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Combinatorics is the branch of mathematics studying the enumeration, combination, and permutation of sets of elements and the mathematical relations that characterize their ...

In antiquity, geometric constructions of figures and lengths were restricted to the use of only a straightedge and compass (or in Plato's case, a compass only; a technique ...

For a right triangle with legs a and b and hypotenuse c, a^2+b^2=c^2. (1) Many different proofs exist for this most fundamental of all geometric theorems. The theorem can ...

The two recursive sequences U_n = mU_(n-1)+U_(n-2) (1) V_n = mV_(n-1)+V_(n-2) (2) with U_0=0, U_1=1 and V_0=2, V_1=m, can be solved for the individual U_n and V_n. They are ...

Relates evolutes to single paths in the calculus of variations. Proved in the general case by Darboux and Zermelo in 1894 and Kneser in 1898. It states: "When a single ...

Expanded notation is the term given in elementary mathematics education for the expansion of a positive integer in the form sum_(k)b_k10^k, i.e., as a sum of appropriate ...

A geometric sequence is a sequence {a_k}, k=0, 1, ..., such that each term is given by a multiple r of the previous one. Another equivalent definition is that a sequence is ...

A pyramid with a hexagonal base. The edge length of a hexagonal pyramid of height h is a special case of the formula for a regular n-gonal pyramid with n=6, given by ...

An isozonohedron is a zonohedron whose faces consist of n(n-1) congruent rhombi (Fedorov 1953, pp. 256-266; Chilton and Coxeter 1963). The following table summarizes some of ...

Q_n^((alpha,beta))(x)=2^(-n-1)(x-1)^(-alpha)(x+1)^(-beta) ×int_(-1)^1(1-t)^(n+alpha)(1+t)^(n+beta)(x-t)^(-n-1)dt. In the exceptional case n=0, alpha+beta+1=0, a nonconstant ...