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Introduction E9F5FC
Questions FFFFC0
Software
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
- Absolute geometry
- Affine geometry
- Algebraic geometry
- Analytic geometry
- Archimedes' use of infinitesimals
- Birational geometry
- Complex geometry
- Combinatorial geometry
- Computational geometry
- Conformal geometry
- Constructive solid geometry
- Contact geometry
- Convex geometry
- Descriptive geometry
- Differential geometry
- Digital geometry
- Discrete geometry
- Distance geometry
- Elliptic geometry
- Enumerative geometry
- Epipolar geometry
- Finite geometry
- Fractal geometry
- Geometry of numbers
- Hyperbolic geometry
- Incidence geometry
- Information geometry
- Integral geometry
- Inversive geometry
- Inversive ring geometry
- Klein geometry
- Lie sphere geometry
- Non-Euclidean geometry
- Noncommutative algebraic geometry
- Noncommutative geometry
- Ordered geometry
- Parabolic geometry
- Plane geometry
- Projective geometry
- Quantum geometry
- Riemannian geometry
- Ruppeiner geometry
- Spherical geometry
- Symplectic geometry
- Synthetic geometry
- Systolic geometry
- Taxicab geometry
- Toric geometry
- Transformation geometry
- Tropical 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 structure Which points lie on which lines.
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
- 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.
- 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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