# Entwined Mathematics

## The Work of Edmund Harriss

There was a point where I recognized myself as a Christian,” explains mathematician and artist Edmund Harriss. “It was in realizing the importance of Jesus’ teaching on forgiveness. I was in a position where I had to offer forgiveness. I realized that not only did I *have* to offer forgiveness, but I *could* offer forgiveness. It wasn’t an easy or light decision; it was a tough one, but a necessary one. And it’s clear that love and forgiveness are at the heart of Jesus’ teaching, his Gospel.” This idea that exposure to the core values of a system can lead people to embrace the system has permeated Harriss’s work as both a mathematician and an artist.

Edmund Harriss was born in Worcestershire, England, near a National Reserve whose official name is Malvern Hills Area of Outstanding Natural Beauty. Since the age of twelve, he has been on a quest to discover other “areas of outstanding natural beauty.”

The beauty that Harriss discovers takes a slightly different form than the beauty of the Malvern Hills though. Harriss has a passion to “communicate the beauty and utility of mathematical thinking.”

Laypersons tend to think of mathematical thought as utilitarian. But beautiful? Where is the beauty in a page of numbers, equations, and symbolic notation? If there is any beauty found there, is it only accessible to formally trained experts in the field? What hope do the rest of us have in seeing that beauty?

Furthermore, for those of us who are Christians looking for the beauty in our created world, can an understanding of the beauty of mathematics inform our faith?

“They are entirely entwined,” says Harriss.

## Patterns of the Universe

“When I was twelve, I had a mathematics teacher who gave just a little bit of a task and then provided enough space to start exploring the topic on your own.” Through this experience Harriss realized that mathematics was a system where you could create spaces where it was possible to understand almost anything.

“You create a space with some rules. You look at all the possibilities within that space, and you will be surprised with the richness of what you discover.” The “space” Harriss’s work encompasses includes tilings, the geometry of crystals, robotics, and . . . coloring books.

Whenever I wanted to learn a new area of mathematics, I thought to myself, “What pictures can I make of this?” Pictures are a wonderful way of self-checking as a mathematician. If it works and it is beautiful, you can show it off to people. If it doesn’t work, nobody needs to know about it.

When collaborator and co-author Alex Bellos approached Harriss with a proposal to create a coloring book based on mathematics, Edmund had ten years worth of mathematical drawings to fall back on. Good thing too—the publisher wanted to cash in on the trendy “adult coloring book” market and gave the authors a turnaround time of only a few weeks to create a final draft. It was worth it though: *Patterns of the Universe* has been released for over a year and it is still in the top ten of multiple best-seller lists on Amazon. The highly-anticipated sequel, *Visions of the Universe*, was released in late November 2016.

“This is my Trojan Horse approach into helping people appreciate mathematics,” laughs Harriss. “The highest compliment I received is that it doesn’t look like a math book; it looks like a coloring book.”

In the same way that Edmund’s teacher provided him space to investigate mathematical topics, he hopes that this book opens some of that same space for coloring enthusiasts. “Between the two coloring books, you’ll see a really good overview of modern mathematics. If you were to study the mathematics, go through the book and work out what is happening in each picture and find out the meaning behind the images in the two books . . . well, you would probably have a better knowledge of modern mathematics than many math professors!”

## Curvahedra

Not one to be hindered by two dimensions, Harriss’s most recent project provides space to build mathematical understanding in three dimensions. Curvahedra are flat laser-cut shapes that link together to form various three-dimensional objects. Even the creator of the figures was surprised at their diversity of uses. “Each figure can do a limited number of things locally—but the global behavior is very, very rich. There’s still a lot to be discovered about the behavior of the Curvahedra when different shapes are linked together.”

This discovery about his own Curvahedra emphasizes one of Harriss’s favorite aspects of mathematical thought: the unexpected discoveries that arise from exploration. “The best examples of finding awe and wonder in the mathematical world come when the mathematics ends up revealing something about the world we live in that maybe we wouldn’t have been able to notice without the mathematics.”

He describes the discovery of non-Euclidean geometry—originally discovered when mathematicians set out to prove that the only possible geometry in the universe was Euclidean in nature. Simply by investigating the structures in the universe we learn more about the amazing amount of wonder that exists in creation, and we learn more about how the world works. And discoveries like these are often vital to the development of our understanding of the universe: General Relativity wouldn’t be possible without non-Euclidean space.

## The Spiral That Bears His Name

One such unexpected discovery that hits close to home for Harriss is the Harriss Spiral. Well, it was called the Plastic Spiral, until Bellos named it after Harriss in an article for *The Guardian*. So, what was the exploration that led Harriss to the discovery of the new class of spiral that bears his name?

“The Golden Ratio is a bit of a celebrity. And, like most celebrities, it gets credit for things that it hasn’t actually accomplished. The actual Golden Ratio is approximately 1.618, and it seems like anything that hovers around a ratio of 1.5 or so gets tagged as ‘The Golden Ratio.’ What’s more true is that *logarithmic spirals*—which can be obtained from *any* ratio—provide interesting and appealing designs.” The Golden Spiral is the most well-known of these beautiful spirals. But is it any more beautiful than a spiral with a ratio of 2?

Harriss believes that the Golden Ratio gains its celebrity status by being equal parts accessible and exotic. It’s less mysterious to focus on a spiral built on a ratio of 2, so that tends to get swept aside. To build his spiral, Harriss used a different irrational number called the Plastic Ratio (closer to 1.3 than the Golden Ratio’s 1.6) and demonstrated the aesthetic quality found when a spiral is built along that ratio.

“Take the nautilus shell for example. It is commonly attributed to the Golden Ratio. More accurately, the shells build along a logarithmic spiral. Depending on the individual shell, the species, the conditions of its environment, etc., it tends to hover around 1.3 to 1.4.” Instead of focusing on a single ratio, as many laypeople and mathematicians are fond to do (guilty!), Harriss’s work encourages us to look at a broader field of view.

There are still similarities in the way the spirals are built, and it’s no less thrilling given the fact that they use differing ratios. But there is learning to be done in the realm of spiraling and how different algebraic numbers provide different types of appealing spirals.

## Entirely Entwined

Harriss considers himself a math evangelist. The space he provides for non-adherents to have an entry point into mathematics is informed by a conversation he had with a family member involved in Christian ministry.

My family member was working at a home for neglected youth. Providing meals, shelter, and other essential services. This shelter was not actively trying to convert these youths to Christianity. Rather, in the service of Christ, the workers at this shelter were trying to live out what it meant to be a believer. Any youth that converted did so simply because they were won over by the love, enthusiasm, and selflessness of workers there.

Harriss sees his role as a mathematical evangelist in a similar light. Similar to how he embraced the Christian faith by exposure to two of its core values—love and forgiveness—so too Harriss believes individuals can be won over to mathematics through exposure to its inherent beauty, harmony, and intricacy.

I want to give people the tools to understand mathematics and to see the beauty of it for themselves. I want to allow people into that pathway of thinking. I am simply trying to help them see the beauty and wonder of mathematics to enrich their lives and give them more space to discover for themselves.

We live in an age where Christians aren’t known for their scientific curiosity. Much of American Christendom is in fact characterized by the certainty of its beliefs instead of curiosity and wonder about what new things we might be able to learn. In the area of scientific study, Christians are much more likely to be known for spending $100 million on a model of Noah’s Ark than for attempting to examine new and unknown areas in math and science.

What if we, as Christians, took advantage of some of the spaces provided by Edmund Harriss and his colleagues and started to examine some of the possibilities within those spaces? What if we fell in love with learning about God’s creation by focusing on the beauty of mathematical thought?

We can start with a coloring book.

*Cover image by Clay Banks.*

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