Non-Euclidean Geometry
Therefore, any theorem derived solely from those first four axioms holds across these two disparate branches. On the other hand, hyperbolic geometry was derived from the an alteration of the parallel postulate where for any line R , and any point P which does not lie on R , there exist multiple lines through point P which do not intersect with line R (or to put it more succinctly, given any line it is possible to construct multiple ‘ parallel ’ lines).
As a result of this derivation, hyperbolic space is defined as a space with negative Gaussian curvature; curvature being the measure of the rate of change of direction of a curve with respect to the distance along the curve. 7 The curvature of a sphere can be determined using sectioning planes or a circle of best
fit at a particular point. For the circle method, the curvature at the point of contact is equal to the reciprocal of the radius of the circle. 8 When using sectioning planes which both contain the normal to the surface, one plane gives the maximum and curvature and the other gives the minimum. These planes give us the ‘ principal curvature ’ and the mean curvature (what ismeant when describing the overall curvature) is found using either the sum or half the sum of the principal curvatures. The Gaussian curvature is the product of these principal curvatures. 9
Surfaces have negative curvature if their Gaussian curvature is less than 0 at all points. 10 A typical shape of negative curvature is that of a saddle; it curves away from its tangent at every point. Similar surfaces appear frequently in nature. Salad leaves display a ‘ crinkling ’ which is characteristic of hyperbolic planes. 11 The same can be said about jellyfish tentacles and even our brains. Many artists have even used crocheting to create artwork in hyperbolic space with striking results:
These artworks display a negative Gaussian curvature (or at least an approximation thereof )
7 Beardon 2011. 8 Ibid. 9 Hosch 1998. 10 Beardon 2011. 11 Vincent 2011, NKS 2011, Retvari 2011, McLaury 2015.
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