symmetry

Tiger, tiger, burning bright

In the forest of the night

What immortal hand or eye

Could frame thy fearful symmetry?

As we wander through this world, trying our best to make sense of it, as we try to wrap our little heads around quantum theory and modern physics, we are called home by the concept of symmetry. For truly symmetry provides a key insight into the workings of the physical world – and consequently into our spiritual and mental worlds (the leap from the physical to the spiritual is no leap at all). The power and elegance of symmetry as the underpinnings of the laws of physics is wonderfully illuminated by Symmetry and the Beautiful Universe by Leon Lederman and Christopher T. Hill, a 2004 science book for the moderately educated (or moderately ignorant but still curious) reader.

The great conceptual developments in modern physics begin with Einstein’s Theory of Relativity and with Noether’s Theorem, which in effect states that the Laws of Physics are themselves – almost by definition – symmetrical. The universality of the Laws of Physics through time and space is a manifestation of symmetry. Symmetry as an underlying principle organizes the universe.

The concept of symmetry, which we apparently inherit genetically as a unit (an example of intelligent design), simplifies physics. Symmetry imposes it’s imprint upon the entire universe. How did Blake call it? “Fearful symmetry”? Well, it certainly is a powerful concept.

On the physical plane, in the two dimensions of space, we can see the power of symmetry. We can “see” and enjoy the beauty of symmetry through a sort of identification or reverberation. Symmetrical objects appeal to us. Symmetry is aesthetically pleasing. We’re just built that way.

What Four?

The symmetry of the four. We see it, we use it all the time. Bumbastories continues with the count down… er count up …er…walk through the numbers. Today we feature the four, the quatro, again!

When we look at a tesselation, a repeating geometric pattern, we see, we organize it, in quite predictable ways. A hundred years ago the early Gestalt psychologists first pointed out some of these organizational principles, these intrinsic patterns of perception: the tendency of the mind to automatically group objects, to distinguish automatically between “figure and ground”, and to search for the “whole” (which is more than the sum of its parts, they remind us). More abstractly, I suppose we search for the “main idea”. The mind’s proclivity toward symmetry is another of these universal phenomena of perception- and what a lovely one it is.

The aesthetic pleasure we receive from a beautiful pattern is perhaps a serendipitous benefit, nay a bounty, of our perceptual wiring. We enjoy order. We love it. As living creatures we already defy the law of entropy: we actually decrease entropy, we organize atoms and molecules while we’re alive – into some really complicated and awesome patterns I may add (that’s a model of a DNA molecule to the right, based on the five by the way).

It’s a human thing. We enjoy order and the order of beautiful patterns.

What good fortune (for us) that the sultans and Nasrid potentates who built the Alhambra palace in Granada, Spain were forbidden by Biblical edict to worship graven images. As a result they decorated their palaces with extraordinary geometric tile work, these tesselated decorations – which Bumba had the great privilege to visit and photograph last week.

Presented here are some of the four-based tilings. The artisans of the Alhambra also used the three and the six, the eight (they used the eight a lot but that’s just two fours) and even used five-fold symmetry to construct their opulent decorations. We’ll just stick to the four today.

The four, as a square, represents stability. Clearly it tesselates easily. The cross, the four seasons, the four questions, the four letters of God’s name are not total coincidences the mystics tell us. Personally I think that coincidence is largely underrated. However, the four cardinal directions seems to call to something deep within us. The American Indians have shown the power and beauty of the four directions and the importance of proper orientation in life. So stay on the Red Road, brothers. And don’t be no square!

Take Five

Five is my favorite number. Sounds funny, a bit nerdy. You know, to have favorite numbers. But it’s true. I confess a certain affection for all the numbers. After all, they’re all “sacred expressions of the transcendent”. But the five is truly special. Its symmetry is perhaps the richest of all.

The golden proportion phi, rooted in fivefold symmetry, is a phenomenon of great beauty and importance to all living things. Phi — the square root of five minus one/divided by two — or 1.618…..is the golden proportion, the proportion most pleasing to the eye. Plants grow using the phi proportion. The DNA molecule is arranged according to five or tenfold symmetry. Seeds are most efficiently packed into a flower using phi. And phi is based on the five.

The pentagram with its pentagon inscribed – which circumscribes another smaller pentagram – which in turn generates another little pentagon – then another pentagram, and so on ad infinitum – will also expand in the same proportions ad infinitum (and if you add all those infinitums together you get an awful lot of infinitums).

The diagonals of the pentagram intersect in phi proportions: the big piece is to the little piece as the entire piece is to the big piece. It’s the “golden mean”. Artists like Leonardo have used it for ages. Phi’s aestheticly pleasing proportions are apparently wired into us. Our senses are naturally attuned to it, and we find it’s proportion of 5 to 8 or thereabouts a simple way to compose a painting or a building or nearly anything. The Parthenon is so constructed. The Great Pyramid too. The application of the phi principle is fundamental to Cheop’s pyramid; it’s dimensions are laid out in accordance to the golden mean. It’s an ancient piece of knowledge. Expressed algebraically phi is represented as AB/AC= (AB+AC)/AB — which reduces to the square root of five minus one/divided by two. Phi squared equals phi plus one. The mathematical wonders of phi never cease.

The five pointed star is a universal and elegant symbol of course. Although neither the star nor the pentagon tesselate cleanly like the square or the triangle or hexagon, figures based on phi angles nonetheless create patterns like these – Penrose tilings – a tesselation discovered by the physicist and mathematician Roger Penrose – which is a lucky coincidence because it would have been weird if someone called Moskowitz discovered it and named it Penrose tiling (Sorry ’bout that one).

I’ve included a song I wrote that starts off with the fifth. I wrote the song many years ago back in the Bronx, but still enjoy playing it. It tells a bittersweet story, a love story of course. The song is in D, but it starts off in A, the fifth of D. Now, how the musical fifth ties in with what I’ve been talking about is beyond me, but this song is the first one I thought of. I have a weakness for songs that start off in the fifth. Not to mention my weakness for the number five. OK, I won’t mention it (oops I just did).

Fast Fourward

The sidewalk outside the L.A. County Art Museum.

The symmetry of the Four again (set to the music of Bumba’s harmonica fourths if you click).

The Quatro: the tesselations of the square, the grid. Everything laid out on graph paper, a world made of little boxes.
The usefulness (to humans, not so great for all the other species) of patterns based on the four has been astounding. Pyramids, temples, and skyscrapers arise square and straight from their cornerstones. Roads and cities laid out on grids. Brick by brick, square by square. Maps of distant lands, maps reaching to the heavens.

Analytic geometry, the great bridge between algebra and geometry – so kindly revealed to us four hundred years ago by the Frenchman Descartes – allows us to visualize patterns and mathematical functions. Equations are made manifest in two or three dimensional space. Functions come to life. Newton’s calculus arises. Motion is finally described via a set of laws. The scientific age, the Industrial Revolution follow. All thanks to poor Rene Descartes, who, lying on his back in bed watching a fly walk across the ceiling, suddenly conceived of the grid defined by the two dimensions we now call the x and y axes. What a guy that Rene Descartes. Here’s to Rene Descartes and the symmetry of the four.

Symmetry of the Four

The whole numbers (1, 2, 3, 4, 5, 6): Each number carries a certain spiritual import. The 5 is perhaps the most beautiful, the 3 with it’s holy trinity…. The symmetry of the four isn’t just for squares – or maybe it is.