Science, maths and computers.
Stare at this painting for a while. Regardless of what you perceive its artistic merit to be (“my little sister could do that…” some of you will no doubt be saying), you have to admit there is some strange aesthetic quality to it. It guides your eye from one focal point to another, giving your brain a plane of seemingly infinite detail to explore and it seems to - in some sense - resonate with your gaze. Artsy bullshit? Or science?
Jackson Pollock was a controversial figure. His work has been described as “mere unorganized explosions of random energy” and even “a joke in bad taste”. However it just so happened that his unconventional painting technique generated what a mathematician would instantly recognise as a fractals. We see fractals a lot in nature and mathematics and their most distinct feature is their display of self-similarity on many scales. Bronchi of a lung, branches of a tree and the Sierpinski triangle are examples of fractals (have a look and you’ll get a good idea).
If we zoom/enhance a portion (say, around a quarter) of the painting above, we get something that looks a lot like the original painting itself!
This is what is meant by self similarity - zoom in and you see the same thing again. We can quantify the ‘fractalness’ of an image by calculating its fractal dimension. Though I won’t go in to the maths here, it will suffice to say that while a 1D line has a dimension of D=1 and a 2D square has one of D=2, fractals can have non integer dimensions such as D=1.71, or anything!
Now, why does the Pollock look pretty? There have been interesting studies on how the human eye assess an image. In one, subjects were asked to look at pictures while their eye motion was tracked. It was found that the eye trajectories themselves were fractals! This makes sense. A 1-dimensional eye motion (below, left) does not give us enough information about a scene, however a fully 2-dimensional gaze (below, right: where the eye attempts to perform a full scan of every point in front of it) is way too much for the visual cortex to process at any instance. Instead, the eye has come up with a much more efficient method of looking - doing an intensive 2D scan of a small area before making a 1D leap to another area. The overall result (below, middle) is a kind of 1.5-dimensional fractal scan; it allows isolated ‘islands’ of detail to be put together by the brain to make a complete image!
It turns out that there is also a ‘resonance’ effect that occurs when viewing an image with a fractal dimension similar to that of the eye. Subjects were asked to view a series of computer generated fractal images and rate their aesthetic quality. It was found that images with a dimension of around 1.5 (same as the eye) had by far the most positive response. Furthermore, fMRI scans indicated that viewing such images activated regions of the brain associated with emotions such as happiness.
It is quite easy to analyse a Pollock painting and calculate its fractal dimension. You can kinda tell where I’m going with this… It turns out that his artwork does indeed possess a fractal dimension of around 1.5! Haters can hate, but science proves that Pollock is good. Who ever said that art was subjective?
Though it may be stretching it a little bit too far to attribute all of Pollock’s success to a mathematical quirk, there’s one thing for sure: Your little sister can’t do it.
Above is a resin cast of a lung. Notice its fractal nature - how it displays self similarity at smaller and smaller scales. You might also notice how this structure is quite ‘tree-like’. Why are they so similar?
Well actually, both lungs and trees want to maximise the surface area of their functional components while constrained to some maximum volume. For lungs this strict constraint is the size of the thorax, but for trees is more relaxed and is to do with the mass they can achieve through photosynthesis and mineral uptake and density of trees around them.
Interestingly, nature has solved both these mathematical problems of optimisation using the mathematical solution of fractals. This is a great example of complexity and universality. Complex structures such as trees and lungs emerge from very simple mathematical rules, laws and constraints. The result is some kind of universality to the structures that we humans see and assume to be very different, though they are fundamentally the same.
Check out this animation I made of a simple fractal construct being transformed into a ‘tree-like’ (or ‘lung-like’!) structure.
This is a question that at first sounds a bit stupid, but the observation that the night sky is dark is in fact a deeply profound one that provides much of the basis for modern cosmology.
The question which has now come to be known as Olbers’ paradox goes something like this: “In an infinite and static universe with an infinite amount of stars, why is the night sky dark?”
The argument was that if you looked at any point in the sky and drew your line of sight, it would eventually reach a star. In other words, along every possible direction, there should be a star, and hence light should be coming from every point in the sky.
No, really, why?
This is a bit of a wishy-washy argument when posed in terms of words, so let’s try some maths:
Imagine that throughout the universe, the density of stars (number per cubic lightyear, say), let’s call it n, remains roughly constant. Now, imagine that we construct a series of spherical shells surrounding the Earth, and that each has a thickness dr. See the main picture to see what I mean.
What we want to do is count up the number of stars, N, in a shell. For a shell a distance r away, we multiply its volume by the star density:
Now let’s work out how bright that shell is. We can assume that each star has a total luminosity of L, but we have to take into account the fact that the further away a star is the fainter it appears. In fact, the apparent brightness, F, of any star varies like:
The brightness of a thin shell - which we’ll call dJ- is just the number of stars times the brightness of each!
Now we integrate over all space, i.e., add up the contribution from every consecutive shell all the way to infinity.
In other words, the total brightness of the sky, J, is infinite!
Okay but WHY?
The essential reason for this is the fact we said that the brightness of a star decreased by an inverse square law, but the number of stars increased by a regular square law. The two r^2 terms cancelled each other out and we found that each shell had the same brightness! Therefore when you add up an infinite number of same-brightness shells the answer you get is ∞.
Well, this is obviously not true when we look up at the sky, so there must be a problem somwhere. Like most things in science, the problem lies within our initial assumptions, namely: ‘the universe is static and infinite’. We have shown that this just can’t be true! The night sky being dark forces the universe to have a finite size and age!
Edgar Allen Poe was eerily accurate when he postulated that no light reaches Earth from beyond a certain distance - corresponding to the age of the oldest stars. Cosmology caught on to this idea and introduced the concepts of the big bang, universal expansion, and the cosmic horizon in order to account for this seemingly trivial darkness problem.
Think of this next time you look at a starry sky. We see faint objects as they were hundreds, thousands, millions and billions of years ago (the time it has taken light from them to reach our eyes). At the farthest depths of what our most powerful telescopes can make out are objects from the beginning of the universe itself, and beyond that… nothing.
We can see the edge. It’s black.
This is because an icosahedron is the optimum way of assembling a closed shell out of identical sub-units. A virus is constructed in just this way - identical triangular units called capsomers arrange themselves to form a protective closed shell around the virus’ genetic material. And they do this in the most mathematically optimum way, as if they’ve studied calculus.
Yay for maths!
Lorenz attractor by pretendy
The Navier-Stokes equation fully describes the flow of a fluid. It is essentially an application of Newton’s second law - that a force acting on a body is equal to its rate of change of momentum. Obviously, a fluid is much harder to model than a single solid body with a given mass and velocity. Instead, the liquid or gas is given a density ρ and a velocity vector field, v(x, y, z, t), which describes the speed and direction of flow at the point (x, y, z) and at a time t.
The left-hand side of the equation then is essentially the ‘mass times acceleration’ part of Newton’s second law, and the right-hand side is the total force - in this case, the sum of a pressure gradient, a stress divergence and an external force.
This equation will fully describe the mixing of milk and tea as well as the dynamics of hurricanes.
Unfortunately though, in almost all circumstances it is impossible to solve or even compute to a high level of accuracy! In fact, one of the Clay Institute’s $1m prizes awaits anyone who can gain a great deal of insight into Navier-Stokes existence and behaviour.
However, the amazing thing about this equation is that it tells us (in theory) everything we need to know about the flow of a fluid. The complex behaviour of a rushing river or a swirling vortex is all locked up in these five simple(ish) terms.
Think about that the next time you stir your tea.
Mathematical Mountains is a collection of generative art pieces by Steve Brunton. These are bifurcation diagrams, which map the stable and chaotic dynamics of a system. Bifurcation essentially means changing a parameter of a system until one stable point forks (or bifurcates) into two. These forks get smaller and smaller and denser and denser until the system has no stable points at all an exhibits chaotic dynamics.