Saccheri convincingly achieved his reductio for the first possibility with the innocent assumption that straight lines are infinite [cf. Jeremy Gray, Ideas of Space Euclidean, Non-Euclidean, and Relativistic, Oxford, 1989; p. 64]. Straight lines would be Euclidean straight, but the properties specified by non-Euclidean axioms would be satisfied. Nevertheless, since Bernhard Riemann (1826-1866), non-Euclidean manifolds are said to be “curved,” and only Euclidean space itself is called “flat.” Contradiction #1 above produces “positively” curved space (“spherical” or “elliptical” geometry, first described by Riemann himself), and contradiction #2 “negatively” curved space (“hyperbolic” or Lobachevskian geometry). To Euclid, this doubtlessly would seem to prove his point: the parallel postulate is about straight lines, so using curved lines hardly produces an honest non-Euclidean geometry. Euclidean geodesics “straight” and generalized straight lines “geodesics”. What “curvature” would have meant to Euclid is now “extrinsic” curvature: that for a line or a plane or a space to be “curved” it must occupy a space of higher dimension, i.e. that a curved line requires a plane, a curved plane requires a volume, a curved volume requires some fourth dimension, etc. Now “intrinsic” curvature has nothing to do with any higher dimension. » Read more: The Metaontology of Universe
Posts Tagged ‘of’
The Universal Cycles of Change: patterns in Nature Translated to Human Behavior
February 26th, 2009
We can learn so much about ourselves from observing the lives of cells, the growth of trees, the waves in the ocean, along with other universal and biological patterns of growth and self creation. Nature is our greatest teacher. Since humans are a part of nature and our universe is 15 billion years old, it seems obvious that there must be some kind of connection between the patterns that exist outside of ourselves and the unconscious patterns » Read more: The Universal Cycles of Change: patterns in Nature Translated to Human Behavior
