Search results
Results from the WOW.Com Content Network
Absolute geometry is an incomplete axiomatic system, in the sense that one can add extra independent axioms without making the axiom system inconsistent. One can extend absolute geometry by adding various axioms about parallel lines and get mutually incompatible but internally consistent axiom systems, giving rise to Euclidean or hyperbolic ...
Non-Euclidean geometry is an example of a scientific revolution in the history of science, in which mathematicians and scientists changed the way they viewed their subjects. [24] Some geometers called Lobachevsky the "Copernicus of Geometry" due to the revolutionary character of his work. [25] [26]
In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called sides or edges and three points called angles or vertices. Just as in the Euclidean case, three points of a hyperbolic space of an arbitrary dimension always lie on the same plane. Hence planar hyperbolic triangles also ...
Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin( α + β ) = sin α cos β + cos α sin ...
A basic example in topology is lifting a path in one topological space to a path in a covering space. [1] For example, consider mapping opposite points on a sphere to the same point, a continuous map from the sphere covering the projective plane. A path in the projective plane is a continuous map from the unit interval [0,1]. We can lift such a ...
An example of such a geometry is the Dehn plane. Non-Archimedean geometries may, as the example indicates, have properties significantly different from Euclidean geometry . There are two senses in which the term may be used, referring to geometries over fields which violate one of the two senses of the Archimedean property (i.e. with respect to ...
In mathematics, a property is any characteristic that applies to a given set. [1] Rigorously, a property p defined for all elements of a set X is usually defined as a function p: X → {true, false}, that is true whenever the property holds; or, equivalently, as the subset of X for which p holds; i.e. the set {x | p(x) = true}; p is its indicator function.
Subtraction is an anticommutative operation because commuting the operands of a − b gives b − a = −(a − b); for example, 2 − 10 = −(10 − 2) = −8. Another prominent example of an anticommutative operation is the Lie bracket.