axis of symmetry, 4 with rotational symmetry - three of them through 180 degrees, one complete, 3 others with two axes of bilateral symmetry, and the rest with one axis of bilateral symmetry: 2 diagonal, 4 horizontal and 7, like us, vertical. Of the digits, 2 have two axes of bilateral symmetry, 1 has a horizontal axis, 1 has full rotational symmetry and 1 rotates through 180 degrees into another. Not much of a pattern. In fact, the distribution is just about random, as though the idea were to get the maximum variety from simple signs, which seems practical.
Words: they are produced by symmetrical organs. The shapes those organs make - the cave of "ae", the tube of "oo", the closed gates of "n", the glug of "g" - are just as bilaterally symmetrical as us, but there is no symmetry at all in words, not even "Malayalam": play it backwards, it doesn't sound the same. They're borne on breath, spoken in time, and can't be taken back. A word on time: it seems it was thinking about right and left that led Kant to say that space and time were intuitions. Blinkers, more like. Time is a syndrome quarks may not suffer from, but we do; like speech, it is not a palindrome. It's a fuite en avant, the devil in the driving mirror. Symmetry is a sign of ecstasy. Make love, "the beast with two backs" (in the words of Othello's adviser) and that's another sinuous axis of it. Make love or, better still, play football! since while the converse goes without saying, it must be emphasized here that thoughts about symmetry or any other activity are at best irrelevant to love-making; also, the rerouting of blood from the brain during intercourse is likely to have an adverse effect on thought. On the other hand, some consideration during your next match with Hamilton (NZ) Academicals ("the beast with two forwards") as to what your opposite number thinks your next move is going to be, and vice versa, might just improve your game. Symmetries, like axioms and the line-up, are for what is settled; the operative part is asymmetric. Some verse doesn't get as far as symmetry: there's only one team on the park, triumphant in its haletotic bardic breath. Some verse can't break out of symmetry, like two collusive teams whiling away 90 minutes. Some transcends it, though, like the human face. "Soccer isn't a matter of life and death: it's much more important than that."

Ports of Perception
If something is apprehended on its axis, if the axis is a port of perception, any symmetry it might have would not be directly perceived. You'll see the symmetry in the shape of a leaf, but not in its colour, because you're at the end of a beam that's undergone filtration and reflection in the leaf and in your eyes. You can feel (and hear some) symmetry in forces and objects, but not in their textures, since the axis is at the meeting of (say) fingertip and surface. The same applies to olfactory receptors and taste buds, where molecules join. There aren't symmetrical smells. And smell is intimate with taste. And taste, like speech, runs along the tongue, sweet at the tip, salt on the blade, boiled broccoli at the uvula.
Are we ourselves the axes of some further symmetry, drawn towards a moment of balance, strained and used by it till we have neither shape, nor humanity, nor name? If time were a fourth dimension, desire could be a sixth sense - but it doesn't get us closer than the other five to answering that question. What desire has it doesn't want: itself, this moment, part of me. It yaws like a compass needle when the lode is on a plane it can't turn to. I want to be this, to have that, but not to have been or be had; I want to go and do things, not be gone or done in; I want to stay, I don't want left behind. All the conjugations tell the same story: desire is centrifugal, contradictory, and as absolutely time-bound as speech. What I'm after is something that does with existence what poetry does with utterance: not reduce the world to symmetries but adumbrate the patterns we're in: it's more a prayer than a power game.
This spiel is mostly mineral, punctate, logical in its way. It's crystallising outwards from the word "symmetry". It's about to be twisted by the word

Chirality
I can't believe that the spiral - the word itself suggests circle and parallel - isn't symmetrical. Find yourself a spiral staircase. There is a good one on Bridge Street in Glasgow, over the Glaswegian pub, I think. A steep flight of stairs from Nelson Street takes you to the circular floor of a cylindrical stairwell. A hanging stair winds widdershins up the wall. Stand at the top and look down. In fact, raise the building another ten floors to make the point quite clear. The banister spirals in towards the centre of the floor of the stairwell, though the staircase has a circular plan - which gives it infinite rotational or mirror symmetry - and in elevation, if we ignore the fact that this particular example levels out into a landing once in every full turn, it is a sine curve set on its side, which gives it two axes of symmetry for every floor it rises (one for each peak and trough of the sine curve).
You may object that I have found symmetry not in a spiral but in the plan and the elevation of a helix. Very good. What I have said is that the spiral can be analysed into two components that have symmetry. The same is true of other obvious examples: cyclones and snail shells. But before we return to specifics, a word on dimensions. For a long time I exercised my trivial imagination on quadrivial space. I tried to see what happened to a party of particles whizzing up and down the four axes (x, y, z and t) of length, breadth, depth and time. It's only this morning that I have finally realized that that's as pointless as trying to visualise quantities in their progress through a long division sum. Feynman diagrams? I'll no more see a particle in there than I'd see the newscaster's scalp if I sawed the top off my TV (and I don't have one). To me, space is nothing and nothing doesn't have a shape, but I won't be bothered by the next physicist who tells me that space-time is curved: all he means is that his co-ordinate system is screwy. If he can live with that, so can I.

Dimensions
But there is still less to dimensions than meets the eye. I cannot visualize one dimension either. Judging from the problems that generations of Greek geometers had in deciding on the nature of the point - Zeno reduced it to the absurd but no-one quite reduced it to unity or set it at naught - there is no shame in that inability. John Burnet thought Leukippos had solved it with the atom, but them Burnet was writing when Rutherford (who visualised atoms as billiard balls) seemed to have done the same. Heisenberg seems to have worried at the same bone (wave or particle? verb or noun?). Worse still - put it down to stereoscopic vision if you like - I am not at all sure that I can see in two dimensions. A line always implies a space. I see nothing but surfaces and I assume, always, that there is depth to them. That is why the spiral is so obviously symmetrical. That is what painters play on: when a surface doesn't make sense the viewer looks into it. When that doesn't make sense either, stay cool: it's the twentieth century. The artist might just be making the point that I am making here: there is no point; don't trust appearances; don't trust realities either.
So if I can't see four dimensions, or one dimension or even two, then what is the point in talking of three? Dimensions are for graph paper. They set things straight: they put vision and dementia behind bars, which isn't always a bad thing, though always is a bad thing: somebody has to take a shillelagh to Occam's razor gang from time to time, and defend different interpretations of the world: a sycamore has one, morning glory has another, the black redstart has a third - to say nothing of barnacle geese, goose barnacles, beautiful hooktips etc.. In other words, respect not only what divides one realm, one science from another (crystals have no pentagonal lattice while cinquefoil plants are common) but what distinguishes each creature from the next. Name them. I do not deny the use of categories: in learning a language, for example, paradigms help distinguish por from para in Spanish, and indicate when to use which aspect of a double imperfective verb in Russian. But there are so many exceptions to the rules, and flaws in the pattern of exceptions, that you have to choose eventually either language, languages, worn stone stairs, eroded reason, most irregular in the most used words, or the practice of weeding out linguistic illogicalities and ambiguities so you can express things more clearly, make do with clarity, and miss out on symmetries that don't add up (see "Insomnia" below).

Mirrors
Symmetry round a vertical plane didn't exist before animals. And it wasn't confused before the invention of mirrors. Reflection in water distinguished opposites: up from down, sea from sky, root from branch, sun from seed; reflection in the glass compared and confused similarities: right and left, and wrong; dexterous, gauche and sinister. They can be both similar and opposite, things of the heart, hope and despair. The intimate but unseen schism between handedness and heartedness, heart and hand, and the way that opposites rotate into similes, unsettle. There's a flat contradiction between two senses, sight and touch. Mirror imagery is not to be taken at face value.

Insomnia

The moon in the bureau mirror
looks out a million miles
(and perhaps with pride, at herself,
but she never, never smiles)
far and away beyond sleep, or
perhaps she's a daytime sleeper.

By the Universe deserted,
she'd tell it to go to hell,
and she'd find a body of water,
or a mirror, on which to dwell.
So wrap up care in a cobweb
and drop it down the well

into that world inverted
where left is always right,
where the shadows are really the body,
where we stay awake all night,
where the heavens are shallow as the sea
is now deep, and you love me.
(Elizabeth Bishop)

Writing is an up-and-down reflection: I think things up and write them down, and there is no more danger of mistaking the words for the world (though it happens), than of taking sea for sky. The cinema, though, is a back-to-front reflection, a mirror image, and film very often is mistaken for life.
The mirror is neither me nor you.
The word is both of us.
So how were we divided?
We could start with two sets of 23 chromosomes, join them up, and follow cell division till it produce up and down, right and left, and another creature with another set of 23 to offer, but that's not really the beginning. I'll take another story. Adam was divided from the earth, Eve from Adam, settler from nomad, hairy Esau from gentle Jacob, builder from planner. Babel! The ziggurat, helical sign of division. With that kind of history between us, any communication is fraught. Clarity isn't the most important quality here, it's charity, trust that ambiguities are honest and illogicalities are true. It's an interpersonal thing and the only important axis of symmetry is the page.

I turned to symmetry hoping for help
in staking out my heart, but it's the knotted
cord itself that sets the measure.
"Quelle que soit la fécondité d'une méthode,
son office est surtout de consolidation
et, si l'on veut, de prolongement,
mais sur un terrain préalablement fixé.
Elle met en ordre l'acquis et, ce faisant,
comble les lacunes et exploite les percées,
mais elle n'inaugure rien d'essentiellement neuf"
(Robert Blanché, l'Axiomatique).

Mozart kicked away the ladder Bach had come up by,
and I've had enough of broken chords and scales.
There's one way and one way only to go about it:
find an angel and pick a fight.