The preoccupations of mathematicians can seem curious and strange to normal people. They sometimes expend great energy proving results that appear glaringly obvious. One such result is called the Jordan Curve Theorem. We all know that a circle has an inside and an outside, and that this property also holds for a much larger collection of closed curves [TM165 or search for “thatsmaths” at irishtimes.com].
A “simple closed curve” is a closed continuous loop in a plane that does not intersect itself. Such a distorted version of a circle is also called a Jordan curve, after the French mathematician Camille Jordan, who first proved some of its key properties.
The Jordan Curve Theorem states that every simple closed curve, no matter how complicated or convoluted, divides the plane into two regions, an inside and an outside. The theorem appears so trivial that it does not require a proof. But results like this can be much more profound than a first glance might suggest and, on occasions, things that appear obvious can turn out to be false.
A Bohemian Savant
Bernhard Bolzano, a Bohemian mathematician, philosopher and priest, was the first person to pose the curve problem in explicit form. He was convinced of the need to introduce rigour into mathematical analysis. He claimed that, for a closed loop in a plane, a line connecting a point enclosed by the loop (inside) to a point distant from it (outside) must intersect the loop. This seems obvious enough, but Bolzano realized that it was a non-trivial problem.
The first proof of the curve theorem appeared in Camille Jordan’s influential book Cours d’analyse, first published in 1882. The theorem is easily stated and easy to prove for curves that are polygons, consisting of straight line segments.
However, for general curves it is quite difficult to prove since “simple” curves can have some bizarre properties, such as being jagged everywhere with no definite direction, or as being fractal in nature like the boundary of a snowflake. This makes it difficult to distinguish which points are inside and which are outside. The proof uses advanced ideas from the branch of mathematics known as topology.
Jordan Curves Inspire Art
The Travelling Salesman Problem, or TSP, seeks the shortest route a salesman can choose to visit a number of cities and return to his starting point. The solution of this optimization problem is a Jordan curve. Such curves have served as inspiration for artists. Professor Robert Bosch of Oberlin College Ohio uses results from optimization theory to produce artistic images using these simple loops.
The illustration above shows a sculpture ― Embrace ― by Bosch, where a simple closed curve separates two regions, represented by metals of different colours. Bosch solved a TSP problem with 726 cities to form the boundary in this sculpture. Embrace was awarded First Prize at the 2010 JMM Mathematical Art Exhibition in San Francisco.
More complex curves, with up to 200,000 cities, have been used to simulate several classical works of art. The example shown below is a detail from Botticelli’s Birth of Venus. It was downloaded from the website www.math.uwaterloo.ca/tsp/data/art . This Jordan curve is an optimal route for a TSP with 140,000 cities.
To illustrate how results that appear obvious can be false, we mention an extension of the Jordan Curve Theorem to higher dimensions. In three dimensional space, the surface of a sphere has an inside and an outside, and both are simply connected: a loop in either region can be shrunk continuously to a point.
A distortion of the surface to a form called Alexander’s horned sphere still divides the remainder of space into inside and outside parts. But now the outside is no longer simply connected, and has a much more complex structure. So, our intuition has led us astray. This indicates the need for rigour in proving seemingly-obvious results.
Intuition serves as an invaluable means of discovering new mathematics, but it can also lead us astray. There is always a need for rigour in proving seemingly-obvious results. For Jordan curves, the great German mathematician Felix Klein expressed the problems thus: “Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions.”
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