Exploring Frontiers of Mathematics

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From ancient Greece to quantum computers

Ancient Greek mathematicians knew about the existence of prime numbers — whole numbers greater than 1 that are divisible only by 1 and themselves, such as 3, 19 or 7,753. But while modern mathematicians have calculated staggeringly large primes, we still don’t understand if there is some underlying structure or pattern to how they appear in the number line. It’s a problem that continues to plague the field of number theory — the mathematics of whole numbers — today.

“One thing I love about number theory is that it’s a really old field. The questions are easy to explain but the solutions are really complicated,” said Associate Professor of mathematics Elena Fuchs.

For the last few years, Fuchs’ special interest has been another ancient Greek puzzle, Apollonian circle packing. Around 200 BCE, Apollonius of Perga was drawing circles with a compass and ruler. He showed that if he drew three circles touching each other, there were only two ways to draw a line (or circle) that touched all three circles — a large circle outside, or a small circle in the middle of the other three.

Appollonian circles
An example of an Apollonian circle packing, or gasket, made from circles mutually touching circles. (Via Wikimedia Commons).

Over the centuries, many famous mathematicians have tinkered with these circle packing problems. In 1643, the French philosopher and scientist René Descartes discovered patterns in the size of circles in an Apollonian packing; Gottfried Liebniz also worked on them.

The size of circles is represented by their curvature, which is 1/radius of the circle. Therefore, the bigger a circle is (the higher the radius) the smaller the curvature. In the early 20th century, Frederick Soddy, 1921 Nobel laureate in chemistry, discovered that if the first three or four circles have curvature that is a whole number (or integer), every circle in the series will have whole-number curvature.

In other words, an infinite series of whole numbers emerges from this particular arrangement of circles.

“As number theorists, we get really excited about this sort of thing,” Fuchs said. “We want to ask questions such as, ‘are there prime numbers in the series and if so where?’”

The early 2000s saw a fresh explosion of interest in Apollonian circle packings, including Fuchs’ own work, which has connected the problem to other mathematical objects through something called Thin Group Symmetry. And this connection might lead to a whole new kind of computer — the quantum computer.

Digital devices handle data as bits that can be either one or zero. Quantum computers use qubits that can be one and zero simultaneously, thanks to the rules of quantum mechanics. Such computers should be extremely fast at addressing some types of problems, including calculating prime numbers and cracking encrypted data.

Thin Group Symmetry might be a candidate for the building blocks of quantum computers, Fuchs said.

“This is exactly what I love about number theory — there are all sorts of connections to other fields and you can go from really simple to really deep very quickly,” she said.

To explain the role of mathematics, Fuchs cites Andrew Wiles, the mathematician who proved Fermat’s Last Theorem after almost 350 years: 

“You’re in a dark room with a bunch of furniture and you’re kind of feeling around with your hands trying to figure out what’s there, and suddenly you find a light switch and you see everything as it has been the whole time. I think that’s a really accurate description.”