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The golden ratio, also known as the divine proportion, golden mean, or golden section, is a number often encountered when taking the ratios of distances in simple geometric ...

It is always possible to "fairly" divide a cake among n people using only vertical cuts. Furthermore, it is possible to cut and divide a cake such that each person believes ...

Given a rectangle having sides in the ratio 1:phi, the golden ratio phi is defined such that partitioning the original rectangle into a square and new rectangle results in a ...

The dimension e(G), also called the Euclidean dimension (e.g., Buckley and Harary 1988) of a graph, is the smallest dimension n of Euclidean n-space in which G can be ...

The Schur number S(k) is the largest integer n for which the interval [1,n] can be partitioned into k sum-free sets (Fredricksen and Sweet 2000). S(k) is guaranteed to exist ...

Any four mutually tangent spheres determine six points of tangency. A pair of tangencies (t_i,t_j) is said to be opposite if the two spheres determining t_i are distinct from ...

The anticomplementary triangle is the triangle DeltaA_1^'A_2^'A_3^' which has a given triangle DeltaA_1A_2A_3 as its medial triangle. It is therefore the anticevian triangle ...

Q(n), also denoted q(n) (Abramowitz and Stegun 1972, p. 825), gives the number of ways of writing the integer n as a sum of positive integers without regard to order with the ...

Szemerédi's theorem states that every sequence of integers that has positive upper Banach density contains arbitrarily long arithmetic progressions. A corollary states that, ...

The triangle DeltaM_AM_BM_C formed by joining the midpoints of the sides of a triangle DeltaABC. The medial triangle is sometimes also called the auxiliary triangle (Dixon ...