Hyperbolic Spaces
Hyperbolic Spaces as 'surfaces of the possible'.
"Rolling inward enables rolling outward; the shape of life’s motion traces a hyperbolic space, swooping and fluting like the folds of a frilled lettuce, coral reef, or bit of crocheting."
See: Donna Haraway, Staying with the trouble (2016)
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Euclidean geometry is located
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at the intersection of metric and affine geometry.
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It is based on 5 axioms:
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- A straight line can be drawn between any two points.
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- A straight line drawn between two points can be continued infinitely
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- A circle is defined as all of the points a certain distance (radius) from any point.
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- All right angles are equal.
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- Parallel lines will maintain an equal distance from one another
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Non-euclidean geometry is what happens when any of the 5 axioms do not apply.
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It arises when either the metric requirement is relaxed,
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or the parallel postulate is replaced with an alternative one.
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In the latter case one obtains hyperbolic geometry
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and elliptic geometry, the traditional non-Euclidean geometries.
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When the metric requirement is relaxed,
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then there are affine planes associated with the planar algebras
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which give rise to kinematic geometries.
https://en.wikipedia.org/wiki/Euclidean_geometry#Axioms + https://en.wikipedia.org/wiki/Non-Euclidean_geometry