What is a good example of a topological structure?
A
good example of a topological structure is called the Möbius strip (also
called the twisted cylinder). It was invented by German mathematician and
astronomer August Ferdinand Möbius (1790–1868) in September 1858,
although he did not publish his findings until 1865. The strip was also developed
independently by German mathematician Johann Benedict Listing (1808–1882),
who discovered it in July 1858. Although Listing published his findings in 1861,
Möbius’s name is used for the strip.
This one-sided surface is easy to visualize: Take a long single strip of paper,
give one of the two ends a twist (that gives the strip a half twist in its length, or a
180 degrees twist), then attach the two ends. Amazingly, you can verify that it
has only one side—it is not possible to paint it with two colors. Even if you cut
the strip in half (down the middle), you will still have a one sided-surface. In
fact, ants would be able to walk on the Möbius strip’s single surface indefinitely,
since there is no true edge in the direction of their movement.ellipse that has no asymptotes. (For more information about hyperbolas, ellipses, and
asymptotes, see elsewhere in this chapter).Whodeveloped the two alternativesto Euclidean geometry?
Hyperbolic geometry was first announced by Russian mathematician Nikolai
Ivanovich Lobachevski (1792–1856; also seen as Lobatchevsky) in 1826. He challenged
Euclid’s fifth postulate that one and only one line parallel to a given line can be drawn
through a fixed point external to the line. Instead, he developed a self-consistent sys-
tem of geometry in which the “flawed” postulate was replaced by one allowing more
than one parallel through the fixed point.
This idea was already developed independently by Hungarian János (or Johann)
Bolyai (1802–1860) in 1823 (after several attempts to prove the Euclidean parallel pos-
tulate, he developed his system by assuming that a geometry could be constructed
without the parallel postulate) and German mathematician, physicist, and astronomer
Karl Friedrich Gauss (1777–1855; also seen as Johann Carl [or Karl] Friedrich Gauss)
in 1816. But, as often happens in science and mathematics, the person who publishes
an idea first gets most of the credit; this time, Lobachevski was the first to publish.What is topology?
In general, topology is a branch of mathematics that examines patterns of geometric
206 figures based on position and relative position without regard to size. It has also been