Reaction to Chapter 9 of Beyond the Third Dimension

Andrew Miller


  • 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.
  • Questions:

    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 bottle: #1 + #2

    Link to Cosmology Group Introduction Page