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A finite geometry is any geometric system that has only a finite number of points. The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the
When the model becomes accurate, it is just as difficult to understand as the real-world processes it represents. Buttered cat paradox: Humorous example of a paradox from contradicting proverbs. Intentionally blank page: Many documents contain pages on which the text "This page intentionally left blank" is printed, thereby making the page not ...
The Fano plane is an example of a finite incidence structure, so many of its properties can be established using combinatorial techniques and other tools used in the study of incidence geometries. Since it is a projective space, algebraic techniques can also be effective tools in its study.
Infinite sets are so common, that when one considers finite sets, this is generally explicitly stated; for example finite geometry, finite field, etc. Fermat's Last Theorem is a theorem that was stated in terms of elementary arithmetic, which has been proved only more than 350 years later.
In standard notation, a finite projective geometry is written PG(a, b) where: a is the projective (or geometric) dimension, and b is one less than the number of points on a line (called the order of the geometry). Thus, the example having only 7 points is written PG(2, 2).
In hyperbolic space of dimension at most 2, every geometrically finite polyhedron has a finite number of sides, but there are geometrically finite polyhedra in dimensions 3 and above with infinitely many sides. For example, in Euclidean space R n of dimension n≥2 there is a polyhedron P with an infinite number of
Numerous other constructions of both finite and infinite non-Desarguesian planes are known, see for example Dembowski (1968). All known constructions of finite non-Desarguesian planes produce planes whose order is a proper prime power, that is, an integer of the form p e, where p is a prime and e is an integer greater than 1.
The Fano plane, the projective plane over the field with two elements, is one of the simplest objects in Galois geometry.. Galois geometry (named after the 19th-century French mathematician Évariste Galois) is the branch of finite geometry that is concerned with algebraic and analytic geometry over a finite field (or Galois field). [1]