# Beauty in numbers

23:37, Dec 28 2013

Renowned as an artist, Puniho's Dale Copeland has just published a new and quite unexpected book: Complex Numbers in Graphs. She gives maths-baffled reporter Matt Rilkoff a simplified version of what it's all about.

Q Complex Numbers in Graphs? You're better known as an artist. How did this book come about?

A It's a book that came from my interest in Chaos theory. I've been playing with complex numbers and what happens when you include them in graphs since the early 80s, when it would take about 10 hours for an early desktop computer to plod through a program I'd written. I'd leave it going all night, and then have to photograph the screen as the old dot- matrix printers couldn't handle printing pictures.

Q What are complex numbers anyway?

A It's a pity about the name. They're no more (or less) complex or imaginary than other numbers. Complex numbers are all numbers. What we call the real number line is just a tiny slice of the complex numbers. If you ask your calculator to find the square root of -1, an old calculator will just give an error message. A fancy new one might give as the answer the letter i. Complex numbers have got two parts, a real bit and some of that i. So when you plot graphs like y=x2 there are two parts to the x values and two parts for the y values so you really need a four- dimensional graph. Difficult to draw on a flat piece of paper.

Q How did you start working with these real-looking shapes that come from numbers that aren't really real?

A I used to work out these shapes in my head while driving (it probably didn't help with the driving, but was fun), until computer software let me show them so much more quickly. And when I got hold of some software which allows you to set equations into a page of text, I started putting together this book, instead of just scribbling in notebooks.

Q Who is this book aimed at?

A The book is pitched at the level of undergraduates, or enthusiastic Year 13 students. Anyone for whom complex numbers are something more than just another maths topic, just another set of tricks to learn for passing exams. It's such a rich and beautiful topic. It really is.

Q How can numbers be beautiful. They are just numbers, right?

A Ah, and a Rembrandt is just paint on canvas. And all of Shakespeare can be found in a dictionary. Maths is what happens when you use numbers; some of it is tedious like accounting, and some of it is absolutely beautiful, like calculus.

Q You were talking about Chaos before. Can you expand on what that is?

A It's lovely. From a simple formula, applied over and over, hundreds of times, you can get these amazing patterns. The 'chaos' description comes from the immensely convoluted areas, where the tiniest of differences in starting point leads to a huge difference in outcome. Hence the saying about a butterfly flapping in Asia leading to a tornado in Kansas.

Computers are necessary to work through the thousands of iterative computations to come up with images which are now so well known. The Mandelbrot set and the associated Julia sets. For those who might not have come across these, ask Google. Look at the images rather than the formulae. Lovely things.

Earlier this year I was immensely excited, and still am, to discover some other formulae which also give shapes with infinitely convoluted edges. Think of looking at the coastline of New Zealand. First on a map of the whole country, then at high definition maps, then walking it, then looking closely at the pebbles and the rock pools, then taking a microscope to the sand, then looking at the atomic structure. But even further. If you expanded your rock pool to the size of the known universe these formulae would still be giving infinitely convoluted shapes to the edges.

Q When and why did you first develop a passion for numbers?

A High school, I guess. It felt so good, because you knew if it was right. Not like writing an essay where you're not sure how an examiner will react. With maths you know if it's right, because there's a method and a beauty, and it just works.

Q Maths and art don't seem like a natural relationship. Or are they?

A There's a beauty which feels the same, whether it's poetry or maths or art. If it's right, then there's an elegant simplicity.

Q What do you think makes people afraid of, or intimidated by, numbers?

AUsually just a teacher somewhere in the past, someone who didn't explain something in the way that student needed. Someone who said or implied, "you're useless at maths".

Q Do you have to be a logically minded brain-box to understand maths?

A Do you have to know the chemistry and biology of plants to appreciate flowers? Sure, maths can become intimidating, tough-looking. So can architectural plans or a diagram of how the internal combustion engine works. You learn the language of symbols and it makes sense.

Taranaki Daily News