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Glossary

8 Geometries Beyond Euclid

 

Axiomatic Systems

The system that Euclid used in The Elements—beginning with the most basic assumptions and making only logically allowed steps in order to come up with propositions or theorems—is what is known today as an axiomatic system.

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Euclid's Postulates

1. Any two points can be joined by a straight line.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

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General Relativity

Einstein's famous theory of General Relativity relates gravity to the curvature of spacetime.

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Geometry

Geometry is the mathematical study of space. The geometry of a space goes hand in hand with how one defines the shortest distance between two points in that space.

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Hyperbolic Geometry

In this type of geometry the angles of a triangle add up to less than 180 degrees. In such a system, one has to replace the parallel postulate with a version that admits many parallel lines.

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Non-Euclidian Geometry

Non-Euclidean geometries abide by some, but not all of Euclid's five postulates.

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Poincaré Disk

The Poincaré Disk is a flat map of hyperbolic space.

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Principal Curvatures

Principal curvatures are a way to extrinsically measure the curvature of a surface by looking at a given point and finding the contour line with the greatest curvature and the contour line with the least curvature.

Principal Curvatures

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Spherical Geometry

In this type of geometry the angles of a triangle add up to more than 180 degrees. In such a system, one has to replace the parallel postulate with a version that admits no parallel lines as well as modify Euclid's first two postulates.

Spherical Geometry

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Stereographic Projection

This method can create a flat map from a curved surface while preserving all angles in any features present.

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The Kissing Circle

The Osculating Circle, or "Kissing" Circle, is a way to measure the curvature of a line.

Kissing Circle

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