Beautiful math often turns out to be useful math, write Joan Licata and Anthony Licata.
What does connect-the-dots have to do with watching a Pixar movie? More than you think.
A connect-the-dots page starts with nothing more than a few labeled dots. As each dot is joined to the next, however, an image emerges. Each step is simple – just add a line segment between two points – but the resulting image can be extremely complex.
Toddlers can produce masterpieces this way, but so can computers. When a computer needs to draw a curve, it begins by connecting a sequence of points. Using just a few stitches can result in a zigzag with sharp corners, but increasing the initial number of stitches makes the resulting curve smoother. With enough points and line segments, we can approximate even the most complicated curves.
But maybe you are more ambitious. Suppose you are not interested in drawing curves, but rather in constructing two-dimensional surfaces. Can you use an analogous approach to build an airplane or a sphere or something more elaborate?
Triangles – the simplest two-dimensional objects – serve as building blocks for more complex surfaces. Just as we can connect a pair of points with a line segment, we can connect three points through a triangle. And, just as we can create complicated curves by gluing many segments to their ends, we can create complicated surfaces by gluing many triangles along their edges. We can approximate extremely complex surfaces as long as we use enough triangles.
Mathematicians began to think seriously about constructing surfaces from triangles in the late 19th century, hoping to classify surfaces. In particular, they wanted to understand when two surfaces could be deformed to look the same without cutting or sticking. They developed mathematical tools to study this question, and a century later it became clear that they had also laid the foundations for an important technique in computer graphics.
Imagine trying to model moving fabric, perhaps a flag flapping in the breeze. Since the flag changes shape as it moves through space, this is a much more difficult problem than simulating the motion of a rigid object like a table. If the flag is approximated by triangles, however, modeling becomes possible because the computer only needs to track sets of three points. When the points move, they drag the triangles with them.
Abstract? Applied? Both!
Mathematicians certainly weren’t thinking about computer graphics in the 1890s. They were studying abstract questions about two-dimensional geometry and developing beautiful mathematics. Nevertheless, the techniques they invented to ask precisely this question and then answer it proved to be extremely useful. In fact, this theme recurs throughout human history: mathematics developed to solve abstract problems prove useful. Maybe not always and certainly not quickly, but it happens again and again.
The first mathematical objects that most people encounter are the counted numbers 1, 2, 3… Most counted numbers are formed by multiplying smaller numbers, but not all of them. Some numbers have only 1 and themselves as factors, and these are called prime numbers. For example, the numbers 2, 3, and 5 are prime, but 4 = 2 x 2 is not.
Prime numbers act as building blocks for the entire number system. Centuries ago, mathematicians who studied prime numbers didn’t think their efforts would defend a castle or build a better steam engine, but they were intrigued by the search for patterns and patterns.
Some of the questions they posed continue to capture the mathematical imagination; today, the most famous unsolved problem in mathematics is the Riemann hypothesis, which deals with the distribution of prime numbers among the numbers to be counted.
Thinking about prime numbers can seem like an intellectual game separate from “real world” concerns. But suppose you ask yourself, “Why is it safe to use my credit card to buy something online?” (Or, perhaps better: “Is it safe to use my credit card online?”) In fact, the basic techniques for sending data securely over the Internet rely on what we know on the factorization of a number into prime numbers. Every time you enter your credit card number on a website and hit “send,” you have a number theorist to thank.
Likewise, Persian mathematicians began to develop the subject we now call algebra in the Middle Ages. This field has evolved over centuries and today it underpins internet search algorithms and Netflix recommendations.
Fourier analysis, which was developed as part of calculus in the late 1700s, provides the basic mathematical tools for signal processing in telecommunications and medical imaging.
Algebraic topology – a branch of mathematics that wasn’t even created until the 20th century – is being used in the 21st century to study artificial intelligence and cancer genomics.
The list is long, but what is striking about all of these examples is that the original researchers could not foresee what applied questions their work would require. Many important mathematics are developed specifically to solve real-world problems, but curiosity-driven research is as important today as it ever has been.
But back to triangles
New applications of old math are exciting, but progress also comes in the form of new math.
The infographic uses flat triangles to approximate smooth surfaces, but if you’re willing to allow the triangles to bend a bit, you can build any surface by sticking enough together. This is equivalent to saying that you can cut any surface into curved triangular pieces.
These curved triangles are an important tool for generalizing what we know about geometry to higher dimensions – after all, mathematicians, scientists and engineers don’t just care about surfaces.
Higher-dimensional spaces occur not only in pure mathematics, but also in nature as patterns in large data sets, as relationships between physical quantities, and in descriptions of the universe itself.
Mathematicians develop formal techniques to study them, compensating for the fact that a seven-dimensional space is more difficult to represent than a flag. Fortunately, the idea of gluing triangles is generalized to all dimensions! In three dimensions, for example, the analog of a triangle is a tetrahedron, and just as gluing triangles builds surfaces, gluing tetrahedra builds new three-dimensional objects. In higher dimensions, the analog of a triangle is called an n-simplex, and gluing n-simplices together creates n-dimensional objects.
Since any surface can be cut into curved triangles, and it is reasonable to ask whether the analogous fact holds in higher dimensions: can any n-dimensional space be cut into n-simplices?
Mathematicians first hypothesized that the phenomena they observed in dimensions one and two would generalize to all dimensions. This belief has become known as the triangulation conjecture.
Decades of research have failed to provide proof. Then breakthroughs in the 1980s revealed examples of four-dimensional objects that cannot be cut into 4-simplices. But in 2012, the triangulation conjecture was finally proven wrong for all dimensions greater than four. The mathematical universe has very strange shapes!
Disproving the triangulation conjecture is a triumph of curiosity-driven research, and the truth now lies in the library of humanity. He can rest there peacefully. Then again, maybe a 26-dimensional space without triangulations will lead to a cure for cancer.
We never know.
About the authors:Joan and Anthony Licata are mathematicians at the Mathematical Sciences Institute of the Australian National University. They will teach classes at the next AMSI Winter School on algebra, geometry and physics. Joan’s research focuses on topology and Anthony’s on representation theory. They don’t do math for nominations, just groupies and fame.
Published on June 10, 2015