# On the cover: Harriss spiral

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.

Left: The large rectangle is golden. If a square (blue) is removed, then the remaining rectangle (green) is also golden. Right: A golden spiral. Image: Chalkdust.

Numbers that are a solution of a polynomial equation with integer coefficients are called algebraic numbers: the golden ratio is algebraic as it is a solution of $x^2=x+1$. At this point, it’s natural to wonder whether you can create interesting spirals like this with other algebraic numbers. Unsurprisingly (as otherwise, we wouldn’t be writing this article), there are other numbers that lead to pretty pictures.

The plastic ratio, $\rho=1.3247179…$, is the real solution of the equation $x\hspace{.5pt}^3=x+1$. Its exact value is
$$\rho=\sqrt[3]{\frac{9+\sqrt{69}}{18}}+\sqrt[3]{\frac{9-\sqrt{69}}{18}}.$$

A plastic rectangle—a rectangle with side lengths in the plastic ratio—can be split into a square and two plastic rectangles. If this splitting is repeated on the smaller plastic rectangles and two arcs are drawn in each square, a spiral is formed. These particular arcs are chosen so that they line up with the corresponding arcs drawn in the smaller rectangles.

Left: The large rectangle is plastic and can be split into a square (blue) and two plastic rectangles (red) and (green). Centre: The two arcs drawn in each square. Right: A Harriss spiral.

This spiral is called the Harriss spiral, and is named after its creator Edmund Harriss. It is the shape that appears on the cover of this issue of Chalkdust, and we think its resemblance to a tree in bloom makes it perfect for spring-time. We also believe that its beauty shows that the golden ratio is a gateway into a world of mathematical creativity, not an end point. There must be other nice algebraic spirals out there, buried in the roots of polynomials. If you unearth a prize-winning specimen, let us know. You may even see it on the cover of a future issue!

# On the cover: Hydrogen orbitals

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.

In school, you’re taught that electrons orbit the nucleus of an atom like a planet orbiting a star. This is mostly wrong. The main problem is that electrons, protons and neutrons aren’t little billiard balls, they exist as ‘clouds’ of probability.

To understand what a hydrogen atom really looks like, imagine a cloud of something whizzing around a single proton. The proton’s positive charge attracts and traps the negatively-charged something in what we call the proton’s potential well. Imagine that cloud is denser in some places and sparser in others. That cloud of something can be just one electron whose position has been smeared out. The density of the cloud at a point represents the probability of finding the electron at that point in space. The electron’s position may be smeared out over all space, but it has different odds of being found at different points in space. In fact, it’s usually exponentially less likely to be found outside the small, confined volume of the potential well.

The mathematical explanation for this is that our system is obeying the Schrödinger equation. For our case, it looks like this:

\begin{equation*}
\Big(\dfrac{-\hbar^{\hspace{0.3mm}2}}{2m}\nabla^2 + V(\mathbf{r})\Big)\psi(\mathbf{r}) = E\psi(\mathbf{r}).
\end{equation*}

The Schrödinger equation is the foundational equation of quantum mechanics. It’s used to determine the wavefunction, $\psi(\mathbf{r})$, and the energy, $E$, of the components of the system. In this case the wavefunction represents the electron (with mass $m$) trapped in the electric potential well of the proton, which is represented by $V(\mathbf{r})$. The reduced Planck constant, $\hbar$, (often called “h-bar”) is a fundamental physical constant, and $\nabla^2$ is the Laplacian operator, which sums second derivatives over all the coordinates. The modulus squared of the wavefunction, $\vert\psi(\mathbf{r})\vert^2$, tells you what the density of that probability cloud is like: where are you more likely to find the electron?

Most people who do quantum mechanics for a living spend their time solving this equation and its variants, myself included. The problem is that this is really, really hard. The Schrödinger equation for a hydrogen atom has analytic solutions you can write down, but with almost all other physical systems, you aren’t so lucky. Once you have more than one electron, the complexity skyrockets. Understanding the analytic solutions form an important part of a physics undergraduate’s introduction to quantum mechanics, especially in my field of research. I work on finding approximate solutions to the Schrödinger equation for more complex systems.

To solve the Schrödinger equation, you can separate the wavefunction to get a radial part which is a function of the distance from the nucleus, $r$, and an angular part which is a function of the angles $(\theta, \phi)$. Both parts have multiple solutions, and it turns out that you need three labels to identify these solutions. We call these labels quantum numbers. Here, the three are called $n$, $l$, and $m$. Putting these two concepts together, we can say:

\begin{equation*}
E\psi_{nlm}(\mathbf{r}) = R_n(r)Y_{lm}(\theta, \phi).
\end{equation*}

Plots of the solution with $n=0$, $l=5$ and $m=0$ to $m=5$

There’s lots of constraints on the allowed values of $n$, $l$, and $m$, but the most important one is that each number take whole number values only. This is where the ‘quantum’ in quantum mechanics comes from!

The quantum numbers each have physical interpretations: they loosely correspond to the three spatial coordinates. Here, $n$ corresponds to energy. Higher values of $n$ mean the electron has a larger amount of energy, which, due to how electric fields work, also exactly corresponds to a larger distance from the nucleus. That means $n$ is associated with the radial coordinate: the higher $n$ is, the further from the nucleus the electron can be.

Meanwhile, $l$ and $m$ correspond to angular momentum, and so they are associated with the angular coordinates. Roughly speaking, higher values of $l$ correspond to the electron ‘orbiting’ around the nucleus with greater energy (in a weird, quantum mechanical way that doesn’t really look like a planet orbiting a star). Changing $m$ means changing exactly how it orbits for a given value of $l$.

What this all means in practice is that by varying the three quantum numbers you get a huge variety of electron distributions. For instance, $n=1$, $l=0$, $m=0$ means that the electron isn’t orbiting the nucleus at all, so it’s most likely to be found right on top of the nucleus – opposite charges attract! When $n$, $l$, and $m$ are all large you get things like concentric sets of lobes of varying shapes and sizes.

Bringing it back to the cover, the pictures were all generated by making a 2D slice through the full 3D distribution at $y=0$. The brighter a given point is shaded, the higher the value of $\vert\psi(\mathbf{r})\vert^2$ is there—the higher the odds of finding the electron there are. The full 3D versions look like spheres, balloons, lobes, and other wild shapes. The 2D slices have a different sort of haunting beauty to them. The distributions can be concentric rings, orange slices, weird lobes, insect-like segments, and more.

The front cover is the 2D slice of the solution for $n=9$, $l=4$, $m=1$. The back cover contains all the allowed solutions from $n=1$, $l=0$, $m=0$ up to $n=9$, $l=7$, $m=7$.

These orbitals are beautiful by themselves as pieces of abstract maths, but they also provide profound insights into the strange quantum nature of our reality. They’re a testament to the amazing power physics and mathematics can have when they work together to help us understand our universe.

# On the cover: Chladni figures of a square drum

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.

# On the cover: Euclidean Egg III

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.

A bilaterally symmetric scorpion. Image: Rosa Pineda, CC BY-SA 3.0

In nature, we often see this ordering principle manifest itself as various kinds of symmetry or repetition.  Most animate creatures exhibit external bilateral symmetry; insects, crustaceans, fish, birds and animals including ourselves all tend to be bilaterally symmetric.

In common with other sentient creatures, we humans navigate and comprehend the world both spatially and temporally through pattern recognition, and being highly social creatures we are particularly attuned to reading expression and meaning in faces and bodies. It is therefore no surprise that bilaterally symmetric shapes seem to have a unique sense of potential meaning and emotional impact for us.

Whilst mirror image symmetry gives structure, the actual pattern being reflected is often far more chaotic. Like a kaleidoscope, the coloured shards are arranged at random; order is created by repetition of these random arrangements. Think of the patterns on moths, butterflies, shield bugs, ladybirds and beetles, there is often very little order in the arrangement of marks on one half, the exquisitely satisfying order of the whole is created by reflection.

Euclidean Egg III, our featured cover art this issue. Image: Peter Randall-Page

In the ‘Euclidean Egg’ series of drawings, as with much of my other work, I have chosen to use a working process which has an inherent element of chance and randomness.

There are two ordering principles in these drawings: one is bilateral symmetry, the other is Euclidean geometry. I constructed a series of geometric egg shapes in such a way as to create a seamless curve where two arcs meet. The result is a faint line drawing of an egg shape together with the construction lines needed in order to create such a taut and smooth curve. These geometric eggs by their very nature have mirror image symmetry around a vertical axis.

Folding the paper along this vertical axis and using paint introduces an element of chance. Using a pipette dropper, I spread ochre paint onto one of the areas between the construction lines on one half of the drawing. Folding the paper in half along the axis of symmetry creates two identical blobs of paint which, whilst roughly contained within the construction lines, inevitably have a somewhat random outline, reminiscent of the inlets and peninsulas of a Scandinavian island. I then add another blob of paint and continue the process, gradually building the drawing; blot by blot, fold by fold.

This process is akin to the psychoanalytic evaluation technique developed by the Swiss psychoanalyst Hermann Rorschach in 1921. Rorschach’s theory was predicted on our psychological sensitivity to bilateral symmetric shapes. He developed a series of 10 mirror image ink blots which are shown to the subject, who is then asked to say what they see in them. Their observations are then used as a way of analysing the subjects subjective response to what are effectively totally random, but highly symmetric, shapes.

Rorschach’s ink blot test has gone in and out of favour as a psychoanalytic tool during the last century but for me, our reaction to his ambiguous symmetric forms reveals something about the way in which our perception of the world is driven by subjective projection of feeling as well as objective analysis and observation. We read meaning into the world as well as taking meaning from what we perceive.

A construction of the simplest Euclidean egg

My fundamental concern in making art is an exploration of what makes us tick, the emotional subtext to our everyday experience.  The world enters our consciousness as emotion and expression as well as information and knowledge. We respond to shapes and colours, forms and spaces, poetry and music in ways which can be difficult to analyse or quantify.
Whilst we have so many ways of communicating with one another (not least language itself), the medium of visual art is uniquely capable of exploring these often intangible emotional responses.

In this particular drawing I am attempting to reconcile order and randomness, Euclid and Rorschach. My attention is concentrated on making a satisfactory balance between the ‘theory’ of pure abstract geometry with the ‘practice’ of what happens in the real world (in this case, the viscosity of the paint as well as the texture and absorbency of the paper are all determining factors).

Being preoccupied with my attempt to reconcile these polarities is strangely liberating. The task involves innumerable decisions and appraisals which is conducive to a spontaneous and playful approach. In fact, play is an important concept for me. Play can be unselfconscious and create fresh associations and ideas. In order to play well, however, one needs a playground. Football without rules and a finite pitch would neither be fun to play nor interesting to watch.

Although rooted in a study of natural phenomena, my work is less concerned with reproducing existing forms than with trying to grasp the underlying dynamics which determine the shapes and forms we see around us and to use these dynamic processes to create new objects which are both novel and familiar.

In the words of the philosopher and art historian Ananda K Coomaraswamy in his 1956 essay The Transformation of Nature in Art, “art is ideal in the mathematical sense like nature, not in appearance but in operation.”

# On the cover: dragon curves

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

Continue reading

# Mathematics: queen of the arts?

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.

The image Central Quadratic explains itself, I hope, as a celebration of analytic geometry.

So artists and mathematicians share the having of ideas’. But what then? Mathematicians communicate their ideas—yes. But ideas in maths take the form of theorems or conjectures about numbers, space or other abstract entities. The quality of these ideas is first assessed by proof. Can the idea be shown to be true? And second, if the idea is true, is it interesting? That is, does it usefully contribute to the mass of existing mathematics? Communication of mathematical ideas may require the invention of new symbols or diagrammatic forms, etc, but these are in the nature of being a new language, not art.

In my view then, art and mathematics share the magical process of idea getting’ but essentially differ in where they go with those ideas. If maths is to be considered an art, it would have to be a sort of super-art’ or art to a higher power’. Easier, I think, to class mathematics as the science of number, space, shape and structure, etc—the abstract entities that exist in our minds.

Imagine an intelligent alien’s perception of our arts and our mathematics. Our art would be more or less incomprehensible, depending on how alien the being was; but our maths would be as true for the alien as it is for us. Furthermore, good mathematics will not diminish with time or go out of fashion.

There is an affinity between some mathematicians and some artists. Certainly, it is a most pernicious error that scientific and artistic talent exclude each other—an idea unfortunately common among school counsellors. The common ground between art and science/maths that leads us to the getting of ideas’ is the activity we call play. The thing of it—the thrilling thing, the magical thing—is the moment when one discovers a new idea, or pattern, or conceptual framework, or whatever: the eureka moment! And are these moments not usually approached through playing in the mind with new combinations and orderings of existing mental constructs?

In ray tracing, each ray is used to decide the colour of a pixel on the image plane.

Spheres was created using ray tracing.

I had one of my most memorable eureka moments sometime in 1971 while sitting on a dead tree in Epping forest. At that time, I had been collaborating on an automatic projective line drawing program with hidden line removal’, going where Autocad later arrived. I was considering algorithmic approaches to colouring surfaces in projective drawings. I realised that if I thought of objects in the scene as being represented mathematically as arrays of vertices and planes in some coordinate space, then I could solve for the equation of the line going from an eyepoint through a particular pixel in the image plane and into the scene (as in the diagram). From the equation of the line, I could find the closest surface along the path and then compute the colour and illumination value for that pixel based on the defined colour on the surface, along with its relationship to any light source or other light-emitting surfaces. And so I had invented ray tracing—the foundation of all computer generated synthetic imaging for special effects in cinema, television and gaming. Of course, I neither invented it first nor alone—and I certainly had neither the persistence nor vision to pursue ray tracing to practical or rewarding development. But its discovery was a thrill, as were the few simple pictures I made using the technique in a primitive manner on the pen plotter available.

As spheres have become my most persistent motif, I will end with two more related works that play on the division and articulation of spherical surfaces: Sphere Architecture and Star Sphere.

[ Pictures: Central Quadratic: Used with permission from UCL Art Museum, University College London; Spheres, Sphere Architecture and Star Sphere: Used with permission from John Crabtree ]

# Spherical Dendrite by Mark J Stock

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

# Fermat Point by Suman Vaze

Fermat Point by Suman Vaze

Suman Vaze sits on her small balcony in crowded, bustling Hong Kong, with a view, just about, of a beautiful Chinese Banyan tree tenaciously growing on a steep stony slope, and paints mathematics. Inspired by the abstract expressionism of Rothko, the radical and influential work of Picasso, and the experimental models of Calder, she fully embodies Hardy’s belief that mathematicians are ‘maker[s] of patterns’. Our front cover is one of her pieces: the bold colours proclaim the eponymous Fermat Point – the point that minimises the total distance to each vertex of a triangle – along with its geometrical construction. Add an equilateral triangle to each side of the original triangle then draw a line connecting the new vertex of the equilateral triangle to the opposite vertex of the original: the intersection of these lines gives the Fermat point. Not only do these lines all have the same length, but the circumscribed circles of the three equilateral triangles will also intersect at the Fermat point.

# The Symposium of the Muses

This issue’s cover picture is a creation of Anthony Lee, a young British artist, who has always been fascinated by exploring the possibilities of creating images through light. In Anthony’s eyes, this experimental process is the result of  “the idea of an ephemeral substance or state, the idea that the captured moment was never intended to last or be repeated. In my light images neither the light nor the shape can last and yet they stay captured in the image I present.”

It is interesting to notice where both the artistic and scientific processes intersect and interact with each other – and where they do not. The artist, Anthony, is looking for a way to use scientific knowledge to express his personal emotions and inner thrills; and the resultant art is the outcome and purpose that elevates and distinguishes the science. And yet Anthony is bending and filling reality with his own meanings – his “ephemeral” ideas of light and shape – that are changeable and unique to him. Contrast this with the aims of scientists, who look for permanent truths that affect every observer, irrespective of their uniqueness in this space-time continuum.