Can you explain how euclidean geometry is flawed?
I’m not sure what you mean, Euclidean geometry isn’t flawed! But you might mean: a) Euclid derived his principles of geometry from five axioms. But he unsuspectingly made use of other unstated assumptions, the most famous of which is Pasch’s axiom, if you draw a line (infinite line) that passes through one edge of a triangle and misses the corners, it must pass through one of the other sides! Essentially this establishes that you can’t avoid the edges somehow, and thus makes Euclidean space locally simple and flat. But this “flaw” is now fixed. b) Euclid’s fifth axiom, that parallel lines never meet, seemed far more complex than the other four. In trying to prove it from the other four it was realized that setting it false led to no contradictions, and thus to other kinds of geometry, such as spherical geometry, the geometry in the surface of a sphere, where the counterpart of lines are “geodesics” and (non-coinciding) geodesics always meet twice. Lines of longitude are geodesics.
1) “Nineteenth century mathematicians realized that the eighteenth century certainty of geometry was mistaken. Geometry was an empirical science. It reported the way our space happened to to be, not the way it had to be. If that was so, other geometries were possible and our experience of space might well have been different. In the nineteenth century, these were regarded as possibilities that were unrealized. Nature had many choices but, they thought, she chose Euclid’s system. This realization of the mere possibility of geometries other than Euclid’s was shocking. Greater shocks were in store. In the twentieth century, Einstein delivered the final insult to Euclid. He found through his general theory of relativity that a non-Euclidean geometry is not just a possibility that Nature happens not to use. In the presence of strong gravitational fields, Nature chooses these geometries.” Source and further information: http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/non_Euclid/inde
