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In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry.As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement.
A set of dice is intransitive (or nontransitive) if it contains X>2 dice, X1, X2, and X3... with the property that X1 rolls higher than X2 more than half the time, and X2 rolls higher than X3 etc... more than half the time, but where it is not true that X1 rolls higher than Xn more than half the time.
The conventional meaning of Non-Euclidean geometry is the one set in the nineteenth century: the fields of elliptic geometry and hyperbolic geometry created by dropping the parallel postulate. These are very special types of Riemannian geometry, of constant positive curvature and constant negative curvature respectively.
Platonic solids are often used to make dice, because dice of these shapes can be made fair. 6-sided dice are very common, but the other numbers are commonly used in role-playing games. Such dice are commonly referred to as d n where n is the number of faces (d8, d20, etc.); see dice notation for more details.
Saccheri is primarily known today for his last publication, in 1733 shortly before his death. Now considered an early exploration of non-Euclidean geometry, Euclides ab omni naevo vindicatus (Euclid Freed of Every Flaw) languished in obscurity until it was rediscovered by Eugenio Beltrami, in the mid-19th century.
The second sense of the term is the metric geometry over a non-Archimedean valued field, [3] or ultrametric space. In such a space, even more contradictions to Euclidean geometry result. For example, all triangles are isosceles, and overlapping balls nest. An example of such a space is the p-adic numbers.
Pair of chiral dice (enantiomorphs) In three dimensions, every figure that possesses a mirror plane of symmetry S 1 , an inversion center of symmetry S 2 , or a higher improper rotation (rotoreflection) S n axis of symmetry [ 5 ] is achiral.
Regular polyhedra in non-Euclidean and other spaces [ edit ] Studies of non-Euclidean ( hyperbolic and elliptic ) and other spaces such as complex spaces , discovered over the preceding century, led to the discovery of more new polyhedra such as complex polyhedra which could only take regular geometric form in those spaces.