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Squaring The Circle

Squaring the circle

Squaring the circle is the problem proposed by ancient Greek geometers of using a finite ruler-and-compass construction to make a square with the same area as a given circle.

"[The Greek Oenopides] may have been the first to lay down the restriction of the means permissible in constructions as the ruler and compass which became a canon of Greek geometry for all plane constructions."

In 1882, it was proved to be impossible to square a circle using only a straightedge and compass. The term quadrature of the circle is synonymous.

Impossibility

The problem dates back to the invention of geometry and has occupied mathematicians for millennia. It was not until 1882 that the impossibility was proved rigorously, though even the ancient geometers had a very good practical and intuitive grasp of its intractability. It should be noted that it is the limitation to just compass and straightedge that makes the problem difficult. If the straightedge is allowed to be a ruler or if other simple instruments, for example something which can draw an Archimedean spiral, are allowed, then it is not difficult to draw a square and circle of equal area.

Transcendence of π

A solution of the problem of squaring the circle by straightedge and compass demands construction of the number

\sqrt

, and the impossibility of this undertaking follows from the fact that π (pi) is a transcendental number—that is, it is non-algebraic and therefore a non-constructible number. The transcendence of π was proved by Ferdinand von Lindemann in 1882. If you solve the problem of the quadrature of the circle, this means you have also found an algebraic value of π, which is impossible. It is possible to construct a square with an area arbitrarily close to that of a given circle. If a rational number is used as an approximation of π, then squaring the circle becomes possible, depending on the values chosen. However, this is only an approximation and does not meet the constraints of the ancient rules for solving the problem. Several mathematicians have demonstrated workable procedures based on a variety of approximations. Bending the rules by allowing an infinite number of ruler-and-compass constructions or by performing the operations on certain non-Euclidean spaces also makes squaring the circle possible. For example, although the circle cannot be squared in Euclidean space, it can in Gauss-Bolyai-Lobachevsky space.

"Squaring the circle" as a metaphor

The mathematical proof that the quadrature of the circle is impossible has not proved to be a hindrance to the many people who have invested years in this problem anyway; having squared the circle is a famous crank assertion. (See also pseudomathematics.) The futility of undertaking exercises aimed at finding the quadrature of the circle has brought this term into use in totally unrelated contexts, where it is simply used to mean a hopeless, meaningless, or vain undertaking. Aleister Crowley used the metaphor in a different sense, to represent the goal of magick and mysticism. He implicitly associated his system of Thelema with pi. For more information, see Abrahadabra.

References

External links

- [http://www.cut-the-knot.org/impossible/sq_circle.shtml Squaring the Circle] at cut-the-knot
- [http://mathworld.wolfram.com/CircleSquaring.html Circle Squaring] from MathWorld, includes information on procedures based on various approximations of π
- [http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Squaring_the_circle.html Squaring the circle] from the MacTutor archive Category:Pi Category:Ruler-and-compass constructions Category:Recreational mathematics

Greek mathematics

Greek mathematics, as that term is used in this article, is the mathematics developed from the 6th century BC to the 5th century AD around the shores of the Mediterranean. It constitutes a major period of the history of mathematics, fundamental in respect of geometry and the idea of formal proof. Greek mathematics also contributed importantly to ideas on number theory, mathematical analysis, applied mathematics, and, at times, approached close to integral calculus. Mathematical developments took place in Greek-speaking centres as far apart as Sicily and Egypt, and with a high estimation of the intellectual and cultural status of mathematics (for example in the school of Plato).

Origins

Greek mathematics has origins that are presumed to go back to the 7th century BC, but are not easily documented. It is generally believed that it built on the computational methods of earlier Babylonian and Egyptian mathematics, and it may well have had Phoenician influences. Some of the most well-known figures in Greek mathematics are Pythagoras, a shadowy figure from the isle of Samos associated partly with number mysticism and numerology, but more commonly with his theorem, and Euclid, who is known for his Elements, a canon of geometry for centuries. The Sand Reckoner by Archimedes bespeaks a man who made major discoveries, and whose originality and accomplishments are commonly reckoned to be on par with those of Isaac Newton and C. F. Gauss. The most characteristic product of Greek mathematics may be the theory of conic sections, largely developed in the Hellenistic period. The methods used made no explicit use of algebra, nor trigonometry. Those were formulated in the way understood in contemporary mathematics as major parts of Islamic mathematics; the texts of Greek mathematics were for the most part preserved and transmitted in the Islamic world. Among the foremost modern historians of Greek mathematics was Thomas Heath.

Famous Greek mathematicians

- Anaxagoras
- Anthemius of Tralles (conic sections)
- Antiphon (squaring the circle)
- Apollonius of Tyana
- Archimedes
- Archytas
- Aristaeus
- Aristarchus
- Aristotle
- Autolycus of Pitane
- Chrysippus
- Conon
- Democritus
- Diocles (Cissoid of Diocles)
- Diophantus (Diophantine equations)
- Eratosthenes (Sieve of Eratosthenes)
- Euclid (Euclid's Elements)
- Eudoxus of Cnidus (method of exhaustion)
- Heraclides of Pontus
- Heron (Heron's formula)
- Hipparchus
- Hippias
- Hippocrates
- Hypatia
- Leucippus
- Menelaus of Alexandria (Menelaus theorem)
- Nicomachus
- Pappus (Pappus's centroid theorem, Pappus's hexagon theorem)
- Perseus
- Plato
- Porphyry
- Posidonius
- Proclus
- Ptolemy (Ptolemaios' theorem, Almagest)
- Pythagoras (Pythagorean theorem)
- Simplicius of Cilicia
- Thales (Thales' theorem)
- Theaetetus
- Theano
- Theodosius
- Theon of Alexandria
- Xenocrates
- Zeno of Elea (Zeno's paradoxes)

External links

- [http://www-history.mcs.st-andrews.ac.uk/history/Indexes/Greek_index.html Index of Greek Mathematicians] Category:History of mathematics

Square (geometry)

In plane geometry, a square is a polygon with four equal sides and equal angles. Those angles are then necessarily right angles. Squares are regular quadrilaterals, rectangles, rhombi, kites, parallelograms, and isosceles trapezoids/isosceles trapezia. The diagonals of a square are equal and conversely, if the diagonals of a rhombus are proven to be equal, then that rhombus must be a square. The coordinates for the vertices of a square centered at the origin and with side length 2 are (±1, ±1), while the interior of the same consists of all points (x₀, x₁) with -1 < x_i < 1.

External links

- [http://agutie.homestead.com/files/triangle_square0.htm Triangle with two squares ] by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas" Category:Quadrilaterals ko:정사각형 ja:正方形 simple:Square th:รูปสี่เหลี่ยมจัตุรัส

Circle

:This article is about the shape and mathematical concept of circle; for other meanings, see Circle (disambiguation). Circle (disambiguation) In Euclidean geometry, a circle is the set of all points at a fixed distance, called the radius, from a fixed point, called the centre (center). The points can only be those that are part of a conic section; within the set of a plane bisecting a cone. Circles are simple closed curves, dividing the plane into an interior and exterior. Sometimes the word circle is used to mean the interior, with the circle itself called the circumference(C). Usually however, the circumference means the length of the circle, and the interior of the circle is called a disk or disc. An arc is any continuous portion of a circle.

Mathematical definitions

In an x-y coordinate system, the circle with centre (a, b) and radius r is the set of all points (x, y) such that :

\left( x - a \right)^2 + \left( y - b \right)^2=r^2.

If the circle is centered at the origin (0, 0), then this formula can be simplified to :

x^2 + y^2 = r^2.

The circle centered at the origin with radius 1 is called the unit circle. Expressed in parametric equations, (x, y) can be written as :x = a + r cos(t) :y = b + r sin(t). The slope a circle at a point (x, y) can be expressed with the following formula, assuming the center is at the origin and (x, y) is on the circle: :

y' = - \frac.

In the complex plane, a circle with a center at c and radius r has the equation

|z-c|^2 = r^2

. Since

|z-c|^2 = z\overline-\overlinez-c\overline+c\overline

, the slightly generalized equation

pz\overline + gz + \overline = q

for real p, q and complex g is sometimes called a generalized circle. It is important to note that not all generalized circles are actually circles. All circles are similar; as a consequence, a circle's circumference and radius are proportional, as are its area and the square of its radius. The constants of proportionality are 2π and π, respectively. In other words:
- Length of a circle's circumference =

2\pi \times r.

- Area of a circle =

\pi \times r^2.

The formula for the area of a circle can be derived from the formula for the circumference and the formula for the area of a triangle, as follows. Imagine a regular hexagon (six-sided figure) divided into equal triangles, with their apices at the center of the hexagon. The area of the hexagon may be found by the formula for triangle area by adding up the lengths of all the triangle bases (on the exterior of the hexagon), multiplying by the height of the triangles (distance from the middle of the base to the center) and dividing by two. This is an approximation of the area of a circle. Then imagine the same exercise with an octagon (eight-sided figure), and the approximation is a little closer to the area of a circle. As a regular polygon with more and more sides is divided into triangles and the area calculated from this, the area becomes closer and closer to the area of a circle. In the limit, the sum of the bases approaches the circumference 2πr, and the triangles' height approaches the radius r. Multiplying the circumference and radius and dividing by 2, we get the area, π r².

Properties

limit limit

Chord properties

- Chords equidistant from the centre of a circle are equal.
- Equal chords are equidistant from the centre.
- A line from the centre, perpendicular to a chord, bisects the chord.
- The line segment drawn from the centre to the midpoint of the chord is perpendicular to the chord.
- The perpendicular bisector of a chord passes through the centre of a circle.

Tangent properties

- The line drawn perpendicular to the end point of a radius is a tangent to the circle.
- A line drawn perpendicular to a tangent at the point of contact with a circle passes through the centre of the circle.
- Tangents drawn from a point outside the circle are equal in length.
- Two tangents can always be drawn from a point outside of the circle.

Inscribed angle theorem

- If a central angle and an inscribed angle of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed angle.
- If two angles are inscribed on the same chord and on the same side of the chord , then they are equal.
- An inscribed angle subtended by a semicircle is a right angle.
- For a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.

Secant, tangent, and chord properties

- The chord theorem states that if two chords, CD and EF, intersect at G, then

CG \times DG = EG \times FG

. (Chord Theorem)
- If a tangent from an external point D meets the circle at C and a secant from the external point D meets the circle at G and E respectively, then

DC^2 = DG \times DE

. (Tangent Secant Theorem)
- If two secants, DG and DE, also cut the circle at H and F respectively, then

DH \times DG = DF \times DE

. (Corollary of the Tangent Secant Theorem)
- The angle between a tangent and chord is equal to the subtended angle on the opposite side of the chord. (Tangent Chord Property)
- If the angle subtended by the chord at the centre is 90 degrees then l = sqrt(2)
- r, where l is the length of the chord and r is the radius of the circle.

External links

- [http://agutie.homestead.com/files/clifford1.htm Clifford's Circle Chain Theorems.] This is a step by step presentation of the first theorem. Clifford discovered, in the ordinary Euclidean plane, a "sequence or chain of theorems" of increasing complexity, each building on the last in a natural progression. by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
- [http://www.cut-the-knot.org/pythagoras/Munching/circle.shtml Munching on Circles] at cut-the-knot Category:Conic sections ja:円 (数学) simple:Circle

Numerical integration

In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of definite integrals. The term quadrature is more or less a synonym for numerical integration, especially as applied to one-dimensional integrals. Two-dimensional integration is sometimes described as cubature, although this term is much less frequently used and the meaning of quadrature is understood for higher dimensional integration as well. The basic problem considered by numerical integration is to compute an approximate solution to a definite integral: :

\int_a^b f(x)\, dx

This problem can also be stated as an initial value problem for an ordinary differential equation, as follows. :

y'(x) = f(x), \quad y(a) = 0

Finding y(b) is equivalent to computing the integral. Methods developed for ordinary differential equations, such as the Runge-Kutta method, can be applied to the restated problem. In the remainder of this article, we shall discuss methods developed specifically for the problem stated as a definite integral.

Reasons for numerical integration

There are several reasons for carrying out numerical integration. The integrand f may be known only at certain points, such as obtained by sampling. Some embedded systems and other computer applications may need numerical integration for this reason. A formula for the integrand may be known, but it may be difficult or impossible to find an antiderivative. An example of such an integrand is exp(-t²). It may be possible to find an antiderivative symbolically, but it may be easier to compute a numerical approximation than to compute the antiderivative. That may be the case if the antiderivative is given as an infinite series or product, or if its evaluation requires a special function which is not available.

Methods for one-dimensional integrals

Numerical integration methods can generally be described as combining evaluations of the integrand to get an approximation to the integral. An important part of the analysis of any numerical integration method is to study the behavior of the approximation error as a function of the number of integrand evaluations. A method which yields a small error for a small number of evaluations is usually considered superior. Reducing the number of evaluations of the integrand reduces the number of arithmetic operations involved, and therefore reduces the total round-off error. Also, each evaluation takes time, and the integrand may be arbitrarily complicated. It should be noted, however, that a 'brute force' kind of numerical integration can always be done, in a very simplistic way, by evaluating the integrand with very small increments.

Quadrature rules based on interpolating functions

A large class of quadrature rules can be derived by constructing interpolating functions which are easy to integrate. Typically these interpolating functions are polynomials. The simplest method of this type is to let the interpolating function be a constant function (a polynomial of order zero) which passes through the point ((a+b)/2, f((a+b)/2)). This is called the midpoint rule or rectangle rule. :

\int_a^b f(x)\,dx \approx (b-a) \, f\left(\frac\right).

The interpolating function may be an affine function (a polynomial of degree 1) which passes through the points (a, f(a)) and (b, f(b)). This is called the trapezoidal rule. :

\int_a^b f(x)\,dx \approx (b-a) \, \frac.

For either one of these rules, we can make a more accurate approximation by breaking up the interval [a, b] into some number n of subintervals, computing an approximation for each subinterval, then adding up all the results. This is called a composite rule, extended rule, or iterated rule. For example, the composite trapezoidal rule can be stated as :

\int_a^b f(x)\,dx \approx \frac \left(  + \sum_^ f \left( a+k \frac \right) \right)

where the subintervals have the form [k h, (k+1) h], with h = (b-a)/n and k = 0, 1, 2, ..., n-1. Interpolation with polynomials evaluated at equally-spaced points in [a, b] yields the Newton-Cotes formulas, of which the rectangle rule and the trapezoidal rule are examples. Simpson's rule, which is based on a polynomial of order 2, is also a Newton-Cotes formula. If we allow the intervals between interpolation points to vary, we find another group of quadrature formulas, called Gaussian quadrature formulas. A Gaussian quadrature rule is typically more accurate than a Newton-Cotes rule which requires the same number of function evaluations, if the integrand is smooth (i.e., if it has many derivatives.)

Adaptive algorithms

If f does not have many derivatives at all points, or if the derivatives become large, then Gaussian quadrature is often insufficient. In this case, an algorithm similar to the following will perform better: // This algorithm calculates the definite integral of a function // from 0 to 1, adaptively, by choosing smaller steps near // problematic points. // Set initial_h to the initial step size. x:=0 h:=initial_h accumulator:=0 WHILE x<1.0 DO IF x+h>1.0 THEN h=1.0-x END IF IF error from quadrature over [x,x+h] for f is too large THEN Make h smaller ELSE accumulator:=accumulator + quadrature of f over [x,x+h] x:=x+h IF error from quadrature over [x,x+h] is very small THEN Make h larger END IF END IF END WHILE RETURN accumulator Some details of the algorithm require careful thought. For many cases, estimating the error from quadrature over an interval for a function f isn't obvious. One popular solution is to use two different rules of quadrature, and use their difference as an estimate of the error from quadrature. The other problem is deciding what "too large" or "very small" signify. A possible criterion for "too large" is that the quadrature error should not be larger than th where t, a real number, is the tolerance we wish to set for global error. Then again, if h is already tiny, it may not be worthwhile to make it even smaller even if the quadrature error is apparently large. This type of error analysis is usually called "a posteriori" since we compute the error after having computed the approximation. Heuristics for adaptive quadrature are discussed by Forsythe et al. (Section 5.4).

Extrapolation methods

The accuracy of a quadrature rule of the Newton-Cotes type is generally a function of the number of evaluation points. The result is usually more accurate as number of evaluation points increases, or, equivalently, as the width of the step size between the points decreases. It is natural to ask what the result would be if the step size were allowed to approach zero. This can be answered by extrapolating the result from two or more nonzero step sizes (see Richardson extrapolation). The extrapolation function may be a polynomial or rational function. Extrapolation methods are described in more detail by Stoer and Bulirsch (Section 3.4).

Conservative (a priori) error estimation

Let f have a bounded first derivative over [a,b]. The mean value theorem for f, where

x < b

, gives :

(x - a) f'(y_x) = f(x) - f(a)\,

for some y_x in [a,x] depending on x. If we integrate in x from a to b on both sides and the take absolute values, we obtain :

\left| \int_a^b f(x)\,dx - (b - a) f(a) \right|
  = \left| \int_a^b (x - a) f'(y_x)\, dx \right|

We can further approximate the integral on the right-hand side by bringing the absolute value into the integrand, and replacing the term in f' by an upper bound: :

\left| \int_a^b f(x)\,dx - (b - a) f(a) \right| \leq  \sup_ \left| f'(x) \right|

(
- ) (See supremum.) Hence, if we approximate the integral ∫_a^bf(x)dx by the quadrature rule (b-a)f(a) our error is no greater than the right hand side of (
- ). We can convert this into an error analysis for the Riemann sum (
- ), giving an upper bound of :

\sup_ \left| f'(x) \right|

for the error term of that particular approximation. (Note that this is precisely the error we calculated for the example

f(x) = x

.) Using more derivatives, and by tweaking the quadrature, we can do a similar error analysis using a Taylor series (using a partial sum with remainder term) for f. This error analysis gives a strict upper bound on the error, if the derivatives of f are available. This integration method can be combined with interval arithmetic to produce computer proofs and verified calculations.

Multidimensional integrals

The quadrature rules discussed so far are all designed to compute one-dimensional integrals. To compute integrals in multiple dimensions, one approach is to phrase the multiple integral as repeated one-dimensional integrals by appealing to Fubini's theorem. This approach requires the function evaluations to grow exponentially as the number of dimensions increases. Two methods are known to overcome this so-called curse of dimension.

Monte Carlo

Monte Carlo methods and quasi-Monte Carlo methods are easy to apply to multi-dimensional integrals, and may yield greater accuracy for the same number of function evaluations than repeated integrations using one-dimensional methods. A large class of useful Monte Carlo methods are the so-called Markov chain Monte Carlo algorithms, which include the Metropolis-Hastings algorithm and Gibbs sampling.

Sparse grids

Sparse grids were originally developed by Smolyak for the quadrature of high dimensional functions. The method is always based on a one dimensional quadrature rule, but performs a more sophisticated combination of univariate results.

Software for numerical integration

Numerical integration is one of the most intensively studied problems in numerical analysis. Of the many software implementations we list a few here.
- QUADPACK (part of SLATEC): description [http://www.netlib.org/slatec/src/qpdoc.f], source code [http://www.netlib.org/slatec/src]. QUADPACK is a collection of algorithms, in Fortran, for numerical integration based on Gaussian quadrature.
- [http://www.gnu.org/software/gsl/ GSL]: The GNU Scientific Library (GSL) is a numerical library written in C which provides a wide range of mathematical routines, like Monte Carlo integration. Numerical integration algorithms are found in GAMS class [http://gams.nist.gov/serve.cgi/Class/H2 H2].

References

- George E. Forsythe, Michael A. Malcolm, and Cleve B. Moler. Computer Methods for Mathematical Computations. Englewood Cliffs, NJ: Prentice-Hall, 1977. (See Chapter 5.)
- William H. Press, Brian P. Flannery, Saul A. Teukolsky, William T. Vetterling. Numerical Recipes in C. Cambridge, UK: Cambridge University Press, 1988. (See Chapter 4.)
- Josef Stoer and Roland Bulirsch. Introduction to Numerical Analysis. New York: Springer-Verlag, 1980. (See Chapter 3.) Category:Numerical analysis

Compass (drafting)

:This article is about the drafting instrument. For other senses, see compass (disambiguation). compass (disambiguation) A compass (or pair of compasses) is a technical drawing instrument used by mathematicians and craftspeople in for drawing or inscribing a circle or arc. In geometry, ruler-and-compass constructions often involve only theoretical conceptual devices rather than physically drawn circles.

External links

- [http://museum.nist.gov/object.asp?ObjID=9 beam or trammel compass] (variant form); [http://www.leevalley.com/wood/page.aspx?p=32629&c=1 another example] Category:Mathematical tools Category:Technical drawing ko:컴퍼스 ja:コンパス

Ruler

:This article is about the drawing instrument. Ruler can also refer to a statesman in charge or ceremonial head of state of a country or minor politically significant principality; for this meaning see Monarch or Lists of incumbents. Lists of incumbents A ruler is an instrument used in geometry and technical drawing to measure short distances and/or to rule straight lines. Strictly speaking, the ruler is the instrument used to rule and calibrated stick for measurement is called a measure. However, common usage is that a ruler is calibrated so that it can measure, creating ambiguity in what a ruler is allowed to do in ruler-and-compass constructions. For instance, a ruler with measurement capability (e.g. its own length) can be used for angle trisection. This is resolved by referring to an instrument that can only rule as a straightedge. Practical rulers have distance markings along their edges. How these distance markings are applied and calibrated should be described here, including a history of old methods.
- A ruler makes a popular tool for use as a pervertible, i.e. for corporal punishment, usually on the hands (either outstretched palms or knuckles) or thighs. Generally, solid, heavy wooden or metal rulers are used for this purpose, but lighter, often flexible plastic ones can also be used punitively, especially in spanking.
- The ruler (calibrated, though numbers are not shown) appears as a charge in heraldry in the arms of Odouze.

External links

- [http://www.markus-bader.de/MB-Ruler/ Virtual Screen Ruler] Category:Dimensional instruments Category:Metalworking measuring instruments Category:Spanking implements ja:定規 th:ไม้บรรทัด

Archimedean spiral

An Archimedean spiral (also arithmetic spiral) is a curve which in polar coordinates (r, θ) can be described by the equation :

\, r=a+b\theta

with real numbers a and b. Changing the parameter a will turn the spiral, while b controls the distance between the arms. This Archimedean spiral is distinguished from the logarithmic spiral by the fact that successive turnings of the spiral have a constant separation distance (equal to 2πb if θ is measured in radians), while in a logarithmic spiral these distances form a geometric progression. Note that the Archimedean spiral has two arms, one for θ > 0 and one for θ < 0. The two arms are smoothly connected at the origin. Only one arm is shown on the accompanying graph. Taking the mirror image of this arm across the y-axis will yield the other arm. One method of squaring the circle, by relaxing the strict limitations on the use of straightedge and compass in ancient Greek geometric proofs, makes use of an Archimedean spiral. Sometimes the term Archimedean spiral is used for the more general group of spirals :

r=a+b\theta^.

The normal Archimedean spiral occurs when x = 1. Other spirals falling into this group include the hyperbolic spiral, Fermat's spiral, and the lituus. Virtually all static spirals appearing in nature are logarithmic spirals, not Archimedean ones. Many dynamic spirals (such as the Parker spiral of the solar wind, or the pattern made by a St. Catherine's wheel) are Archimedean.

External links

- [http://mathworld.wolfram.com/ArchimedesSpiral.html MathWorld]
- [http://www-groups.dcs.st-and.ac.uk/~history/Java/Spiral.html Page with Java application to interactively explore the Archimedean spiral and its related curves] Category:Curves

Transcendental number

In mathematics, a transcendental number is any real number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. It follows that all transcendental numbers are irrational. However, not all irrational numbers are transcendental; √2 is irrational but is a solution of the polynomial x² - 2 = 0. The set of all transcendental numbers is uncountable. The proof is simple: Since the polynomials with integer coefficients are countable, and since each such polynomial has a finite number of zeroes, the set of algebraic numbers is countable. But the reals are uncountable; so the set of all transcendental numbers must also be uncountable. In a very real sense, then, there are many more transcendental numbers than algebraic ones. However, only a few classes of transcendental numbers are known and proving that a given number is transcendental can be extremely difficult. The existence of transcendental numbers was first proved in 1844 by Joseph Liouville, who exhibited examples, including the Liouville constant: :

\sum_^\infty 10^ = 0.110001000000000000000001000....

in which the nth digit after the decimal point is 1 if n is a factorial (i.e., 1, 2, 6, 24, 120, 720, ...., etc.) and 0 otherwise. The first number to be proved transcendental without having been specifically constructed to achieve this was e, by Charles Hermite in 1873. In 1882, Ferdinand von Lindemann published a proof that the number π is transcendental. In 1874, Georg Cantor found the argument described above establishing the ubiquity of transcendental numbers. See also Lindemann-Weierstrass theorem. Here is a list of some numbers known to be transcendental:
- e^a if a is algebraic and nonzero. In particular, e itself is transcendental.
- π
- e^π Gelfond's constant
- 2^√2, the Gelfond-Schneider constant, or more generally a^b where a ≠ 0,1 is algebraic and b is algebraic but not rational (Gelfond-Schneider theorem and Hilbert's seventh problem).
- sin(1)
- ln(a) if a is positive, rational and ≠ 1
- Γ(1/3), Γ(1/4), and Γ(1/6) (see gamma function).
- Ω, Chaitin's constant.
-

\sum_^\infty 10^;\qquad \beta > 1\; ,

:where

\beta\mapsto\lfloor \beta \rfloor

is the floor function. For example if β = 2 then this number is 0.11010001000000010000000000000001000… Any non-constant algebraic function of a single transcendental number is also transcendental. However, an algebraic function of several transcendental numbers may be algebraic if they are not algebraically independent: π and 1-π are both transcendental, but π+(1-π)=1 is obviously not. It is unknown whether π+e, for example, is transcendental, though at least one of π+e and π e must be transcendental. Indeed, for any two transcendental numbers a and b, both a+b and a b cannot be algebraic. Proof: consider the polynomial (x−a) (x−b) = x² − (a+b) x + a b. If (a+b) and a b were both algebraic, then this would be a polynomial with algebraic coefficients, and so its roots, a and b, would also be algebraic, by definition. But this is a contradiction. The discovery of transcendental numbers allowed the proof of the impossibility of several ancient geometric problems involving ruler-and-compass construction; the most famous one, squaring the circle, is impossible because π is transcendental.

Proof that $e$ is transcendental

The first proof that

e

is transcendental dates from 1873. We will now follow the strategy of David Hilbert (1862–1943) who gave a simplification of the original proof of Charles Hermite. The idea is the following: Suppose that

e

is algebraic. Then

e

is the solution of a non-zero polynomial equation with integers

a,a_,\ldots,a_

a+a_e+a_e^+\ldots+a_e^=0

(1) Define

\int^_

as follows:

\int^_=\int^_z^[(z-1)(z-2)\ldots(z-n)]^e^dz

(2) where

z^[(z-1)(z-2)\ldots(z-n)]^e^dz

is the product of the functions

[z(z-1)(z-2)\ldots(z-n)]^

and

(z-1)(z-2)\ldots(z-n)e^.

When we multiply (1) with (2) we obtain the following:

a\int^_+a_e\int^_+\ldots+a_e^\int^_

which can now be written in the form

P_+P_

where

P_=a\int^_+a_e\int^_+\ldots+a_e^\int^_

P_=a_e\int^_+a_e^\int^_+\ldots+a_e^\int^_

Now, the strategy is to prove that

\frac+\frac\neq0

. We hence prove that

\frac

is a nonzero integer and

\left|\frac\right|<1

. For proving that

\frac

is a nonzero integer we use the relation:

\int^_x^e^=k!

Showing that

\left|\frac\right|<1

requires - among other things - some straightforward estimates. Specifying

k

and making it sufficiently large finally leads to

\frac+\frac\neq0

. For proving that the number

\pi

is transcendental, we almost follow the same strategy. Besides the gamma-function and some estimates as in the proof for

e

, important facts about symmetric polynomials play a vital role in this proof. For detailed information concerning the proofs of the transcendence of

\pi

and

e

see the references and external links.

References

- David Hilbert, "Über die Transcendenz der Zahlen

e

und

\pi

", Mathematische Annalen 43:216–219 (1893).
- Michael Spivak, Calculus. New York, Amsterdam: W. A. Benjamin, Inc. (1967).

External links

- [http://www.mathematik.uni-muenchen.de/~fritsch/euler.pdf Proof that

e

is transcendental (PDF)]
- [http://www.mathematik.uni-muenchen.de/~fritsch/pi.pdf Proof that

\pi

is transcendental (PDF)]
- ko:초월수 ja:超越数 th:จำนวนอดิศัย

Constructible number

A point in the Euclidean plane is a constructible point if, given a fixed coordinate system (or a fixed line segment of unit length), one can construct the point with unruled straightedge and compass. A complex number is a constructible number if its corresponding point in the Euclidean plane is constructible from the usual x- and y-coordinate axes. Note that this is quite a distinct notion from Gödel's constructible universe, L; though every number that is constructible in the sense of this article is in L, the converse fails badly. It can then be shown that a real number is constructible if and only if, given a line segment of unit length, one can construct a line segment of length

|r|

with ruler and compass. It can also be shown that a complex number is constructible if and only if its real and imaginary parts are constructible. The set of constructible numbers can be completely characterized in the language of field theory. This has the effect of transforming geometric questions about ruler-and-compass constructions into algebra. This transformation leads to the solutions of many famous mathematical problems, which defied centuries of attack.

Geometric definitions

The geometric definition of a constructible point is as follows. First, for any two distinct points P and Q in the plane, let L(P, Q) denote the unique line through P and Q, and let C(P, Q) denote the unique circle with center P, passing through Q. (Note that the order of P and Q matters for the circle.) By convention, L(P, P) = C(P, P) = . Then a point Z is constructible from E, F, G and H if either #Z is in the intersection of L(E, F) and L(G, H), where L(E, F) ≠ L(G, H); #Z is in the intersection of C(E, F) and C(G, H), where C(E, F) ≠ C(G, H); #Z is in the intersection of L(E, F) and C(G, H). Since the order of E, F, G, and H in the above definition is irrelevant, the four letters may be permuted in any way. Put simply, Z is constructible from E, F, G and H if it lies in the intersection of any two distinct lines, or of any two distinct circles, or of a line and a circle, where these lines and/or circles can be determined by E, F, G, and H, in the above sense. Now, let A and A be any two distinct fixed points in the plane. A point Z is construtible if either #Z = A; #Z = A #there exist points P₁, ..., P_n, with Z = P_n, such that for all j ≥ 1, P_{j + 1} is constructible from points in the set . Put simply, Z is constructible if it is either A or A, or if it is obtainable from a finite sequence of points starting with A and A, where each new point is constructible from previous points in the sequence. The origin O is defined as follows. The circles C(A, A) and C(A, A) intersect in two distinct points; these points determine a unique line, and the origin O is defined to be the intersection of this line with L(A, A).

Transformation into algebra

All rational numbers are constructible, and all constructible numbers are algebraic numbers. Also, if a and b are constructible numbers with b ≠ 0, then a − b and a/b are constructible. Thus, the set K of all constructible complex numbers forms a field, a subfield of the field of algebraic numbers. Furthermore, K is closed under square roots and complex conjugation. These facts can be used to characterize the field of constructible numbers, because, in essence, the equations defining lines and circles are no worse than quadratic. The characterization is the following: a complex number is constructible if and only if it lies in a field at the top of a finite tower of quadratic extensions, starting with the rational field Q. More precisely, z is constructible if and only if there exists a tower of fields

\mathbb = K_0 \subseteq K_1 \subseteq \dots \subseteq K_n

where z is in K_n and for all 0 ≤ j < n, the dimension [K_{j + 1} : K_j] = 2.

Impossible constructions

The algebraic characterization of constructible numbers provides an important necessary condition for constructibility: if z is constructible, then it is algebraic, and its minimal irreducible polynomial has degree a power of 2, or equivalently, the field extension Q(z)/Q has dimension a power of 2. One should note that it is true, (but not obvious to show) that the converse is false — this is not a sufficient condition for constructibility. However, this defect can be remedied by considering the normal closure of Q(z)/Q. The nonconstructibility of certain numbers proves the impossibility of certain problems attempted by the philosophers of ancient Greece. In the following chart, each row represents a specific ancient construction problem. The left column gives the name of the problem. The second column gives an equivalent algebraic formulation of the problem. In other words, the solution to the problem is affirmative if and only if each number in the given set of numbers is constructible. Finally, the last column provides the simplest known counterexample. In other words, the number in the last column is an element of the set in the same row, but is not constructible.

Ruler-and-compass construction

A number of ancient problems in plane geometry involve the construction of lengths or angles using only an idealized ruler and compass, or more properly a straightedge and compass. The most famous ruler-and-compass problems have been proven impossible, in several cases by the results of Galois theory. In spite of these impossibility proofs, some mathematical novices persist in trying to solve these problems. Many of them fail to understand that many of these problems are trivially solvable provided that other geometric transformations are allowed: for example, squaring the circle is possible using geometric constructions, but not possible using ruler and compass alone. Mathematician Underwood Dudley has made a sideline of collecting false ruler-and-compass proofs, as well as other work by mathematical cranks, and has collected them into several books.

Ruler and compass

The "ruler" and "compass" of ruler-and-compass constructions is an idealization of rulers and compasses in the real world:
- The ruler is infinitely long, but it has no markings on it and has only one edge (thus making it a straightedge instead of what we usually think of as a ruler). The only thing you can use it for is to draw a line segment between two points, or to extend an existing line.
- The compass can be opened arbitrarily wide, but (unlike most real compasses) it also has no markings on it. It can only be opened to widths you have already constructed. Each construction must be exact. "Eyeballing" it (essentially looking at the construction and guessing at its accuracy, or using some form of measurement, such as the units of measure on a ruler) and getting close does not count as a solution. Stated this way, ruler and compass constructions appear to be a parlor game, rather than a serious practical problem. Figuring out how to do any particular construction is an interesting puzzle, but the persistent interest is in the problems derived from what you can't do this way. The three classical unsolved construction problems were:
- Squaring the circle: Drawing a square the same area as a given circle.
- Doubling the cube: Drawing a cube with twice the volume of a given cube.
- Trisecting the angle: Dividing a given angle into three smaller angles all of the same size. For 2000 years people tried to find constructions within the limits set above, and failed. All three have now been proven under mathematical rules to be impossible. The straightedge and compass give you the ability to produce ratios which are solutions to quadratic equations, but doubling the cube and trisecting the angle require ratios which are the solution to cubic equations, while squaring the circle requires a transcendental ratio. Curiously, origami (i.e. paper folding without any equipment) is more powerful and can be used to solve cubic equations, and thus solve two of the classical problems.

Constructible points and lengths

How do you prove something impossible? There are many different ways, but this particular problem we carefully demarcate the limit of the possible, and show that to solve these problems you must transgress that limit. Using a ruler and compass, you can impose coordinates on the plane. Draw two points, and draw the line through them. Call that the x-axis, and define the length between the two points to be one. One construction that you can do is draw perpendiculars, so draw a perpendicular to your x-axis, and call it your y-axis. We now have a Cartesian coordinate system on the plane. You can identify a point (x,y) in the Euclidean plane with the complex number x + y i. In ruler and compass construction, one starts with a line segment of length one. If one can construct a given point on the complex plane, then one says that the point is constructible. By standard constructions of Euclidean geometry one can construct the complex numbers in the form x+yi with x and y rational numbers. More generally, using the same constructions, one can, given complex numbers a and b, construct a+b, a-b, a×b, and a/b. This shows that the constructible points form a field, a subfield of the complex numbers. Moreover, one can show that the given a constructible length one can construct its complex conjugate and square root. The only way to construct points is as the intersection of two lines, of a line and a circle, or of two circles. Using the equations for lines and circles, one can show that the points at which they intersect lie in a quadratic extension of the smallest field F containing two points on the line, the center of the circle, and the radius of the circle. That is, they are of the form x+y√k, where x, y, and k are in F. Since the field of constructible points is closed under square roots, it contains all points that can be obtained by a finite sequence of quadratic extensions of the field of complex numbers with rational coefficients. By the above paragraph, one can show that any constructible point can be obtained by such a sequence of extensions. As a corollary of this, one finds that the degree of the minimal polynomial for a constructible point (and therefore of any constructible length) has degree a power of 2. In particular, any constructible point (or length) is an algebraic number.

Impossible constructions

Squaring the circle

The most famous of these problems, "squaring the circle", involves constructing a square with the same area as a given circle using only ruler and compass. Squaring the circle has been proved impossible, as it involves generating a transcendental ratio, namely

1:\sqrt

. Only algebraic ratios can be constructed with ruler and compass alone. The phrase "squaring the circle" is often used to mean "doing the impossible" for this reason. Without the constraint of requiring solution by ruler and compass alone, the problem is easily solvable by a wide variety of geometric and algebraic means, and has been solved many times in antiquity.

Doubling the cube

Doubling the cube: using only ruler and compass, construct the side of a cube that has twice the volume of a cube with a given side. This is impossible because the cube root of 2, though algebraic, cannot be computed from integers by addition, subtraction, multiplication, division, and taking square roots. This follows because its minimal polynomial over the rationals has degree 3.

Angle trisection

Angle trisection: using only ruler and compass, construct an angle that is one-third of a given arbitrary angle. This requires taking the cube root of an arbitrary complex number with absolute value 1, and is likewise impossible. Specifically, one can show that the angle of 60° cannot be trisected. If it could be trisected, then the minimal polynomial of cos 20° would have to have a power-of-two degree. Using the trigonometric identity cos(3α) = 4cos³(α) - 3cos(α), one sees that, letting cos 20° = y, that 8y³ - 6y - 1 = 0, so, substituting x = 2y, x³ - 3x - 1 = 0. The minimal polynomial for x is a factor of this, but if it were not irreducible, then it would have a rational root which, by the rational root theorem, must be 1 or -1, which are clearly not roots. Therefore the degree for the minimal polynomial for cos 20° is of degree 3, so cos 20° is not constructible and 60° cannot be trisected. Trisection, like many constructions impossible by ruler and compass, can easily be accomplished by the more powerful (but physically easy) operations of paper folding, or origami. Huzita's axioms (types of folding operations) can construct cubic extensions (cube roots) of given lengths, whereas ruler-and-compass can construct only quadratic extensions (square roots). See Mathematics of origami.

Constructing regular polygons

Some regular polygons (e.g. a pentagon) are easy to construct with ruler and compass; others are not. This led to the question: Is it possible to construct all regular polygons with ruler and compass? Carl Friedrich Gauss in 1796 showed that a regular n-sided polygon can be constructed with ruler and compass if the odd prime factors of n are distinct Fermat primes. Gauss conjectured that this condition was also necessary, but he offered no proof of this fact, which was proved by Pierre Wantzel in (1836). See constructible polygon.

Constructing with only ruler or only compass

It is possible (according to the Mohr-Mascheroni theorem) to construct anything with just a compass that can be constructed with ruler and compass. It is impossible to take a square root with just a ruler, so some things cannot be constructed with a ruler that can be constructed with a compass; but (by the Poncelet-Steiner theorem) given a single circle and its center, they can be constructed. A further generalisation of this theorem (due to K. Venkatachala Iyengar) gives constructions using only a ruler given a point and five distinct points equidistant from it.

Recent research

Simon Plouffe has written a paper showing how ruler and compass can be used as a simple computer with unexpected power to compute binary digits of certain numbers.

Reference

- Simon Plouffe. "The Computation of Certain Numbers Using a Ruler and Compass." Journal of Integer Sequences, Vol. 1 (1998), Article 98.1.3.

External links

- [http://wims.unice.fr/~wims/en_tool~geometry~rulecomp.en.phtml Online ruler-and-compass construction tool]
- [http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Squaring_the_circle.html Squaring the circle]
- [http://www.geom.umn.edu/docs/forum/square_circle/ Impossibility of squaring the circle]
- [http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Doubling_the_cube.html Doubling the cube]
- [http://www.geom.umn.edu/docs/forum/angtri/ Angle trisection]
- [http://www.jimloy.com/geometry/trisect.htm Trisection of an Angle]
- [http://mathforum.org/dr.math/faq/formulas/faq.regpoly.html Regular polygon constructions]
- [http://www.math.uwaterloo.ca/JIS/compass.html Simon Plouffe's use of ruler and compass as a computer]
- [http://www.math-cs.cmsu.edu/~mjms/1996.2/clements.ps Why Gauss could not have proved necessity of constructible regular polygons]
- [http://www.cut-the-knot.org/do_you_know/compass.shtml Construction with the Compass Only] at cut-the-knot
- [http://agutie.homestead.com/files/ArchBooLem08.htm Archimedes' neusis construction] by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.
- ko:작도 ja:定規とコンパスによる作図

Euclidean space

In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. The generalization applies Euclid's concept of distance, and the related concepts of length and angle, to a coordinate system in any number of dimensions. It is the "standard" example of a finite-dimensional, real, inner product space. A Euclidean space is a particular metric space that enables the investigation of topological properties such as compactness. An inner product space is a generalization of a Euclidean space. Both inner product spaces and metric spaces are explored within functional analysis. Euclidean space plays a part in the definition of a manifold which embraces the concepts of both Euclidean and non-Euclidean geometry. One mathematical motivation for defining a distance function is the ability to define an open ball around points in the space. This fundamental concept justifies a differential calculus between a Euclidean space and other manifolds. Differential geometry brings such a differential calculus into play, together with a technique of launching a mobile, local Euclidean space, to explore the properties of non-Euclidean manifolds.

Real coordinate space

Let R denote the field of real numbers. For any non-negative integer n, the space of all n-tuples of real numbers forms an n-dimensional vector space over R sometimes called real coordinate space and denoted Rⁿ. An element of Rⁿ is written x = (x₁, x₂, …, x_n) where each x_i is a real number. The vector space operations on Rⁿ are defined by :

\mathbf + \mathbf = (x_1 + y_1, x_2 + y_2, \ldots, x_n + y_n)

a\,\mathbf = (a x_1, a x_2, \ldots, a x_n)

Real coordinate space Rⁿ comes with a standard basis: :

\mathbf_1 = (1, 0, \ldots, 0)

\mathbf_2 = (0, 1, \ldots, 0)

\vdots

\mathbf_n = (0, 0, \ldots, 1)

An arbitrary vector in Rⁿ can then be written in the form :

\mathbf = \sum_^n x_i \mathbf_i

Real coordinate space is the prototypical example of a real n-dimensional vector space. In fact, every real n-dimensional vector space V is isomorphic to Rⁿ. This isomorphism is not canonical however. A choice of isomorphism is equivalent to a choice of basis for V (by looking at the image of the standard basis for Rⁿ in V). The reason for working with arbitrary vector spaces instead of Rⁿ is that it is often preferable to work in a coordinate-free manner (i.e. without choosing a preferred basis).

Euclidean structure

Euclidean space is more than just real coordinate space. In order to do Euclidean geometry one needs to be able to talk about the distance between points and the angles between lines or vectors. The natural way in which to do this is to introduce what is called an inner product or dot product on Rⁿ. This product is defined by :

\mathbf\cdot\mathbf = \sum_^n x_iy_i = x_1y_1+x_2y_2+\cdots+x_ny_n.

The dot product of any two vectors x and y gives a real number. This product allows us to define the "length" of a vector x in the following way :

\|\mathbf\| = \sqrt = \sqrt

This length function satisfies the required properties of a norm and is called the Euclidean norm on Rⁿ. The (interior) angle θ between x and y is then given by :

\theta = \cos^\left(\frac\right)

where cos⁻¹ is the arccosine function. Finally, one can use the norm to define a distance function (or metric) on Rⁿ in the following manner :

d(\mathbf, \mathbf) = \|\mathbf - \mathbf\| = \sqrt.

The form of this distance function is based on the Pythagorean theorem, and is called the Euclidean metric. Real coordinate space together with the above Euclidean structure (dot product and the associated norm and metric) is called Euclidean space often denoted by Eⁿ. (Many authors refer to Rⁿ itself as Euclidean space, with the Euclidean structure being understood). The Euclidean structure on Eⁿ gives it the structure of an inner product space (in fact a Hilbert space), a normed vector space, and a metric space.

Euclidean topology

Since Euclidean space is a metric space it is also a topological space with the natural topology induced by the metric. The metric topology on Eⁿ is called the Euclidean topology. A set is open in the Euclidean topology if and only if it contains an open ball around each of its points. The Euclidean topology turns out to be equivalent to the product topology on Rⁿ considered as a product of n copies of the real line R (with its standard topology). An important result on the topology of Rⁿ, that is far from superficial, is Brouwer's invariance of domain. Any subset of Rⁿ (with its subspace topology) which is homeomorphic to another open subset of Rⁿ is itself open. An immediate consequence of this is that R^m is not homeomorphic to Rⁿ if m ≠ n — an intuitively "obvious" result which is nonetheless difficult to prove. Euclidean n-space is the prototypical example of an n-manifold, in fact, a smooth manifold. For n ≠ 4, any differentiable n-manifold that is homeomorphic to Rⁿ is also diffeomorphic to it. The surprising fact that this is not also true for n = 4 was proved by Simon Donaldson in 1982; the counterexamples are called exotic (or fake) 4-spaces. Euclidean space is also known as linear manifold. An m-dimensional linear submanifold of Rⁿ is a Euclidean space of m dimensions embedded in it (as an affine subspace). For example, any straight line in some higher-dimensional Euclidean space is a 1-dimensional linear submanifold of that space.

Numerical integration

\int_a^b f(x)\, dx

This problem can also be stated as an initial value problem for an ordinary differential equation, as follows. :

y'(x) = f(x), \quad y(a) = 0

Reasons for numerical integration

Methods for one-dimensional integrals

Quadrature rules based on interpolating functions

\int_a^b f(x)\,dx \approx (b-a) \, f\left(\frac\right).

The interpolating function may be an affine function (a polynomial of degree 1) which passes through the points (a, f(a)) and (b, f(b)). This is called the trapezoidal rule. :

\int_a^b f(x)\,dx \approx (b-a) \, \frac.

\int_a^b f(x)\,dx \approx \frac \left(  + \sum_^ f \left( a+k \frac \right) \right)

Adaptive algorithms

Extrapolation methods

Conservative (a priori) error estimation

Let f have a bounded first derivative over [a,b]. The mean value theorem for f, where

x < b

, gives :

(x - a) f'(y_x) = f(x) - f(a)\,

for some y_x in [a,x] depending on x. If we integrate in x from a to b on both sides and the take absolute values, we obtain :

\left| \int_a^b f(x)\,dx - (b - a) f(a) \right|
  = \left| \int_a^b (x - a) f'(y_x)\, dx \right|

We can further approximate the integral on the right-hand side by bringing the absolute value into the integrand, and replacing the term in f' by an upper bound: :

\left| \int_a^b f(x)\,dx - (b - a) f(a) \right| \leq  \sup_ \left| f'(x) \right|

\sup_ \left| f'(x) \right|

for the error term of that particular approximation. (Note that this is precisely the error we calculated for the example

f(x) = x

Multidimensional integrals

Monte Carlo

Sparse grids

Software for numerical integration

References

Crank (person)

"Crank" (or kook, crackpot, or quack) is a pejorative term for a person who writes or speaks in an authoritative fashion about a particular subject, often in science, but is alleged to have false or even ludicrous beliefs. Crank can also be used to describe the opinions of these people (see [http://dictionary.reference.com/search?q=crank American Heritage Dictionary 2000 - noun definition 3]). Usage of the label is often subjective, with proponents of competing theories labeling their opponents cranks, but the term still connotes someone who is well out of mainstream opinion on a matter, as well as connoting a person of dubious mental stability. The belief that the earth revolves around the sun was once considered a crank belief, so the mainstream consensus on crank beliefs can change. On the other hand, while most cranks think of or portray themselves as a new Galileo whose superior insight will be vindicated, for every Galileo, there are thousands of cranks who are just plain wrong.

Crank tactics

Regardless of whether they are acting in good faith, cranks in all subject areas use similar tactics in their attempts to persuade the general public. The following subset of tactics are deployed by science cranks:
- The (apparent) offering of cash rewards to anyone who disproves a (scientific) theory
- Grandiose claims for the validity and scope of the theory
- Stated belief that a conspiracy by the scientific establishment is hindering uptake of the theory
- The use of neologisms without proper definition
- Comparison of the originator with Einstein, Newton, Galileo, or Copernicus
- Direct communication of the idea to the media, typically holding a press conference before going through the usual peer review process of publishing in scholarly journals (see cold fusion)
- Any observable effects of the theory are at or beyond the limits of detection (see N-rays; some skeptics also associate this tactic with superstrings)
- Lack of a working prototype where one might expect it to convince a skeptical audience (see perpetual motion)
- Extensive use of testimonials, e.g. in advertising, where testimonials by users of a crank therapy make claims that would be against FDA anti-fraud regulations if the purveyor were to make the claims outright
- Claims that something is proved to work because there is a patent on it, when the patent is actually a design patent rather than a utility patent, or covers an insignificant aspect of the patented subject matter
- Related: Claims that something must work because it has a patent pending on it, when standard patent office stall tactics have been used to delay the pendency of the patent application, sometimes for up to several decades

Cranks on the Internet

Science fiction author and critic Bruce Sterling noted in his essay in CATSCAN 13 [http://www.eff.org/Misc/Publications/Bruce_Sterling/Catscan_columns/catscan.13]: :There's supposed to be a lot of difference between the hurtful online statement "You're a moron," and the tastefully facetious statement "You're a moron :-)". I question whether this is really the case, emoticon or no. And even the emoticon doesn't help much in one's halting interaction with the occasional online stranger who is, in fact, gravely sociopathic. Online communication can wonderfully liberate the tender soul of some well-meaning personage who, for whatever reason, is physically uncharismatic. Unfortunately, online communication also fertilizes the eccentricities of hopeless cranks, who at last find themselves in firm possession of a wondrous soapbox that the Trilateral Commission and the Men In Black had previously denied them.

Related terminology

"Kook" is a somewhat similar pejorative term that is usually used to describe a person whose areas of interest are perceived to be eccentric, fantastic, or insane. A person may be said to be a "kook" if they are seen to hold socially unacceptable beliefs, or perceptions that outrageously conflict with known scientific results, and appear to base their entire world views upon them. The term was coined in 1960 and originates from the word cuckoo, which is also the name of a bird, but which is also pejoratively associated with mental illness. Predictably, "kooks" tend to draw criticism and generate controversy; it has been speculated that some kooks are motivated by a desire for such attention.

Topics typically associated with the "crank" label

Physics, computer science and mathematics

- squaring the circle, doubling the cube and trisecting the angle
- producing unified Theories of Everything, and particularly doing so with high school or undergraduate level physics knowledge, or attempting to extend such theories to extend knowledge gathered by "experiments" such as astral projection
- lossless data compression that can always reduce the size of random data
- unbreakable cryptographic ciphers (other than the one-time pad)
- Perpetual motion, probably the earliest example of kookery
- finding a simple proof for Fermat's last theorem, the Goldbach conjecture, etc
- cold fusion
- Attempts at disproving quantum mechanics, the theory of relativity, Cantor's diagonal argument, Gödel's incompleteness theorem and other accepted but counterintuitive theories
- Time Cube as an archetypal example of the work of a science crank
- VMSK and other "ultra narrowband" modulation schemes whose claims violate the Shannon-Hartley theorem
- Alchemy
- Flat Earth Society which asserts that the earth is flat.
- Modern geocentrism
- Bogdanov Affair
- Global warming conspiracy theorists, who believe that global warming is only an idea that was invented and promoted by a conspiracy.

Medicine

- Alternative Medicine or Holistic Health practitioners who seek to completely substitute their treatment modality for mainstream medicine (and particularly in the treatment of potentially fatal conditions such as cancer and AIDS), or who practice a fortiori empirically unfounded modalities (ie: Ear candling).
- Results (and particularly purported cures) that are not reproducible by third parties
- Mainstream Physicians who, during the mid-twentieth century, systematically dismissed mounting scientific evidence as to the effects of diet on heart disease.
- Homeopathy
- Crystal healing
- Anti-vaccination (and particularly claimed connections between vaccination and autism)
- Faith healing (see this same entry under the topic of Paranormal and spiritual)
- Phrenology, especially when used in (pseudo-)psychiatric diagnosis.
- Psychic surgery
- Chiropractic, when considered as a total replacement for medical science or when it claims that unrelated conditions are caused by spinal subluxations, or curable by correcting the same.
- Medical illnesses neither recognised nor under consideration for inclusion by the World Health Organization into its International Classification of Diseases. Examples include Mucoid plaque.
- Psychiatric illnesses neither recognised nor under consideration for inclusion into the DSM-IV. For example, some cranks still believe that homosexuality, which was de-listed from the DSM in 1973, is a form of mental illness.
- Magnetic therapy.
- "Q-Ray" therapy, e.g. with emissions from a "Q-Ray" bracelet.
- Mud therapy, wherein therapeutic effects are supposed to be gained by drinking mud.
- Attempts at treatment (and particularly treatment conditions such as perceived penis size concerns which are likely to be purely psychological on the part of patients) when no attempt at prior diagnosis is made. This is another gray area — for example, physicians faced with a child with strep throat will often start antibiotics without waiting for positive diagnosis, because the potential benefit if strep throat is present outweighs any harm of the throat culture comes back negative.

Nutrition

- Diets that forbid or severely limit one or more necessary nutrients. Examples include extreme low-carbohydrate or low-fat diets.
- Those who consider common foods, such as milk or wheat, harmful or even fatal to everyone, including those with no history of food allergies or of anaphylaxis.
- Macrobiotics
- Water fluoridation opponents
- Raw food diets to the extent that cooked food is believed to be actively harmful (as opposed, for example, to claims that certain vitamins are lost in cooking, which are not disputed).
- Those who believe cancer is caused by deficiency of an unknown vitamin.

Politics, economics, and law

- some conspiracy theories
- currency crankism
- Federal Reserve conspiracies
- social credit
- tax protesters
- Holocaust denial
- supporters of Lyndon LaRouche
- supporters of Bo Gritz
- supporters of the Posse Comitatus
- certain supporters of a gold standard as opposed to fiat money (goldbugs)
- New World Order / Trilateral Commission / Council on Foreign Relations conspiracy theorists
- Spectral evidence in which legal testimony based on apparitions and dreams are considered admissible in a court of law

Paranormal and spiritual

- immortality rings
- New Age movement adherents' denial of the laws of Physics -- examples of this denial include telekinesis and yogic flying
- Faith healing particularly when advocating it as a replacement for regular visits to a licensed physician
- Divine Revelation and prophecy only when it attempts to trump Science
- Religious fundamentalism when it disguises faith as science (ie: Creationism, and Intelligent design).
- Feng Shui
- Numerology
- End times and end of the world prophecy
- Anglo-Israelism and Christian Identity
- The belief that a planet called Hercolubus or "Planet X" is drawing closer to Earth.
- flying saucers and little green men
- Certain alleged Marian apparitions, especially those that have been investigated and not found credible by the Roman Catholic Church

External links

- [http://home.pacifier.com/~dkossy/kooksmus.html Kooks Museum], a humorous collection of kook ideas, by Donna Kossy
- [http://www.crank.net/ Crank Dot Net], a collection of Web sites related to cranks, created by Erik Max Francis
- [http://www.LART.com/ the Almighty LART.com], more kook appreciation
- [http://insti.physics.sunysb.edu/~siegel/quack.html Are you a quack?], page dedicated to quacks
- [http://math.ucr.edu/home/baez/crackpot.html John Baez' crackpot index], in the author's words: A simple method for rating potentially revolutionary contributions to physics.
- [http://angel.1jh.com./nanae/kooks/ The Unofficial NANAE Kooks Kollection], a collection of articles on the Usenet kooks particular to the NANAE newsgroup.
- [http://www.quackwatch.org/ Quack Watch] Lists varieties of quackery and health fraud. Category:Pseudoscience category:Pejorative terms for people

Hope

Hope is a belief that things are obtainable regardless of the remoteness of the probabilities. That means "I can [get well, rich, happy etc] no matter how [sick, poor, miserable etc] I am." It does not mean "I will get well, rich, happy etc." - which is "positive thinking" not always based in facts. According to the above definition, hope recognizes probabilities, but is not attached to the outcome. Hope also implies a certain amount of perseverance, believing that something is possible even when there is some evidence to the contrary. Hope may be directed toward something minor or towards something extremely significant. "False hope" does not exist - according to the definition there is always space for hope. "False hope" may be an example of positive thinking. Examples include:
- hoping to catch a train
- hoping to get some object or a job, or to get rich
- being in love and hoping that this is reciprocal
- hoping for somebody to be cured of a disease
- hoping for happiness
- hoping for a blessing from God
- hoping for the ability to fly (a false hope) The absence of hope is despair. In some religions, despair is considered to be a sin. Hope was personified in Greek mythology. When Pandora opened Pandora's Box, she let out all the evils except one-- Hope. Apparently the Greeks considered Hope to be as dangerous as all the world's evils. But without hope to accompany all their troubles, humanity was filled with despair. It was a gre