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This is geometry that is based on ordinary space-- space as we normally consider it. This is the ordinary space of 3 dimensions as you can imagine them in the coordinate system. Planes are flat, and parallel lines on any plane never ever meet, parallel planes never meet... you get the point. There are geometries that involve other kinds of space and they are called "non-Euclidean" geometries.
Some of these non-Euclidian geometries are very real and not just theoretical in nature. For example, in the relativistic world, the space in and around very strong gravitational forces is distorted. This has been observed and verified in several ways. Euclidean proofs and the methods of analytical geometry do not work without accounting for these spacial distortions.
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Does the Pythagorean theorem work with Euclidean and Hyperbolic geometry?
It works in Euclidean geometry, but not in hyperbolic.
Why don't parallel lines exist in elliptical geometry?
Elliptical geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry was replaced by the statement that through any point in the plane, there exist no lines parallel to a given line. A consistent geometry - of a space with positive curvature - was developed on that basis.It is, therefore, by definition that parallel lines do not exist in elliptical geometry.
What are some real world applications of geometry?
Euclidean geometry has become closely connected with computational geometry, computer graphics, convex geometry, and some area of combinatorics. Topology and geometry The field of topology, which saw massive developement in the 20th century is a technical sense of transformation geometry. Geometry is used on many other fields of science, like Algebraic geometry. Types, methodologies, and terminologies 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 Euclidean geometry Finite geometry Geometry of numbers Hyperbolic geometry Information geometry Integral geometry Inversive geometry Inversive ring geometry Klein geometry Lie sphere geometry Non-Euclidean geometry Numerical 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
What tools allowed the Greeks to exploit the five basic postulates of Euclidean geometry?
compass and straightedge
In non-Euclidean geometry triangles on a sphere have more than 180 degrees true or false?
true apex
