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Isoperimetry literally means "having an equal perimeter". In mathematics, isoperimetry is the general study of geometric figures having equal boundaries. The perimeter is the distance around a given two-dimensional object. ...
Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ...
The isoperimetric problem in the plane The classical isoperimetric problem dates back to antiquity. The problem can be stated as follows: Among all closed curves in the plane of fixed perimeter, which curve (if any) maximizes the area of its enclosed region? This question can be shown to be equivalent to the following problem: Among all closed curves in the plane enclosing a fixed area, which curve (if any) minimizes the perimeter? In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...
This problem is conceptually related to the principle of least action in physics, in that it can be restated: what is the principle of action which encloses the greatest area, with the greatest economy of effort? The 15th-century philosopher and scientist, Cardinal Nicholas of Cusa, considered rotational action, the process by which a circle is generated, to be the most direct reflection, in the realm of sensory impressions, of the process by which the universe is created. German astronomer and astrologer Johannes Kepler invoked the isoperimetric principle in discussing the morphology of the solar system, in Mysterium Cosmographicum (The Sacred Mystery of the Cosmos, 1596). The principle of least action was first formulated by Pierre-Louis Moreau de Maupertuis, who said that Nature is thrifty in all its actions. See action (physics). ...
Physics (from the Greek, (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space and time. ...
Nicholas of Cusa Nicholas of Cusa (1401 – August 11, 1464) was a German cardinal of the Catholic Church, a philosopher, jurist, mathematician, and an astronomer. ...
A sphere rotating around its axis. ...
Circle illustration In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. ...
Johannes Kepler Johannes Kepler (December 27, 1571 – November 15, 1630), a key figure in the scientific revolution, was a German mathematician, astronomer, astrologer, and an early writer of science fiction stories. ...
Mysterium Cosmographicum, (The Sacred Mystery of the Cosmos [Explained]) (alternately translated Cosmic Mystery, The Secret of the World or some variation) is an astronomy book by the German astronomer Johannes Kepler, published at Tübingen in 1596. ...
Although the circle appears to be an obvious solution to the problem, proving this fact is rather difficult. The first progress toward the solution was made by Swiss geometer Jakob Steiner in 1838, using a geometric method later named Steiner symmetrisation.[citation needed] Steiner showed that if a solution existed, then it must be the circle. Steiner's proof was completed later by several other mathematicians. Jakob Steiner (18 March 1796 – April 1, 1863) was a Swiss mathematician. ...
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Steiner begins with some geometric constructions which are easily understood; for example, it can be shown that any closed curve enclosing a region that is not fully convex can be modified to enclose more area, by "flipping" the concave areas so that they become convex. It can further be shown that any closed curve which is not fully symmetrical can be "tilted" so that it encloses more area. The one shape that is perfectly convex and symmetrical is the circle, although this, in itself, does not represent a rigorous proof of the isoperimetric theorem (see external links). Look up Convex set in Wiktionary, the free dictionary. ...
The theorem is usually stated in the form of an inequality that relates the perimeter and area of a closed curve in the plane. If P is the perimeter of the curve and A is the area of the region enclosed by the curve, then the inequality states that The feasible regions of linear programming are defined by a set of inequalities. ...
 For the case of a circle of radius r, we have A = πr2 and P = 2πr, and substituting these into the inequality shows that the circle does indeed maximize the area among all curves of fixed perimeter. In fact, the circle is the only curve that maximizes the area.
There are dozens of proofs of this classic inequality. Several of these are discussed in the Treiberg paper below. In 1901, Hurwitz gave a purely analytic proof of the classical isoperimetric inequality based on Fourier series and Green's theorem. Hurwitz is the last name of several famous people. ...
In structural proof theory, an analytical proof is a proof whose structure is simple in a special way. ...
The Fourier series is a mathematical tool used for analyzing an arbitrary periodic function by decomposing it into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ...
In physics and mathematics, Greens theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. Greens theorem was named after British scientist George Green and is a special two-dimensional case of...
Modern formulations of isoperimetric problems are sometimes given in terms of sub-Riemannian geometry; Dido's problem specifically finds expression in terms of the Heisenberg group: given an arc connecting two points, the "height" z of a point in the Heisenberg group corresponds to the area subtended by the arc. In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. ...
In mathematics, the Heisenberg group is a group of 3×3 upper triangular matrices of the form Elements a,b,c can be taken from some (arbitrary) commutative ring. ...
The isoperimetric theorem generalises to higher dimensional spaces: the domain with volume 1 with the minimal surface area is always a ball. Attempting to understand the nature of space has always been a prime occupation for philosophers and scientists. ...
This article explains the meaning of area as a Physical quantity. ...
See also In mathematics, the isoperimetric dimension of a manifold is a notion of dimension that tries to capture how the large scale behavior of the manifold resembles that of a Euclidean space (unlike the topological dimension or the Hausdorff dimension which compare different local behaviors against those of the Euclidean space). ...
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