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Intuitively, a d-dimensional discrete percolation model is said to be long-range if direct flow is possible between pairs of graph vertices or graph edges which are "very ...

The Buffon-Laplace needle problem asks to find the probability P(l,a,b) that a needle of length l will land on at least one line, given a floor with a grid of equally spaced ...

All the propositions in projective geometry occur in dual pairs which have the property that, starting from either proposition of a pair, the other can be immediately ...

A transformation of the plane which transforms collinear points into collinear points. A projective collineation transforms every one-dimensional form projectively, and a ...

A compactum (plural: compacta) is a compact metric space. An example of a compactum is any finite discrete metric space. Also, the space [0,1] union [2,3] is a compactum, ...

Let P=p:q:r and U=u:v:w be distinct points, neither lying on a side line of the reference triangle DeltaABC. Then the P-cross conjugate of U is the point ...

Let P=p:q:r and U=u:v:w be distinct trilinear points, neither lying on a sideline of DeltaABC. Then the crossdifference of P and U is the point X defined by trilinears ...

A plane parallel to two (or more) skew lines, also called a director plane. The orientation of a director is fixed, but it is specified uniquely only if a point lying on it ...

Given a triangle DeltaA_1A_2A_3, the points A_1, I, and J_1 lie on a line, where I is the incenter and J_1 is the excenter corresponding to A_1. Furthermore, the circle with ...

The Johnson midpoint is the point of concurrence of the line segments joining the vertices of a reference triangle with the centers of a certain set of circles (that resemble ...