| Hyperbolic Geometry |
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Hyperbolic geometry exists on a negatively-curved infinite surface. One way it can be pictured is like a saddle. The edges of the saddle go to infinity, and every point on the surface acts as if it were the saddle center -- curving up in one direction and curving down in the other direction. The first four axioms are valid in hyperbolic geometry, but the fifth axiom states there are infinitely MANY parallel lines. |
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AXIOM
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COMMENTS
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| 1. A straight line can be drawn between any two points | Choose any two points and draw a straight line between them. The shortest distance between two points tends to look like a concave arc. |
| 2. A finite line can be infinitely extended in both directions | Since the hyperbolic surface is infinite, the line between any two points can be extended infinitely in both directions. |
| 3. A circle can be drawn with any center or radius | Choose a center and find all the points a given distance (radius) from the center. In hyperbolic geometry, a circle looks pretty much the same. Think of putting a large warm pancake on a horses saddle -- it will conform to the surface without much distortion. |
| 4. All right angles are equal to each other | Right angles can be drawn anywhere on the hyperbolic surface. While appearing perpendicular at the intersection, the lines curve away from each other in the distance. |
| 5. There are infinitely MANY parallel lines | Parallel lines never intersect. Draw a line across the horizontal ridgeline of the saddle. Now choose any two points on the lower front edge of the saddle. A line connecting these two points will arch across the side of the saddle but never rise high enough to cross the line at the ridge. We can draw many lines that will not cross -- MANY parallel lines. |
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