A diagonal can refer to a line joining two nonconsecutive vertices of a polygon or polyhedron, or in contexts any upward or downward sloping line. In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit What is a polyhedron? We can at least say that a polyhedron is built up from different kinds of element or entity each associated with a different number of dimensions The word "diagonal" derives from the Greek διαγώνιος (diagonios)^[1], used by both Strabo^[2] and Euclid^[3] to refer to a line connecting two vertices of a rhombus or cuboid,^[4] and is formed from dia- ("through", "across") and gonia ("angle", related to gony "knee. Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly In Geometry, a rhombus (from Ancient Greek ῥόμβος - rrhombos “rhombus spinning top” (plural rhombi or rhombuses In anatomy the Cuboid bone is a bone in the foot See also Hyperrectangle Oblong "), later adopted into Latin as diagonus ("slanting line").

In mathematics, in addition to its geometric meaning, a diagonal is also used in matrices to refer to a set of entries along a diagonal line. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally

Contents 1 Non mathematical uses 2 Polygons 3 Matrices 4 Geometry 5 See also 6 External links 7 References

Non mathematical uses

In engineering, a diagonal brace is a beam used to brace a rectangular structure (such as scaffolding) to withstand strong forces pushing into it; although called a diagonal, due to practical considerations diagonal braces are often not connected to the corners of the rectangle. Engineering is the Discipline and Profession of applying technical and scientific Knowledge and

Diagonal pliers are wire-cutting pliers defined by the cutting edges of the jaws intersects the joint rivet at an angle or "on a diagonal", hence the name. Diagonal pliers (or wire cutters or diagonal cutting pliers) are Pliers intended for the Cutting of Wire (they are generally not used

A diagonal lashing is a type of lashing used to bind spars or poles together applied so that the lashings cross over the poles at an angle. A lashing is an arrangement of Rope used to secure two or more items together in a somewhat rigid manner

In association football, the diagonal system of control is the method referees and assistant referees use to position themselves in one of the four quadrants of the pitch. Association football, more commonly known as football or soccer, is a Team sport played between two teams of eleven players and is widely considered The diagonal system of control is the system of positioning used by match officials ( referees and Assistant referees in association football (soccer

Polygons

As applied to a polygon, a diagonal is a line segment joining any two non-consecutive vertices. In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points Therefore, a quadrilateral has two diagonals, joining opposite pairs of vertices. In Geometry, a quadrilateral is a Polygon with four sides or edges and four vertices or corners. For any convex polygon, all the diagonals are inside the polygon, but for re-entrant polygons, some diagonals are outside of the polygon. A re-entrant, or concave Polygon is one in which at least one interior Angle is more than 180 degrees (i

Any n-sided polygon (n ≥ 3), convex or concave, has

diagonals, as each vertex has diagonals to all other vertices except itself and the two adjacent vertices, or n − 3 diagonals.

Matrices

In the case of a square matrix, the main or principal diagonal is the diagonal line of entries running from the top-left to bottom-right corners. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally For example, the identity matrix can be defined as having entries of 1 on the main diagonal, and 0s elsewhere. In Linear algebra, the identity matrix or unit matrix of size n is the n -by- n Square matrix with ones on the Main The top-right to bottom-left diagonal is sometimes described as the minor diagonal or antidiagonal. A superdiagonal entry is one that is directly above and to the right of the main diagonal. If otherwise unqualified, it refers to the one adjacent to the main diagonal. Likewise, a subdiagonal entry is one that is directly below and to the left of the main diagonal. A diagonal matrix is one whose off-diagonal entries are all zero. In Linear algebra, a diagonal matrix is a Square matrix in which the entries outside the Main diagonal (↘ are all zero

Geometry

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 identity relation. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. 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 Mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function

In geometric studies, the idea of intersecting the diagonal with itself is common, not directly, but by perturbing it within an equivalence class. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X This is related at a deep level with the Euler characteristic and the zeros of vector fields. In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. For example, the circle S¹ has Betti numbers 1, 1, 0, 0, 0, and therefore Euler characteristic 0. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Algebraic topology, the Betti number of a Topological space is in intuitive terms a way of counting the maximum number of cuts that can be made without dividing 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 Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar 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. In Mathematics, the Lefschetz fixed-point theorem is a formula that counts the number of fixed points of a continuous mapping from a compact

See also

• Jordan normal form
• Main diagonal
• Diagonal functor
• Face diagonal
• Space diagonal

External links

• Diagonals of a polygon with interactive animation
• Polygon diagonal from MathWorld. In Linear algebra, Jordan normal form (often called Jordan canonical form)shows that a given square matrix M over a field K In Linear algebra, the main diagonal (sometimes leading diagonal or primary diagonal) of a matrix A is the collection of cells A_{ij} In Category theory, for any object a in any category C where the product a × a exists there exists the diagonal morphism In Geometry, a face diagonal of a Polyhedron is a Diagonal on one of the faces, in contrast to a Space diagonal passing through the In a rectangular box or a Magic cube, the four space diagonals are the lines that go from a corner of the box or cube through the center of the box or cube to the MathWorld is an online Mathematics reference work created and largely written by Eric W
• Diagonal of a matrix from MathWorld. MathWorld is an online Mathematics reference work created and largely written by Eric W
• Diagonal media from Diagonal.

References

1. ^ Online Etymology Dictionary
2. ^ Strabo, Geography 2. 1. 36-37
3. ^ Euclid, Elements book 11, proposition 28
4. ^ Euclid, Elements book 11, proposition 38

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