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All of the area formulas for general convex quadrilaterals apply to parallelograms. Further formulas are specific to parallelograms: Further formulas are specific to parallelograms: A parallelogram with base b and height h can be divided into a trapezoid and a right triangle , and rearranged into a rectangle , as shown in the figure to the left.
Vectors involved in the parallelogram law. In a normed space, the statement of the parallelogram law is an equation relating norms: ‖ ‖ + ‖ ‖ = ‖ + ‖ + ‖ ‖,.. The parallelogram law is equivalent to the seemingly weaker statement: ‖ ‖ + ‖ ‖ ‖ + ‖ + ‖ ‖, because the reverse inequality can be obtained from it by substituting (+) for , and () for , and then simplifying.
4 parallelograms The parallelepiped with O h symmetry is known as a cube , which has six congruent square faces. The parallelepiped with D 4h symmetry is known as a square cuboid , which has two square faces and four congruent rectangular faces.
In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that defines a lattice in the complex plane. This type of lattice is the underlying object with which elliptic functions and modular forms are defined. Fundamental parallelogram defined by a pair of vectors in the complex plane.
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into () or 1 / 2 m(m − 1) parallelograms. These tilings are contained as subsets of vertices, edges and faces in orthogonal projections m -cubes . [ 7 ]
As affine geometry deals with parallel lines, one of the properties of parallels noted by Pappus of Alexandria has been taken as a premise: [9] [10] Suppose A, B, C are on one line and A', B', C' on another. If the lines AB' and A'B are parallel and the lines BC' and B'C are parallel, then the lines CA' and C'A are parallel.
Based on ancient Greek methods, an axiomatic system is a formal description of a way to establish the mathematical truth that flows from a fixed set of assumptions. Although applicable to any area of mathematics, geometry is the branch of elementary mathematics in which this method has most extensively been successfully applied.
Manifolds whose holonomy groups are proper subgroups of O(n) or SO(n) have special properties. One of the earliest fundamental results on Riemannian holonomy is the theorem of Borel & Lichnerowicz (1952), which asserts that the restricted holonomy group is a closed Lie subgroup of O(n). In particular, it is compact.