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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
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.
Angle trisection is the construction, using only a straightedge and a compass, of an angle that is one-third of a given arbitrary angle. This is impossible in the general case. For example, the angle 2 π /5 radians (72° = 360°/5) can be trisected, but the angle of π /3 radians (60°) cannot be trisected. [8]
The summit angles of a Saccheri quadrilateral are acute if the geometry is hyperbolic, right angles if the geometry is Euclidean and obtuse angles if the geometry is elliptic. The sum of the measures of the angles of any triangle is less than 180° if the geometry is hyperbolic, equal to 180° if the geometry is Euclidean, and greater than 180 ...
Thus, for every finite projective plane there is an integer N ≥ 2 such that the plane has N 2 + N + 1 points, N 2 + N + 1 lines, N + 1 points on each line, and N + 1 lines through each point. The number N is called the order of the projective plane. The projective plane of order 2 is called the Fano plane. See also the article on finite geometry.
Mathematicians can now explain how many people would need to be invited to a party so at least 4 people always know one another. It only took 90 years to solve.
In finite geometry, PG(3, 2) is the smallest three-dimensional projective space. It can be thought of as an extension of the Fano plane. It has 15 points, 35 lines, and 15 planes. [1] It also has the following properties: [2] Each point is contained in 7 lines and 7 planes. Each line is contained in 3 planes and contains 3 points.
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).