A doughnut and coffee mug have
the same topology because one
can be stretched into the other
without any gluing or tearing.
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.
Topology tackles Königsberg and the entire universe
To vote for one of these topics to appear on From the Archive,
A. How “wiggleworms” might learn to do science
B. First scanning electron microscope, introduced as “seeing eye”
C. Living coelacanth discovered in South African waters