Is Visual Art Mathematical?

Siddiq Islam

Everyone knows and loves talking about how connected music is to mathematics. It’s a well-established concept. Musicians’ brains are compared to mathematicians’ brains. We revel in Bach’s meticulous canons, augmentations, and retrogrades, which can be thought of like translations, stretches, and reflections as though the music were a Cartesian graph (see video below). There is even a ‘math rock’ genre (although pretty unconventional).

While music is viewed as having inherently mathematical aspects, the opposite is often said about visual art. Drawing, painting and sculpture are thought of as the antithesis of rigid mathematics: they capture beauty and emotions which cannot be defined by equations. So, I pose the question: could visual art be linked to mathematics the same way music is?

The golden ratio is one piece of mathematics said to be found in many artworks. Called in Renaissance times the ‘Divine Proportion’, it is supposedly exhibited in Da Vinci’s Mona Lisa and The Last Supper, as well as other artists’ work. Modern photographers might use a golden spiral to compose their pictures and legend has it the most beautiful faces have eyes, noses and mouths aligned in this magical ratio. Even galaxies and shells grow in ‘golden spirals’.

Unfortunately, the beauty of the golden ratio is probably only a mathematical one. Online, you will find golden rectangles and spirals superimposed onto Renaissance paintings, but they are grabbing at straws. As for the shells and galaxies, they can produce logarithmic spirals, but are never in this exact ratio, and are in fact usually quite far off. Photographers may use it in their photographs, but so many beautiful pictures don’t require mathematical levels of precision and people definitely don’t need to conform to such a bizarre beauty standard in order to look good!

Whilst Da Vinci did know about the golden ratio and used it in his lovely mathematical diagrams, he did not need it for his art. So, can artists do without maths, then?

When you look at medieval art, you might find it disproportionate and flat, as though everyone is facing the wrong way. By the Renaissance, artists were using a trick to aid composition that is still taught in art class today – picking a vanishing point at infinity where all lines converge. This takes care of the foreshortening effect that the farther things get from the viewer, the smaller they appear.

Jesus’s face is placed at the vanishing point in The Last Supper so it becomes the centre of focus. Everyone else’s eyes converge near the horizon line.
Photographers use the rule of thirds trick more often than the golden ratio. Although both ratios appear to fit Da Vinci’s painting, it is not clear whether he did this on purpose. Perhaps he used them subconsciously.

So, geometry can help us to depict things more realistically. However, this still isn’t the whole truth of what we see…

If you are indoors right now, take a look at the edges of a wall in the room. It is a rectangle made of straight sides, so the corners should be right angles, correct? Now look at the corners of the room. They do not appear as 90 degrees. They are slightly larger. What happened?

Definitely not a right angle??!

Perspective is something our brains deal with constantly without us realising. The section of the wall directly in front of you is closer, so it appears larger than the section of wall in the corners of the room. To our eyes, the wall starts off small in the corners of the room, gets larger in the middle, and shrinks again as it approaches the next corner. Our brains fill in the gaps to tell us the wall is a rectangle!

This is all the consequence of trying to project a rectangular room onto the sphere of vision around us. If we want to keep all the straight lines in the picture straight on our page, we might have to stretch some things around the edges.

When you project a cuboid room onto your sphere of vision, the corners have to add up to 360 degrees, which three 90 degree angles can’t do!

You might have noticed this if you have ever taken a photograph with people near to the edges. Their faces become distorted. This is a result of rectilinear projection, the correction that a wide-angle lens makes to help things look straighter. You can see the alternative reality if you use a fish-eye lens or take a panoramic photo. This kind of image is a curvilinear projection. Everything is the right proportion, but straight lines appear curved.

The bending of our visible world is not a problem most of the time – it only occurs when we are trying to capture things very far apart in our vision – but realist artists should be aware of such optical paradoxes nonetheless. Once again, understanding geometry can help us to create more realistic images.

Rectilinear lenses correct the distortion. Everything in my bedroom looks straight but near the corners, the image has been stretched.
This is a panoramic, consisting of multiple photos stitched together. Curvilinear lenses like fish-eye lenses produce a similar effect. Everything is the right size, but straight lines appear curved.
This is also made from multiple photos stitched together. I moved from one end of the room to the other (which is why it is a bit wobbly), so everything is facing forwards. I think the bookshelf in the corner shows the clearest difference. You would never be able to see images like this by standing in one place, but it is still a valid artistic representation of my bedroom.

So, what about artists who aren’t realists? What about abstract, wavering paintings that disregard dimension? Take Jackson Pollock’s Birth or Picasso’s Weeping Woman. Is there any way that these painters could benefit from geometric discoveries?

These artists have to have a sense of space on the page the same way musical composers do. The painter must know their way around 2D space. They have to leave enough room for the eyes of an ambiguous monster even if they are not halfway (or phi-way) up the face, and while the colour and texture of individual brushstrokes might be out of their control, the shapes and forms they represent are mathematical objects in 2D space (or 3D space for sculptors), much in the same way that a composer cannot control the texture of their instruments but can arrange notes to perfection on the stave.

Art is distinct from mathematics and the abstract concepts it often symbolises – whether love, conflict or sorrow – are hard to describe with numerical formulas. But, the media through which visual artists communicate requires mathematical knowledge, and can benefit particularly from geometry.

Is there art which is wholly mathematical? I will try to assess whether mathematics on its own can be considered art in the next article – see you there!

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