called “rubber-sheet” geometry because objects examined in topology are under con-
tinuous transformation (also called topological transformation or homeomorphism).
The shapes can be stretched, bent, and twisted in various ways, as long as they are not
torn or cut as they are changed. For example, a circle is considered to be topologically
equivalent to an ellipse (it can be deformed by stretching) and a sphere is equivalent
to an ellipsoid. This includes changing an object based on a one-to-one correspon-
dence. (For more about changing such shapes and one-to-one correspondence, see
elsewhere in this chapter.)
207
GEOMETRY AND TRIGONOMETRY
Who further developed the ideas in non-Euclidean geometry?
I
n 1854 German mathematician Georg Friedrich Bernhard Riemann (1826–
1866) presented several new general geometric principles, laying the founda-
tions of a non-Euclidean system of geometry called elliptical, or Riemann geom-
etry. In this he represented elliptic space and generalized the work of German
mathematician Karl Friedrich Gauss in differential geometry. This would even-
tually provide the basic tools for the general theory of relativity’s mathematical
expression. (For more about Riemann, see “History of Mathematics.”)