Geometric Perfection

The beauty of geometric objects fascinates us. Sure, the concepts are useful, and they cross over into the practical world in a surprisingly diverse variety of applications, but the pure abstractions of circle, cube, line, etc. only exist only in descriptions or visualizations of the minds eye. These ideal shapes have a certain spiritual luster to them that cannot be easily corralled into words.

Socrates did an alright job at describing the attractiveness of pure geometric shapes when he described the them as having "peculiar pleasure, quite unlike the pleasures of scratching."

Hopefully you get that rash checked out soon, Soc... Here is the full quote:
 

I do not mean by beauty of form such beauty as that of animals or pictures, which many would suppose to be my meaning; but, says the argument, understand me to mean straight lines and circles, and the plane or solid figures which are formed out of them by turning-lathes and rulers and measures of angles; for these I affirm to be not only relatively beautiful, like other things, but they are eternally and absolutely beautiful, and they have peculiar pleasure, quite unlike the pleasures of scratching.

— The Philebus, 51

The reason for our attraction to these shapes may be the same reason that they cannot exist. The geometric forms are perfect, too perfect for this world. We, squishy sacks of thrown together meat and bone, yearn for perfection while writhing in the dirt. Wherever something physical even mildly evokes feelings of perfection, there you will find the humans gathered, drawn to it like bacteria to a ripe carcass. Perhaps processes of natural selection have implanted the perfection-striving function in the human psyche, finding it a convenient proxy for successful reproduction.

Approximation of Circle

Although all geometric shapes are perfect, they are not all equally perfect. Our biases have led us to particularly fetishize a select few. The point, line, circle, square, cube, and sphere each have been awarded a special position in both mathematical literature and everyday vernacular.

Where do these preferences come from? Are they truly some perfect ideal that humans have successfully conceptualized? Or are they arbitrary obsessions of the human mind, which are only not called into question because we all share them?