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 Rd, 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.