Web edition: October 7, 2011
Print edition: October 22, 2011; Vol.180 #9 (p. 32)
January 6, 1940 | Vol. 37 | No. 1
Mathematicians think of everything as rubber
Topology, which you won’t find defined in the ordinary dictionary, was on the tip of mathematical tongues at the Columbus science meetings. This new geometry is as popular with the mathematicians as exploration of the atom is with physicists.
To those who are used to Euclidean geometry such as taught in school, this relatively new branch of mathematics, bulking large in the science meetings, will seem strange.
As explained on behalf of the American Mathematical Society by Prof. G. Baley Price of the University of Kansas, you must think of all sorts of objects in the Land of Topology as made of rubber. It is not necessary to keep the distance between each two points unchanged when two figures are compared. It is expected that the two figures will be stretched and distorted in any manner so long as they are not torn or glued together in new places. Any two figures which can be made to coincide by such stretchings and distortions are said to have the same topological properties.
In Euclidean geometry there are right angled triangles and equilateral triangles, but in topology all triangles are the same. If the two triangles be thought of as cut from a sheet of rubber, they can be stretched until they coincide. The surface of a sphere is topologically different from the surface of a doughnut, because no deformation without tearing will change a sphere into the surface of a doughnut. The fact that a figure is made up of several disconnected pieces is a topological property; such a figure is distinct from one consisting of a single piece, for it is not permitted to glue the parts together when they are compared. Although distinct in Euclidean geometry, a sphere and an egg-shaped surface are the same in topology.
UPDATE | October 22, 2011
Topology tackles Königsberg and the entire universe
A doughnut and coffee mug have the same topology because one can be stretched into the other without any gluing or tearing.
Mathematical curiosities related to the field of topology have existed since at least the 1700s. A well-known puzzle of that time asked whether a person could follow a path through the city of Königsberg crossing each of its bridges only once and returning to the starting point. In 1735, Leonard Euler found that such a trip would be impossible, later publishing his proof in a paper titled “Solutio problematis ad geometriam situs pertinentis,” or “The solution of a problem relating to the geometry of position.” Euler seems to have recognized early on that the problem was not about geometry as it relates to distances and angles, but rather about the qualitative shape of the landscape and how different positions are connected.
Still, the application of topology to serious scientific questions was novel enough in the 1940s for Science News to claim a “relatively new branch of mathematics.” Since then, topology has been used by scientists working in a surprisingly broad range of fields, from cancer research to quantum computing. Some of today’s topological thinkers are even exploring the idea of different possible shapes for the universe. While Einstein’s general theory of relativity describes the geometry of spacetime, the theory allows different topologies. Instead of having the simplest possible topology, the universe may twist, loop or have doughnut holes, with implications for whether the cosmos is finite or infinite.
Understanding such complexity would allow researchers to think of the whole wide universe the way Euler thought of Königsberg’s crosstown connections. Like a Prussian trying to traverse the city by passing once over all its bridges, could a spaceship headed off in one direction eventually return to its starting position? —Elizabeth Quill