## The Colour Purple

Imagine you had to explain the concept of purple to a being from another planet, one who experiences life using an entirely different set of senses and in wildly different ways. How could you do it?

One way would be to use hydrogen, which makes up 75 per cent of all the elements in the universe. Hydrogen atoms consist of a central proton, *p*^{+} and a single electron, *e*^{–}, that can occupy one of many different orbits. These orbits are numbered; n=1 is the first orbit, n=2 the second orbit and so on. The electron can move from one orbit to another.

When the electron falls from a higher to a lower orbit, it emits a package of energy called a photon. The amount of energy–the colour of the photon–depends on the size of the jump. Jumps of many different sizes are possible but in hydrogen, only four produce visible light. These jumps correspond to specific shades of the colours red, cyan, blue and purple.

When astronomers analyse light from the universe, they see each of these colours as single spectral lines. H-delta is the name of the fourth line – our view of the electron falling from hydrogen’s sixth orbit to its second orbit. The Rydberg formula calculates the wavelength, λ, of the light produced in this way for any two orbits or values of *n*. *R _{H}* is a number called the Rydberg constant. For the H-delta line,

*n*

_{1}is 2 and

*n*

_{2}is 6 so the wavelength is 410.1734 nanometres. To our eyes, pure purple.

Because hydrogen is so common, these four colours fill the cosmos. The universe is bathed in purple. That’s why nebulas have spectacular pinky-purplish colouring. Cosmologists believe the universe must look more or less the same, wherever you happen to be in it. So our alien friend ought to be as familiar with the H-delta line as we are, even though they might experience it in an entirely different way.

*For Jacky, June 2021** Digital print on Hahnemuhle smooth fine art paper, **42 x 30 cm*

## Incantation of truth No 1

If I write down a statement, how do you know whether it is true or false?

*May 2021** Digital print on Hahnemuhle smooth fine art paper** 30 x 42 cm*

## Zeno’s forgotten dream

How do we know what is true and what is false? Zeno wrestled with this infinite series, which would take longer than the age of the universe to write out in full. But its sum is straightforward to see…if you believe your eyes.

*May 2021** Digital print on Hahnemuhle smooth fine art paper** 30 x 42 cm*

## What’s the time, Mr Wolf?

If it’s 9 o’clock now, what time will it be in 6 hours? Even mathematical truth can mean one thing for one person and another for somebody else. Truth can unite but it can also divide.

*February 2020** Digital print on Hahnemuhle smooth fine art paper** 30 x 42 cm** Edition: 30*

## Teacher in profile

Imagine a box of numbers, the type that can be affixed to front doors. If I pick out a number three and then pick out a number one, how many numbers have I picked out?

*February 2020** Digital print on Hahnemuhle smooth fine art paper** 30 x 42 cm** Edition: 30*

## Half plus a quarter

Two plus four equals six, right?

*February 2020** Digital print on Hahnemuhle smooth fine art paper** 30 x 42 cm** Edition: 10*

## Decimal places

*February 2020** Digital print on Hahnemuhle smooth fine art paper** 30 x 42 cm** Edition: 10*

## Veteran

A sign carried by a man sitting with a cup in Times Square

*2005** Digital print on Hahnemuhle smooth fine art paper** 30 x 42 cm** Edition: 500*

## Young boy with paper

There are limitless proofs of the Pythagorean theorem but this is the one most often taught at school. It has probably inspired, bemused and terrorised millions.

*March 2019**Digital print on Photorag stock paper**60 x 42 cm*

## An evening in Hannover

I once spent an evening in Hannover with a mathematically-minded friend laughing uproariously at this joke while discussing the mathematics behind it.

*March 2019**Digital print on Photorag stock paper**60 x 42 cm*

## The burden of life

A formula for compound interest, a phenomenon that condemns millions to lifetimes of debt and misery.

*March 2019**Digital print on Photorag stock paper**60 x 42 cm*

## Family life

These equations show an entangled state consisting of five photons. These photons are so deeply linked that they form a single quantum entity and remain connected despite being in different parts of the gallery.

*March 2019**Digital print on Photorag stock paper**5 x 42 x 30 cm*

## Working man with a grin

As a student, I spent months studying contour integrals, fascinated by their power and elegance. But in all the years since, the only time I have ever needed this mathematical knowledge was when somebody told me this joke, which seemed all the funnier for the work I had put into understanding it in advance!

*March 2019**Digital print on Photorag stock paper**60 x 42 cm*

## Middle-aged man with paper

Pythagoras’ theorem is one of the great relations in mathematics. There are an infinite number of proofs to choose from, some 367 of them published in *The Pythagorean Proposition* by Elisha Loomis. This one–a proof using vector geometry– appears in slightly different form at the end of Loomis’ book.

*March 2019**Digital print on Photorag stock paper**60 x 42 cm*

## THE SHOCK OF SUDDEN DEATH

Entanglement can sometimes end in an unexpected way called sudden death. These matrices show the difference between entangled states that fade away and those that end in sudden death.

*March 2019**Digital print on Photorag stock paper**60 x 42 cm*

## Carving on a quantum oak tree

The connections between ordinary objects are fleeting and superficial. Two atoms may collide and separate, never to meet again. Others can stick together by virtue of the chemical bonds they form, until the day that bond is broken.

But there is another type of connection that is far more powerful and romantic. Certain objects can become linked by a mysterious process called entanglement. Particles that become entangled are deeply connected regardless of the distance between them. If they become separated by the width of the Universe, the bond between them will remain intact. These particles J and S are so deeply linked that it’s as if they somehow share the same existence.

*March 2019**Digital print on Photorag stock paper**60 x 42 cm*

## FALLACY?

Mathematics evokes many emotions but the most powerful often come from errors. This derivation contains an error that is easy to make but difficult to spot generating puzzlement, frustration, amusement and more.

*2007**Digital print on Photorag stock paper**60 x 42 cm*

## EULER’S RELATION

One of the great wonders of mathematical world, at once surprising, insightful, awe-inspiring and profound.

*1998**Digital print on Hahnemule fine art paper**60 x 42 cm**Edition: 500*

## Gabriel and Sophia in sunlight

When electron and a positron meet, they can interact in a way that creates two particles of light.

*March 2019**Digital print on Photorag stock paper**60 x 42 cm*

## BACK TO THE FUTURE

There are no laws of physics that prevent time travel. In the quantum world, the probabilistic nature of these laws means that particles can interact with their former selves. Physicists have created quantum time travel machines that perform this experiment and this diagram shows the quantum time machine’s circuit diagram.

*March 2019**Digital print on Photorag stock paper**30 x 42 cm*

## Benedict at GOSH 2016

One of the surprising discoveries about entanglement is that it can whither and die in an unexpected way that physicists call sudden death. Just how this happens isn’t clear but the quantum environment seems to play a crucial role. Even more surprising is that after sudden death, entanglement can sometimes be restored in carefully controlled circumstances. This equation sets out the circumstances in which two particles can be entangled and the plot shows how the connection between them, , can die and then come back to life.

*March 2019**Digital print on Photorag stock paper**60 x 42 cm*

###### About the art of Justin Mullins

For many people, their main experience of mathematics is sheer blind terror. Show them an equation and cold beads of sweat appear on their foreheads as they succumb to the icy grip of fear. For others, the experience is quite different. Some are bemused or irritated; others feel a surge of curiosity and a powerful sense of achievement when the hieroglyphics have been conquered.

Then there is the sense of beauty, elegance and power that mathematics conveys. Many mathematicians have remarked on this exquisiteness as well as on its inevitable counterpoint: a tortured ugliness that is sometimes almost suffocating.

All this points to an extraordinary but rarely remarked upon role for mathematics: as a vehicle for social, emotional and cultural exchange. That’s where my interest lies.

I am not interested in mathematics as it is often portrayed: as a silver thread of logic that leads from hypothesis to proof. This is a kind of ivory castle of mathematics, a perfect but ultimately unreachable world.

For me, mathematics is a human activity—at times it is awe inspiring and mind blowing but it is also infuriating, puzzling, unsatisfactory and often wrong (at least when I think about it). Reuben Hersh describes it refreshingly in his book *What is mathematics, really?* “Mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context.”

My work attempts to capture this element of mathematics. I often use it to remember people, things and important moments. I try to use it to capture vignettes of ordinary moments, to create portraits of people I know or snapshots of mathematical landscapes that have inspired or terrified me. I too know the icy grip of fear! For me, it is an emotional experience, indeed it is a roller coaster ride.

Let me say upfront that I am not a mathematician. I lay no claim to the equations I have selected in my work. Those are the discoveries of the philosophers and scientists who spend their lives exploring the mathematical world and revealing its great wonders. For me they are like the great explorers returning from distant shores with tales of fantastic lands and magical creatures.

If mathematicians are explorers, then my role is that of a photographer who retraces their steps. During my journey, I photograph what I find. By that I mean I frame it, record it and later present it.

There is nothing particularly special about this process. In the same way that an ordinary photograph is a snapshot of an area of outstanding natural beauty, a mathematical photograph is a snapshot of mathematical beauty.

**Justin Mullins** is an artist and writer. He has been producing and exhibiting his artwork in the UK and US since 1998. His art has been covered by New Scientist, The Guardian and various radio and TV shows.

Email: justin@justinmullins.com