The intersection of a polytope with a tangent hyperplane; it is itself a polytope of some lower dimension. The 0-dimensional faces are called vertices, the 1-dimensional faces are called edges, the (d-2)-dimensional faces are called ridges, and the (d-1)-dimensional faces are called facets.

A representation of a planar graph as a planar straight line graph in which no two edges cross.

A sequence of numbers defined by the
recurrence relation * F(n) = F(n-1) + F(n-2) * and the
initial values
* F(0)=1 * and * F(1)=1*.
The Fibonacci numbers are supposed to give the number of pairs of rabbits
in an enclosure *n* months after a breeding pair is placed
in it, if newly
born pairs of rabbits begin breeding themselves in their second month.

An algebraic entity consisting of a set of elements and two binary associative operations on those elements, usually called addition and multiplication. If the multiplication operation is not commutative, we call it a skew field or division ring. If there are non-zero elements which don't have multiplicative inverses, we have a ring.

In Combinatorics the fields of greatest interest are finite. A finite
field must have size equal to a power of a prime, and are usually
denoted *GF(p)* (for Galois Field). The finite field with
two elements, *GF(2)*, is of particular importance. *GF(2)*
consists of the two elements 0 and 1, connected by the operations +
and * as in the following tables:

+ | 0 1 * | 0 1 --+-------- ---+-------- 0 | 0 1 0 | 0 0 | | 1 | 1 0 1 | 0 1

A technique for analysing sequences, from a given sequence *f(n)*, a new
sequence *df(n)* is formed which consists of the difference of successive
terms of *f(n)*. This process is iterated and if eventually the resulting
sequence is constant, the original sequence can be reconstructed from
the finite differences evaluated at some fixed *n*.

For instance the perfect cubes have the following finite difference table:

0 1 8 27 64 125 216 343 512 729 1000 . . . 1 7 19 37 61 91 127 169 217 271 . . . 6 12 18 24 30 36 42 48 54 . . . 6 6 6 6 6 6 6 6 . . .

The cubes could be reconstructed from the numbers (0,1,6,6) using only addition.

In a polytope or simplicial complex, a collection of faces, one of each possible dimension, all having a common nonempty intersection.

In R^{d}, a set formed either by translating an affine subspace
or by intersecting a collection of hyperplanes.

An (undirected) graph without cycles. Hence it is a union of trees.

A classical problem asking if it is possible to always color a map with just four colors and never have countries that border one another be colored the same. The question has been settled in the affirmative by computer.

A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z

An On-line Dictionary of Combinatorics