The Mathematics of Art

I’ve heard about this a few times, but everytime I learn it again, it never fails to make me uncomfortable. Mathematics and art, 2 very disparate subjects (of which I am fond of neither), are actually quite intricately linked! You would think that mathematics, being all rigid and systematic, would have nothing to do with the fluid randomness of art. However, it doesn’t seem to be always the case.

If you’ve heard of the Golden Ratio, you’d know how it relates to beauty. The Golden Ratio relates 2 numbers such that the ratio of the sum of the numbers to the larger number is equal to the ratio of the larger number to the smaller one. So basically, for 2 numbers a and b, where a > b,

(a + b) / a = a / b

Quite a simple equation to grasp, but such numbers have far-reaching implications. Artists and architects believe that structures with such a ratio of length to breadth is aesthetically pleasing. Luca Pacioli himself wrote 3 books to show just how harmonious and pleasing golden-ratio structures are, but later on it was found that his interpretation was traced to an error, and that Pacioli actually advocated the Vitruvian system of rational proportions. Pacioli also found Catholic religious significance in the ratio. Even in book designs at the time, books produced between 1550 and 1770 had these “beautiful” page proportions. Even in music, Bartok and other musicians might also have used the Golden Ratio to compose their musical scales.

Another relation between mathematics and art is in the form of fractals, which is relatively related to technology. Fractals are “rough or fragmented geometric shapes that can be split into parts, each of which is a reduced-size copy of the whole”. It’s hard to talk about this without illustrations, but basically it posits that fractal triangles may actually be made up of an infinite number of triangles, so minuscule that they form the shape of a line when zoomed out. Great shapes can be made when you invert 1 part of the fractal, or by magnifying it to show the complex little shapes that make up the final shape.

There are other stranger concepts floating out there about beauty in geometric shapes. Francis Hutcheson says that shapes have to have uniformity and variety in order to be most beautiful. For instance, among regular shapes, the hexagon would be more beautiful than the pentagon, which is more beautiful than square, because the more beautiful shapes have a greater number of sides. On the other hand, among shapes with the same number of sides, a square would surpass the rhombus, which is more beautiful than the trapezium, due to their irregular curved sides. Is this true or is it the theory of a 1726 old fogey?

But of course, who cares about shapes? When it comes to people, those with facial symmetry are more attractive than those who don’t. It’s in fact played on in World of Darkness’ Changeling: The Lost. Changelings, no matter how weird and boorish they look, always have a strange charm to them, which I think can be attributed to innate facial symmetry. In fact, facial symmetry has been suggested as a possible physical manifestation of the Big 5 personality traits, especially extraversion and openness. This is because hormones such as testosterone and oestrogen are associated with development of facial features during puberty, and are thus the cause for these individual differences. So not only are symmetrical people beautiful, they may also have a beautiful personality too — if you like extroverted open people, that is.

Wouldn’t you say mathematics and art interweave in many parts of our lives then?

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