**March 2015**

## Hyperbolic Spaces

**005**

**Hyperbolic Spaces**

**2016**

**Possible Bodies, Kym Ward**

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