BBC's Why Factor has a fascinating segment
on Symmetry. They missed a couple of points that would have made it even more fascinating.
(1) They mentioned Gershwin's symmetrical up/down pattern in I've Got Rhythm, but skipped the fact that Gershwin gave explicit instructions to performers. Those four notes were supposed to be exactly equal in length and stress to bring out the symmetry. This spondaic equality is hard because English is strongly trochaic. Very few performers follow the instructions, and the recording BBC used was typical of this failure.
(2) After discussing the Gershwin thing, which was definitely a case of symmetry, BBC slid over into discussing periodicity
in music or literature, while continuing to call it symmetry. Not the same thing! Symmetry involves starting from a center point or center line, and running out to the end in various directions, exactly once. Periodicity means repeating the same pattern in the same direction, spatially or temporally. Stanzas in a poem, steps in a dance, columns or dormers on a building, movements in a symphony. Some of those repetitions may be symmetrical, like the Gershwin example, but that's rather rare.
(3) Some explorer type noted that we are sensitive to symmetry because animals (i.e. likely threats) are symmetrical. True, but only part of the reality. Full reality is more interesting. Every natural object in the world, living and non-living, is asymmetrical on Y.
Up and down are always different. Plants are radially
symmetrical in the X and Z planes because they stand still. They need to be equally aware of events on all sides and equally available on all sides. They can't run away from a caterpillar or walk toward a sunny area; they have to fight the caterpillar with chemicals or turn flowers toward the sunny side. Sessile animals are also radially symmetrical on XZ for the same reason, but we don't encounter sessile animals on land. They're on the floor of lakes and oceans. Mobile land animals are bilaterally
symmetrical on X (side-side), and strongly asymmetrical on Z (front-back). We move forward along Z, eating in front and leaving poop behind. So we need to scan a wide range on X to know what's ahead of us. We need to be equally capable of turning in either direction, so our limbs are also symmetrical on X but not on Z.
This helps to explain why mirrors are hard to deal with. A mirror doesn't have any preference for Y or X. It simply reflects each point straight back. Because nothing in the natural world has Y-axis symmetry, our sensory system doesn't care or know about Y-axis symmetry. So we don't see anything unusual about the mirror's straight-back action on Y. But our senses are deeply interested in X-symmetry, because that's the direction that tells us whether an object is a threat. We empathize with a facing human. We can get inside his head and "feel" which is his weapon hand. The mirror's version of a facing human is wrong on X. His weapon hand is on the wrong side, which gets us all jangly when we try to think about it.