Imagine three mice equally distanced from each other, ie at the vertices of an equilateral triangle. If at the same time, all three mice start chasing their neighbour clockwise, then each of their paths would be a logarithmic curve. But this is rather hard to draw, especially if we want to restrict ourselves to only using a ruler.
Instead, let us imagine that the mice can only run in a straight line and need to stop to reassess their direction. If at a given stage we draw their intended path, and assume that the mice cover a tenth of the distance to the next mouse before stopping and reassessing their direction, we get the picture below. While these pictures have been drawn using straight lines only, we see three logarithmic spirals emerging:
Stages 1, 2, 3, and 20.
But why stop there? Why not start with $4$, $5$ or $n$ mice on the vertices of a regular square, pentagon or $n$-gon? The following instructions show a very algorithmic approach to drawing these patterns:
What can you do with this space? So asks Andrew Stacey. ‘Fill it’ is the prompt reply, but fill it with what? Maybe like Andrew you want to use a single curve, but I want to use circles. If you do this in the way shown above in blue, the result is called an Apollonian packing, a variant of which can be seen on the cover of this issue.
Here we shall explore the history of this entrancing object, which spans over 2000 years, and percolates into a surprising variety of mathematical disciplines. Starting in the familiar world of Euclidean geometry, Apollonian packings extend into fractal geometry and measure theory; Möbius transformations and the hyperbolic plane; and then on into the distant reaches of geometric group theory, number theory, orbital mechanics, and even ship navigation. Continue reading
From the exquisite patterns of the Alhambra palace in Spain to a jigsaw puzzle on a rainy day, tessellations (tilings of the plane using shapes with no overlaps or gaps) are everywhere. They are sometimes used for practical reasons: providing durable and water-resistant surfaces, or for efficiencies of space (like hexagons in a honeycomb). And sometimes they are there for aesthetic reasons: tessellations are known to have been used in architecture since at least 4000BC when the Sumerians decorated walls with patterns of clay tiles. Continue reading
The golden ratio (1.6180339…) has a rather overblown reputation as a mathematical path to aesthetic beauty. It is often claimed that this number is a magic constant hidden in everything from flowers to human faces. In truth, this is an exaggeration, but the number does however have some beautiful properties.
The golden ratio, often written $\phi$, is equal to $(1+\sqrt5)/2$, and is one of the solutions of the equation $x^2=x+1$. The other solution of the equation is $(1-\sqrt5)/2$, or $-1/\phi$. One of the nicest properties of the golden ratio is self-similarity: if a square is removed from a golden rectangle (a rectangle with side lengths in the golden ratio), then the remaining rectangle will also be golden. By repeatedly drawing these squares on the remaining rectangle, we can draw a golden spiral. Continue reading
Quantum mechanics has a reputation.
It’s notorious for being obtuse, difficult, confusing, and unintuitive. That reputation is… entirely deserved. I work on quantum systems full time for my job and I feel like I’ve barely scratched the surface of the mysteries it contains. But one other feature of quantum mechanics that’s often overlooked is how beautiful it can be.
So, for the cover of this issue, I wanted to share one aspect of quantum mechanics that I think is stunning. It’s a certain set of solutions to a differential equation: the orbitals of an electron in a hydrogen atom. Continue reading
The 18th century—the age of enlightenment. Ernst Chladni travelled around Europe demonstrating his ‘musical curiosities’. The star attraction was a novel technique to expose the various modes of vibration of a rigid surface.
Throughout my life I have made an informal study of natural phenomena, through drawing or just looking, in a spirit of curiosity. This long but unsystematic practice has given me an impression of the world around us as a dynamic and fertile system, driven by a ubiquitous tendency for spontaneous pattern formation (best understood in terms of the laws of physics) mitigated by an equally strong tendency for seemingly random variation.
It could be argued that the evolutionary process itself is driven by this tension between pattern and randomness, structure and chaos, order and disorder, theme and variation; without random mutation there would be stasis. Continue reading
Take a long strip of paper. Fold it in half in the same direction a few times. Unfold it and look at the shape the edge of the paper makes. If you folded the paper $n$ times, then the edge will make an order $n$ dragon curve, so called because it faintly resembles a dragon. Each of the curves shown on the cover of issue 05 of Chalkdust, and in the header box above, is an order 10 dragon curve.
Left: Folding a strip of paper in half four times leads to an order four dragon curve (after rounding the corners). Right: A level 10 dragon curve resembling a dragon
In the brief tradition of Chalkdust cover articles there is a developing discussion of how mathematics and art are related.
Art is simply the making of representations. Art happens when a person has an idea or a vision that exists in their imagination (the mind’s eye) and is impelled to communicate said idea by making a visible manifestation (representation) of it in the material world. The idea or vision on its own is not art. Art occurs amid the struggle to make a representation of the idea that the artist can show to other people. Art may be relatively `fine’ or popular, conceptual or objective, highbrow or applied, yet still fall within this definition. Judgements about the quality of art are made largely by consensus among the cognoscenti in a given art milieu. These judgements are subject to change over time as the perception of works of art are always modified by the current `cultural environment’ and fashion. Continue reading
We are surrounded by complex structures and systems that appear to be lawless and disorderly. Mathematicians try to look for patterns in the seemingly chaotic behaviour and build models that are simple, and yet have the capacity to accurately predict the reality around us. But can a scientific or mathematical model have any artistic value? It seems that the answer is yes. There is a group of digital and algorithmic artists that use science and computational mathematics to create visual art. However, there is an even smaller group of people whose art and science coincide. Meet Mark J Stock. Continue reading