World Globe

... • Lines extend indefinitely • All points equidistant from a given point in a plane form a circle ...

... • Lines extend indefinitely • All points equidistant from a given point in a plane form a circle ...

Modern Geometries: Non-Euclidean, Projective, and Discrete

... Pythagoras of Samos (569-475 BC) Hippocrates of Chios (470-410 BC) Plato (427-347 BC) Theaetetus of Athens (417-369 BC) Eudoxus of Cnidus (408-355 BC) Euclid of Alexandria (325-265 BC) Archimedes of Syracuse (287-212 BC) Hypatia of Alexandria (370-415 AD) ...

... Pythagoras of Samos (569-475 BC) Hippocrates of Chios (470-410 BC) Plato (427-347 BC) Theaetetus of Athens (417-369 BC) Eudoxus of Cnidus (408-355 BC) Euclid of Alexandria (325-265 BC) Archimedes of Syracuse (287-212 BC) Hypatia of Alexandria (370-415 AD) ...

Chapter 7: Hyperbolic Geometry

... 2. A finite straight line may be produced to any length. 3. A circle may be described with any center and any radius. 4. All right angles are equal. 5. If a straight line meet two other straight lines so that as to make the interior angles on one side less than two right angles, the other straight l ...

... 2. A finite straight line may be produced to any length. 3. A circle may be described with any center and any radius. 4. All right angles are equal. 5. If a straight line meet two other straight lines so that as to make the interior angles on one side less than two right angles, the other straight l ...

Euclidean/non-Euclidean Geometry

... On a sphere, there are no straight lines. As soon as you start to draw a straight line, it curves on the sphere. In curved space, the shortest distance between any two points (called a geodesic) is not unique. In curved space, (for spherical (riemannian) or hyperbolic geometry)the concept of perpend ...

... On a sphere, there are no straight lines. As soon as you start to draw a straight line, it curves on the sphere. In curved space, the shortest distance between any two points (called a geodesic) is not unique. In curved space, (for spherical (riemannian) or hyperbolic geometry)the concept of perpend ...

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language.For more than two thousand years, the adjective ""Euclidean"" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious (with the possible exception of the parallel postulate) that any theorem proved from them was deemed true in an absolute, often metaphysical, sense. Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Albert Einstein's theory of general relativity is that physical space itself is not Euclidean, and Euclidean space is a good approximation for it only where the gravitational field is weak.Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms to propositions without the use of coordinates. This is in contrast to analytic geometry, which uses coordinates.