Summary of ­čî▓Hyperbolic Geometry (& Imaginary numbers as rotation!)

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00:00:00 - 00:05:00

This video discusses hyperbolic geometry and imaginary numbers. Hyperbolic geometry is used to model the behavior of objects traveling at supersonic speeds, and is also known as anti-deciduous space. The author explains how vectors can be abstracted to rotations, and how multiplying or dividing vectors by complex numbers can produce rotations in space.

  • 00:00:00 In this video, the author discusses hyperbolic geometry and imaginary numbers. Hyperbolic geometry is used to model the behavior of objects traveling at supersonic speeds. When objects are flying close to the ground, they create hyperbolic cones projected into section with the ground. There are three usual ways to cut up a cone in hyperbolic geometry: elliptically, parabolically, and hyperbolically. In minkowski space, space-time is represented by two inverted cones - the top cone represents the future and the bottom cone represents the past. Particles travel in both space and time, and each particle traces a path into the future and that path through the light cone is known as the world line. In minkowski space, hyperbolic geometry is considered to be unbent, similar to Euclidean geometry. Hyperbolic geometry is also known as anti-deciduous space, and when dealing with this many dimensions things can get messy. Complex systems bending state space negatively can sometimes result in the appearance of hyperbolas and hyperbolic functions.
  • 00:05:00 In this video, the author explains how vectors can be abstracted to rotations, and how multiplying or dividing vectors by complex numbers can produce rotations in space (or in a coordinate system). The author also warns against thinking of imaginary numbers as simply rotations, as the mechanics involved can often be complicated.

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