Reaction to Chapter 9 of
I am interested in the different
approaches to plane geometry and noncommutative algebra. The first, held
by the ancient Greeks, is that geometric rules exist in themselves and
are revealed slowly by human observation and inspection. This outlook seems
more prone to error by inherent assumptions unconsciously held by mathematicians
who believe they are uncovering a predetermined, cohesive system. Kant
and Gauss' philosophies seems more useful, namely to approach geometry
as nothing more than the artifical laws that mathematicians themselves
create by deriving formulas. It is particularly impressive that Gauss wondered
whether the curved, non-Euclidian geometry might even extend to our space
and was roughly correct even though his experiments were inconclusive.
I'm curious about the fact that
from a point in four dimensional space one could see every point in three
dimensional space just as a point in three dimensional space can see every
point in two dimensional space. A certain amout of power is implied in
this pattern. Each higher perspective in fact does have a superior vantage
point in observing lower dimensions.
1) Have mathematicians developed
non-Euclidean geometries for the fourth and higher dimensions? For example,
have mathematicians ever examined a four dimensional geometry that is curved
along a fifth dimension of time?
2) Is a Klein
bottle in the fourth dimension analogous to a three dimensional Möbius
band? Can a Möbius band be constructed in two dimensions only with the
compromise of self-intersection?
Link: Möbius band within a Klein
to Cosmology Group Introduction Page