Diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word diagonal derives from the ancient Greek διαγώνιος diagonios,[1] "from angle to angle" (from διά- dia-, "through", "across" and γωνία gonia, "angle", related to gony "knee"); it was used by both Strabo[2] and Euclid[3] to refer to a line connecting two vertices of a rhombus or cuboid,[4] and later adopted into Latin as diagonus ("slanting line").
For the avenue in Barcelona, see Avinguda Diagonal. For the Spanish newspaper, see Diagonal (newspaper).In matrix algebra, the diagonal of a square matrix consists of the entries on the line from the top left corner to the bottom right corner. There are also many other non-mathematical uses.
Higher dimensions[edit]
N-Cube[edit]
The lengths of an n-dimensional hypercube's diagonals can be calculated by mathematical induction. The longest diagonal of an n-cube is . Additionally, there are of the xth shortest diagonal. As an example, a 5-cube would have the diagonals:
Its total number of diagonals is 416. In general, an n-cube has a total of diagonals. This follows from the more general form of which describes the total number of face and space diagonals in convex polytopes.[9] Here, v represents the number of vertices and e represents the number of edges.
Geometry[edit]
By analogy, the subset of the Cartesian product X×X of any set X with itself, consisting of all pairs (x,x), is called the diagonal, and is the graph of the equality relation on X or equivalently the graph of the identity function from X to X. This plays an important part in geometry; for example, the fixed points of a mapping F from X to itself may be obtained by intersecting the graph of F with the diagonal.
In geometric studies, the idea of intersecting the diagonal with itself is common, not directly, but by perturbing it within an equivalence class. This is related at a deep level with the Euler characteristic and the zeros of vector fields. For example, the circle S1 has Betti numbers 1, 1, 0, 0, 0, and therefore Euler characteristic 0. A geometric way of expressing this is to look at the diagonal on the two-torus S1xS1 and observe that it can move off itself by the small motion (θ, θ) to (θ, θ + ε). In general, the intersection number of the graph of a function with the diagonal may be computed using homology via the Lefschetz fixed-point theorem; the self-intersection of the diagonal is the special case of the identity function.