Once you start thinking about the nature of space, you soon arrive at the question of what space actually is. Now let's turn to the impact of this realisation on human thought. But as was shown by the mathematician Bernhard Riemann (among others) there are many more non-Euclidean spaces besides the hyperbolic paraboloid, including positively curved spaces and spaces of three or more dimensions (you can find out more here). Carl-Friedrich Gauss, one of the discoverers of this fact, never even got up the nerve to publish his work on this subject. It's this realisation - that space doesn't have to be as Euclid and our intuition suggest, but could be otherwise - that 19th century thinkers found so revolutionary. It turns out that the hyperbolic paraboloid forms a perfectly decent geometric space. What's more, the blue and yellow parallel lines are not everywhere the same distance apart, as you would expect from parallel lines on the plane. But notice that the red and yellow lines are both parallels of the blue line, yet they pass through the same point. The lines drawn on the shape are the "straight lines" of the paraboloid: they are paths of shortest distance between points. (See the first article for an explanation.) If a straight line that falls on two straight lines makes the interior angles on the same side add up to less than two right angles, then the two straight lines, if produced indefinitely, meet on that side.All right angles are equal to one another. A circle can be constructed with any point as its centre.A finite straight line can be extended as long as desired.A straight line can be drawn from any point to any other point.
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