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Mathematics Illuminated

Geometries Beyond Euclid

We live in a world — a reality — ruled by straight lines. Our streets, houses, cubicles — virtually all of our space is parceled into rectilinear grids. Mathematicians were also ruled by straight lines — some would say imprisoned by them — for two thousand years. But what is a straight line? And when is a straight line not "straight"?

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Hyperbolic Disc

Hyperbolic Disc

Our first exposure to geometry is to that of Euclid; in which all triangles have 180 degrees. As it turns out, triangles can have more or less than 180 degrees. This unit explores these curved spaces that are at once otherworldly, yet firmly of this world—and present the key to understanding the human brain.

Unit Goals

  • Geometry is the mathematical study of space.
  • Euclid’s postulates form the basis of the geometry we learn in high school.
  • Euclid’s fifth postulate, also known as the parallel postulate, stood for over two thousand years before it was shown to be unnecessary in creating a self-consistent geometry.
  • There are three broad categories of geometry: flat (zero curvature), spherical (positive curvature), and hyperbolic (negative curvature).
  • The geometry of a space goes hand in hand with how one defines the shortest distance between two points in that space.
  • Stereographic projection and other mappings allow us to visualize spaces that might be conceptually difficult.
  • Einstein showed that curved geometry is a way to model gravitational attraction.
  • The recently proven Geometrization Theorem states that if we live in a randomly selected universe with a uniform geometry, then it is probably a hyperbolic universe.




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