Andrius Kulikauskas

  • m a t h 4 w i s d o m - g m a i l
  • +370 607 27 665
  • My work is in the Public Domain for all to share freely.


  • 读物 书 影片 维基百科

Introduction E9F5FC

Questions FFFFC0


See: Four geometries, Geometry, Geometry theorems, Specifications

What distinguishes geometries?

  • What are the geometries above and beyond the four basic geometries: affine, projective, conformal, symplectic?
  • How are these other geometries distinguished?

  • Systematize a list of geometries.

Kinds of geometry

From nLab overview:

  • Euclidean geometry
  • differential geometry of curves and surfaces
  • Riemannian geometry
  • G-structured differentiable manifolds (differential Cartan geometry)
  • topos-theoretic notions (cf. “geometric logic”) of (higher) functorial geometry
  • algebraic geometry
  • supergeometry
  • arithmetic geometry
  • absolute geometry
  • duality between algebra and geometry
  • noncommutative geometry
  • derived geometry

Incidence geometry

Euclidean space

  • Euclidean geometry: empty space + tools: straightedge, compass, area measurer
    • most important theorem: Pythagoras q=q1+q2
    • (q1+q2+q3)2 = 2(q1^2 + q2^2 + q2^3)
  • Euclidean space - (algebraic) coordinate systems - define left, right, front, backwards - and this often makes sense locally - but this does not make sense globally on a sphere, for example

Ordered geometry

  • Features the concept of intermediacy. It is a common foundation for affine, Euclidean, absolute geometry and hyperbolic geometry, but not projective geometry. Like projective geometry, it omits the notion of measurement.

Absolute geometry

  • Also known as neutral geometry, is based on the axioms of Euclidean geometry (including the first four of Euclid's axioms) but with the parallel postulate removed.

Thomas Lam. An invitation to positive geometries


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