Researcher Highlights
The Life of Dr π

With a reputation as the world's most renowned π expert, University of Newcastle Laureate Professor Jon Borwein has spent more than a quarter of a century pioneering the field of experimental mathematics.

Now he is helping industries like medicine and music unearth further value from his mathematical breakthroughs.

Founder and head of the University of Newcastle's Priority Research Centre for Computer-Assisted Research Mathematics and its Applications (CARMA), Prof Borwein has used the computer as a laboratory for more than a quarter of a century, with just some of his feats including producing the world's largest mathematical picture (of π) and proving a 60+ year-old conjecture incorrect.

When asked what brought him to Newcastle, Prof Borwein jokes that "CARMA brought me here", adding that, "the University of Newcastle is a world-class mathematical hotspot. If you want an interesting life, then you need to follow the interesting work".

The life of Doctor π - a title he has gained for his fascination and breakthroughs in understanding the irrational number calculated by dividing a circle's circumference by its diameter - has certainly been an interesting one.

A finger in every π

A Rhodes Scholar and Oxford Doctor of Philosophy, Prof Borwein has held professorships as a pure mathematician, an applied mathematician, an operations researcher and a computer scientist.

Working with a new method of visualising large strings of numbers through random walks – a path described by a sequence of randomly generated points in the plane – Prof Borwein and his colleagues produce papers and digital images using computers to show information at a glance that would take many pages to set out in written numerical form.

Some of the algorithms visual resemblance to genomes and other structures seen in nature are not lost on Prof Borwein and his team. They are working with a number of other leading researchers and industry to see the potential applications and learnings from CARMA's breakthroughs, in fields as diverse as genetics, cognitive science, the fine arts, operations research and theoretical computer science.

For example, Prof Borwein is collaborating with radiology at the University of Newcastle to explore mathematical equations in spectral analysis in functional MRI. On another project, he is working with the University of Newcastle's Conservatorium to showcase the mathematical patterns of what constitutes beautiful music.

In celebration of π Day on March 14, 2013, Prof Borwein together with a team of researchers produced the world's largest mathematical picture, which was of π. The number of digits for π are infinite and seem random, however as computers have become more powerful we now know many more of the digits than ever before.

Borwein and his team had each digit describe the direction and distance being moved and used colour to designate time. The colours followed the RYPGPOB sequence, with red and yellow being used earlier on. The further the colours went along the spectrum, the later it was in the sequence.

"Since we could not see some of the colours in the image, it raised questions we would not have normally asked in a maths laboratory, like 'Where is the yellow?'. This led us to realise that some `normal' numbers, but not π, actually reroute over the same pathway several times," Prof Borwein said.

Maths in the genes

When you look at Borwein's genetics, the probability that he would end up in mathematics seems almost inevitable. His family tree reads like a who's who of contemporary mathematics - with his father, brother and son-in law all renowned mathematicians in their own right.

Prof Borwein collaborated with his brother, Peter Borwein, together with David H. Bailey and Simon Plouffe, in developing the famous Bailey-Borwein-Plouffe (BBP) formula. The BBP formula calculates the value of π and allows the binary digits of π and other constants to be determined without knowing the previous digits. With IBM, they are also calculating digits of π squared beginning at the 10th trillionth place in both base three and base two.

Each year since 1987 he has also completed at least one paper with his now 89-year old father, David Borwein.

As frequent collaborator, David H. Bailey, describes the Borweins in the foreword of one of two books on selected writings by the father-son duo, "The Borweins are inarguably the world's leading exponents of utilising state-of-the-art computer technology to discover and prove new and fundamental mathematical results."

"As a direct result of the Borwein's influence, hundreds of researchers worldwide are now engaged in "experimental" and computationally-assisted mathematics, and the pace of mathematics has measurably quickened," Bailey said.

Writing in the numbers

So prolific is Prof Borwein's output, Microsoft Academic has listed him as number 21 on their list of top mathematics authors of all time. He has also produced over a dozen books, including a monograph on convex functions which was a Choice 2011 Outstanding Academic Book, and over 450 refereed articles and proceedings.

His latest book, Lattice Sums Then and Now published by Cambridge Press, sits on the boundary between mathematical physics and number theory, and as with many of his written subjects surrounding mathematics, the book surprisingly begins with an unexpected connection – in this case a quote from Olivia Newton-John.

"Olivia Newton-John's grandfather was Nobel prize winner Max Born, an expert in lattice theory. Not many people realise this and it's a great way to engage people and make them think about mathematics in a different way. Her dad, Brin Newton-John, was actually the University of Newcastle's first Pro Vice-Chancellor!"

Changing the perspective on mathematics to include a physics-like inductive component rather than being exclusively a deductive science is a strong passion of Borwein's. 

Case in point is Borwein and his students' use of the computer as a laboratory to prove that the 60 year-old Guiga Conjecture about prime numbers can only fail for numbers with more than 20,000 decimal digits.

On a tour of the CARMA lab, there are clocks for New York, San Francisco, London and more lining the wall, that speak of the laboratory's extensive collaborations with leading mathematics institutions around the world. The group boasts an outstanding international profile in number theory and currently has 36 members, eight associates and seven students.

"The University of Newcastle is the only University to receive a rating of '5' for above world-standard research in the ERA rankings twice in Applied Mathematics," Prof Borwein said.

"In one minute we can now achieve calculations that would have been the subject of an applied maths PhD 40 years ago. I'm conducting intensive computer-assisted research on work that I once thought I had completed 15 years ago and which is providing new insights."

"One of the greatest attractions of science is that you don't know how ignorant you are until you acquire more knowledge. As has been said before, the best is the enemy of the better, we are constantly looking for new breakthroughs on things we previously thought we had solved or explored."