41. Shapes Of Constant Width The length of the boundary of shapes of constant width depends only on R.Honsberger, Ingenuity in mathematics, MAA, New math Library, 1970; H.Rademacher and O http://www.cut-the-knot.org/do_you_know/cwidth.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Shapes of constant width Yes there are shapes of constant width other than the circle No - you can't drill square holes. But saying this was not just an attention catcher. As the applet on the right illustrates, you can drill holes that are almost square - drilled holes whose border includes straight line segments! Now then let us define the subject of our discussion. First we need a notion of width. Let there be a bounded shape. Pick two parallel lines so that the shape lies between the two. Move each line towards the shape all the while keeping it parallel to its original direction. After both lines touched our figure, measure the distance between the two. This will be called the width of the shape in the direction of the two lines. A shape is of constant width if its (directional) width does not depend on the direction. This unique number is called the width of the figure. For the circle, the width and the diameter coincide. The curvilinear triangle above is built the following way. Start with an equilateral triangle. Draw three arcs with radius equal to the side of the triangle and each centered at one of the vertices. The figure is known as the

42. Math Surprises math Surprises An Example. In other words, the average number of crossings per 1 unit of relative length L/D is constant and is equal to 2/p. Color one half of http://www.cut-the-knot.org/ctk/August2001.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Cut The Knot! An interactive column using Java applets by Alex Bogomolny Math Surprises: An Example August 2001 Compte de Buffon (1707-1788) in the 18 th century posed and solved the very first problem of geometric probability. A needle of a given length L is tossed on a wooden floor with evenly spaced cracks, distance D apart. What is the probability of the needle hitting a crack? The answer he discovered with the help of integral calculus is given by the simple formula [ Beckmann Eves Kasner Paulos ... Stein P = 2L/ p D With P approximated by the ratio of hits to the total number of tosses, the formula offers a way of evaluating p , an observation that eventually led Pierre Simon Laplace (1749-1827) to propose a method, known today as the Monte Carlo Method, for numerical evaluation of various quantities by realizing appropriate random events. History records several names of people who applied the method manually to approximate p . A Captain Fox [ Beckmann , p. 77] mentions Wolf from Zurich (1850) who obtained

43. Feigenbaum's Universal Constant FEIGENBAUM S UNIVERSAL constant. it. Last updated 191197. Send comments to berland(a)stud.math.ntnu.no © 1997 Back to homepage. http://www.stud.ntnu.no/~berland/math/feigenbaum/feigconstant.html

F EIGENBAUM'S UNIVERSAL CONSTANT Back to main page Back to homepage Send comment The short answer: However, noone should be satisfied by that. In fact, this number is perhaps the most fantastic aspect of this fractal. There are many many formulas that produce the same tree, but the number is always the same. It is said that mr. Mitchell Feigenbaum called home to his mother when he discovered this universality and said this was going to make him famous. The famous value, comes when you compare the length of one part of the tree, that is a parts between the line divisions/bifurcations. See illustration at right. The first part is from -0.25 to 0.75, and has a length of 1.00. The next part is from 0.75 to 1.25, and has a length of 0.50. The relationship between the two lengths is 1.00/0.50=2.00. Now that is far from the Feigenbaumvalue, but the exact value springs up when you compare two parts as far right as possible, as long as x follows a periodic orbit. I have graphically found the values for the first 6 bifurcations: Bifurc no. Divides at Length This length/next length -0.25 - - 1 0.75 L1=1.0 L1/L2=2.0 2 1.25 L2=0.5 L2/L3=4.25 3 1.3677 L3=0.1147 L3/L4=4.492 4 1.3939 L4=0.0262 L4/L5=4.6208 5 1.39957 L5=0.00567 L5/L6=4.536 6 1.40082 L6=0.00125 L6/L7=?

44. Math::Symbolic::Constant NAME. MathSymbolicConstant Constants in symbolic calculations. SYNOPSIS. Returns a MathSymbolicConstant. Constructor zero. http://steffen-mueller.net/modules/Math-Symbolic/Math-Symbolic-Constant.html

NAME SYNOPSIS DESCRIPTION EXPORT ... SEE ALSO NAME Math::Symbolic::Constant - Constants in symbolic calculations SYNOPSIS DESCRIPTION This module implements numeric constants for Math::Symbolic trees. EXPORT None by default. METHODS Constructor new Takes hash reference of key-value pairs as argument. Special case: a value for the constant instead of the hash. Returns a Math::Symbolic::Constant. Constructor zero Arguments are treated as key-value pairs of object attributes. Returns a Math::Symbolic::Constant with value of 0. Constructor one Arguments are treated as key-value pairs of object attributes. Returns a Math::Symbolic::Constant with value of 1. Constructor euler Arguments are treated as key-value pairs of object attributes. Returns a Math::Symbolic::Constant with value of e, the Euler number. The object has its 'special' attribute set to 'euler'. Constructor pi Arguments are treated as key-value pairs of object attributes. Returns a Math::Symbolic::Constant with value of pi. The object has its 'special' attribute set to 'pi'. Method value object , not just every named variable.

45. World Web Math: Calculus Summary algebraically where in the last equation, c is a constant and in the first two equations, if both limits of f and g exist. integrals For c a constant,. http://web.mit.edu/wwmath/calculus/summary.html

Calculus Summary Calculus has two main parts: differential calculus and integral calculus. Differential calculus studies the derivative and integral calculus studies (surprise!) the integral. The derivative and integral are linked in that they are both defined via the concept of the limit: they are inverse operations of each other (a fact sometimes known as the fundamental theorem of calculus): and they are both fundamental to much of modern science as we know it. Derivatives The limit of a function f x ) as x approaches a is equal to b if for every desired closeness to b , you can find a small interval around (but not including) a that acheives that closeness when mapped by f . Limits give us a firm mathematical basis on which to examine both the infinite and the infinitesmial. They are also easy to handle algebraically: where in the last equation, c is a constant and in the first two equations, if both limits of f and g exist. One important fact to keep in mind is that doesn't depend at all on f a ) in fact

46. World Web Math: Vector Calculus: Partial Differentiation A partial function is a onevariable function obtained from a function of several variables by assigning constant values to all but one of the independent http://web.mit.edu/wwmath/vectorc/scalar/partial.html

Partial Differentiation Prerequisites: Derivatives Bound as we humans are to three spacial dimensions, multi-variable functions can be very difficult to get a good feel for. (Try picturing a function in the 17th dimension and see how far you get!) We can at least make three-dimensional models of two-variable functions, but even then at a stretch to our intuition. What is needed is a way to cheat and look at multi-variable functions as if they were one-variable functions. We can do this by using partial functions . A partial function is a one-variable function obtained from a function of several variables by assigning constant values to all but one of the independent variables. What we are doing is taking two-dimensional "slices" of the surface represented by the equation. For Example: z x y can be modeled in three dimensional space, but personally I find it difficult to sketch! In the section on critical points a picture of a plot of this function can be found as an example of a saddle point. But by alternately setting x =1 (red)

47. Conic Sections distance to the origin is constant, sum of distances to each focus is constant, distance to focus = distance to directrix, difference between distances to each http://www.math2.org/math/algebra/conics.htm

Conic Sections Circle Ellipse (h) Parabola (h) Hyperbola (h) Definition: A conic section is the intersection of a plane and a cone. Ellipse (v) Parabola (v) Hyperbola (v) By changing the angle and location of intersection, we can produce a circle, ellipse, parabola or hyperbola; or in the special case when the plane touches the vertex: a point, line or 2 intersecting lines. Point Line Double Line The General Equation for a Conic Section: Ax + Bxy + Cy + Dx + Ey + F = The type of section can be found from the sign of: B If B - 4AC is... then the curve is a... ellipse, circle, point or no curve. parabola, 2 parallel lines, 1 line or no curve. hyperbola or 2 intersecting lines. The Conic Sections. For any of the below with a center (j, k) instead of (0, 0), replace each x term with (x-j) and each y term with (y-k). Circle Ellipse Parabola Hyperbola Equation (horiz. vertex): x + y = r x / a + y / b 4px = y x / a - y / b Equations of Asymptotes: Equation (vert. vertex): x + y = r y / a + x / b 4py = x y / a - x / b Equations of Asymptotes: Variables: r = circle radius a = major radius (= 1/2 length major axis) b = minor radius (= 1/2 length minor axis) c = distance center to focus p = distance from vertex to focus (or directrix) a = 1/2 length major axis b = 1/2 length minor axis c = distance center to focus Eccentricity: c/a c/a Relation to Focus: p = a - b = c p = p a + b = c Definition: is the locus of all points which meet the condition...

48. S.O.S. Mathematics - Sites Of Interest On The Web Everything you always wanted to know about mathematical constants like the FransénRobinson constant et al. Temporarily unavailable.; math Humor - Enough of http://www.sosmath.com/wwwsites.html

S.O.S. Mathematics - Sites of Interest on the World Wide Web Here is a list of some nice sites on the World Wide Web. If you know of a site that should be listed, please let us know.... General Mathematics Sites The Math Forum @ Drexel . This site is worth visiting. The webmasters did a wonderful job in collecting mathematically related material available on the internet. Ask Dr. Math. Any questions? Math Archives. This is a complete site where you may find virtually anything about Math on the Web. Math2.org . Features a variety of tables in diverse areas of math, a message board and a chat room. Eric Weisstein's Math World , now hosted by Wolfram Research. A very nice encyclopedic resource! Britannica.com Maple . - Created by Jean-Michel Ferrard. The Mathematical Atlas provides an introduction to the areas of modern mathematics, and points to sources of further information. A very nice site! ExploreLearning is an interactive math and science site with lots of Java applets to experiment with. Earliest Uses of Various Mathematical Symbols These pages show the names of the individuals who first used various common mathematical symbols, and the dates the symbols first appeared. A very nice site! MathNerds provides discovery-based, mathematical guidance via an international, volunteer network of mathematicians.

49. World Builders: Math For The Solar Constant In The Biosphere E Viau CSULA Try to understand the basic idea, and don t use my math numbers on any Important Exams unless you have checked them out! The Solar constant. http://curriculum.calstatela.edu/courses/builders/lessons/less/biomes/SunEnergy_

Home Science Notes Web Links Biomes ... Lessons Following the Energy Trail Let's see how the sun's energy flows in the Biosphere. Warning: I am making a lot of assumptions and doing a lot of rounding in the math here. My numbers may be off by several thousand percent, but I think the process is all right. Try to understand the basic idea, and don't use my math numbers on any Important Exams unless you have checked them out! Dr Viau says, "Gerald Nordley and Mark Wistey were kind enough to help with this, but I made the mistakes all by myself!" The Solar Constant How much of the sun's energy gets to the surface of the earth, and what does that mean to a lemming? The earth gets only 2 billionths of the sun's energy, but that is still a lot. However, you can see on the chart that life (through photosynthesis) uses only 023% of the energy that reaches the surface of the earth 34% of the sun's energy is reflected back into space by snow and clouds. This reflective quality of a planet is called its

50. Rydberg Constant Definition Meaning Information Explanation Which gives the Rydberg constant for a certain atom with one electron with the rest mass math m /math and the atomic nucleus mass math M /math . http://www.free-definition.com/Rydberg-constant.html

A B C D ... Contact Beta 0.71 powered by: akademie.de PHP PostgreSQL Google News about your search term Rydberg constant The rydberg constant is named after physicist Janne Rydberg , and is a physical constant discovered when measuring the spectrum of hydrogen , and building upon results from Anders Jonas  ngstr¶m and Johann Balmer. Each chemical element has it's own Rydberg constant, but most commonly referred to is the "infinity" constant. The "infinity" Rydberg constant is: The "infinity" constant appears in the formula: Which gives the Rydberg constant for a certain atom with one electron with the rest mass atomic nucleus As the formula for the Rydberg constant contains no less than five other physical constant s, namely the elementary charge electron rest mass permittivity ... speed of light in vacuum For more information about this formula, see the article on the Rydberg formula See Also Rydberg formula Janne Rydberg Books about 'Rydberg constant' at: amazon.com or amazon.co.uk Note: This article from Wikipedia is made available under the terms of the GNU FDL Further Search within Free-Definition Link from your web site to this article with this HTML tag: top

51. E (mathematical Constant) Definition Meaning Information Explanation The mathematical constant math e /math (occasionally called Euler s number or Napier s constant in honor of the Scottish mathematician John Napier who http://www.free-definition.com/E-mathematical-constant.html

A B C D ... Contact Beta 0.71 powered by: akademie.de PHP PostgreSQL Google News about your search term E (mathematical constant) The mathematical constant Euler's number or Napier's constant in honor of the Scottish mathematician John Napier who introduced logarithms ) is the base of the natural logarithm . It is approximately equal to e It is equal to exp(1) where exp is the exponential function and therefore it is the limit and can also be written in a number of ways as an infinite series, the most important of which is: factorial proof of the equivalence of these definitions is given below. The number e is relevant because one can show that the exponential function exp( x derivative and is hence commonly used to model growth or decay processes. The number e is known to be irrational and even transcendental . It was the first number to be proved transcendental without having been specifically constructed; the proof was given by Charles Hermite in . It is conjectured to be normal . It features (along with a few other fundamental constants) in Euler's identity described by Richard Feynman as " the most remarkable formula in mathematics The infinite continued fraction Proof that e is irrational Proof that e is transcendental In 1975, the Swiss Felix A. Keller discovered the following formula that converges in e ("Keller's Expression"

52. Netscape DevEdge You refer to the constant PI as math.PI and you call the sine function as math.sin(x) , where x is the method s argument. constants http://devedge.netscape.com/library/manuals/2000/javascript/1.3/reference/math.h

DevEdge Skip to: [ content navigation Search Table of Contents Previous Next Index Math A built-in object that has properties and methods for mathematical constants and functions. For example, the Math object's PI property has the value of pi. Core object Implemented in JavaScript 1.0, NES 2.0 ECMA version ECMA-262 Created by The Math object is a top-level, predefined JavaScript object. You can automatically access it without using a constructor or calling a method. Description All properties and methods of Math are static. You refer to the constant PI as Math.PI and you call the sine function as Math.sin(x) , where x is the method's argument. Constants are defined with the full precision of real numbers in JavaScript. It is often convenient to use the with statement when a section of code uses several Math constants and methods, so you don't have to type "Math" repeatedly. For example, a = PI * r*r y = r*sin(theta) x = r*cos(theta) Property Summary Property Description E Euler's constant and the base of natural logarithms, approximately 2.718. Natural logarithm of 10, approximately 2.302.

53. Core JavaScript Reference 1.5: You refer to the constant PI as math.PI and you call the sine function as math.sin(x), where x is the method s argument. constants http://devedge.netscape.com/library/manuals/2000/javascript/1.5/reference/math.h

DevEdge Skip to: [ content navigation Search Previous Contents Index Next Core JavaScript Reference 1.5 Math A built-in object that has properties and methods for mathematical constants and functions. For example, the Math object's PI property has the value of pi. Core object Implemented in JavaScript 1.0, NES 2.0 ECMA version ECMA-262 Created by The Math object is a top-level, predefined JavaScript object. You can automatically access it without using a constructor or calling a method. Description All properties and methods of Math are static. You refer to the constant PI as Math.PI and you call the sine function as Math.sin(x) , where x is the method's argument. Constants are defined with the full precision of real numbers in JavaScript. It is often convenient to use the with statement when a section of code uses several Math constants and methods, so you don't have to type "Math" repeatedly. For example, a = PI * r*r y = r*sin(theta) x = r*cos(theta) Property Summary Property Description E Euler's constant and the base of natural logarithms, approximately 2.718.

54. Math::BigInt::Constant - Arbitrary Sized Constant Integers SiteFelix / Toolbox. MathBigIntconstant Arbitrary sized constant integers. MathBigIntconstant - Arbitrary sized constant integers. Update 2003/04/23. http://www4.kcn.ne.jp/~felix/Softwares/MathPPM/Math-BigInt-Constant.html

Location : Home Softwares Perl Math Packages Title : Math::BigInt::Constant Site:Felix / Toolbox Math::BigInt::Constant - Arbitrary sized constant integers ŠT—v Žg—p—á ƒoƒO Math::BigInt::Constant - ”CˆÓ’·‚̐®”’萔 ŠT—v use Math::BigInt ':constant' ®”’萔‚ªƒRƒ“ƒpƒCƒ‹Žž‚É BigInt ‚̒萔‚ÉŽw’肳‚ê‚邪AƒIƒuƒWƒFƒNƒg‚²‚Æ‚Ì’è‹`‚âA•¶Žš—ñ‚⠂̂悤‚È•‚“®¬”“_‚É‚Í“K—p‚³‚ê‚È‚¢B Žg—p—á @ƒ‰ƒ“ƒ^ƒCƒ€ƒ`ƒFƒbƒN‚ðs‚¢A•¶Žš—ñ‰»‚·‚éB ƒoƒO ¡‚Ì‚Æ‚±‚댩‚‚©‚Á‚Ä‚¢‚È‚¢B Tels http://bloodgate.com in early 2001. Math::BigInt::Constant - Arbitrary sized constant integers Update : 2003/04/23 Site:Felix / Toolbox Felix ( felix@m4.kcn.ne.jp

55. Math . You reference the constant PI as math.PI . constants are defined with the full precision of real numbers in JavaScript....... Navigator 2.0. http://members.ozemail.com.au/~phoenix1/html/ref_m-q.htm

[Previous reference file] Math Object. A built-in object that has properties and methods for mathematical constants and functions. For example, the Math object's PI property has the value of pi. Syntax To use a Math object: 1. Math. propertyName 2. Math. methodName parameters Parameters propertyName is one of the properties listed below. methodName is one of the methods listed below. Property of None. The Math object is a top-level, built-in JavaScript object. Implemented in Navigator 2.0 Description You reference the constant PI as Math.PI . Constants are defined with the full precision of real numbers in JavaScript. Similarly, you reference Math functions as methods. For example, the sine function is Math.sin(argument) , where argument is the argument. It is often convenient to use the with statement when a section of code uses several Math constants and methods, so you don't have to type "Math" repeatedly. For example, a = PI * r*r y = r*sin(theta) x = r*cos(theta) Properties The Math object has the following properties: E PI Methods The Math object has the following methods: abs acos asin atan ... valueOf Event handlers None.

56. Constant Field Values (Math 1.0-dev API) constant Field Values. Contents org.apache.*. org.apache.*. org.apache.commons.math.random.ValueServer. public static final int, constant_MODE, 5. http://jakarta.apache.org/commons/math/apidocs/constant-values.html

Overview Package Class Use Tree Deprecated Index Help PREV NEXT FRAMES NO FRAMES All Classes Constant Field Values Contents org.apache.* org.apache.* org.apache.commons.math.random. ValueServer public static final int public static final int public static final int public static final int public static final int public static final int org.apache.commons.math.stat.univariate. DescriptiveStatistics public static final int Overview Package Class Use Tree Deprecated Index Help PREV NEXT FRAMES NO FRAMES All Classes

57. Constant Problem References. Bailey, D. H. ``Numerical Results on the Transcendence of constants Involving , , and Euler s constant. math. Comput. 50, 275281, 1988. http://icl.pku.edu.cn/yujs/MathWorld/math/c/c612.htm

Constant Problem Given an expression involving known constants, integration in finite terms, computation of limits, etc., determine if the expression is equal to Zero . The constant problem is a very difficult unsolved problem in Transcendental Number theory. However, it is known that the problem is Undecidable if the expression involves oscillatory functions such as Sine . However, the Ferguson-Forcade Algorithm is a practical algorithm for determining if there exist integers for given real numbers such that or else establish bounds within which no relation can exist (Bailey 1988). See also Ferguson-Forcade Algorithm Integer Relation Schanuel's Conjecture References Bailey, D. H. ``Numerical Results on the Transcendence of Constants Involving , and Euler's Constant.'' Math. Comput. Sackell, J. ``Zero-Equivalence in Function Fields Defined by Algebraic Differential Equations.'' Trans. Amer. Math. Soc. Eric W. Weisstein

58. Programming Ruby: The Pragmatic Programmer's Guide m = math.clone, », math. m.constants, », E , PI . m math, », false. Returns the value of the named constant in mod. math.const_get PI, », 3.141592654. http://www.rubycentral.com/book/ref_c_module.html

Programming Ruby The Pragmatic Programmer's Guide Contents ^ class Module Parent: Object Version: Index: constants nesting new ancestors ... public Subclasses: Class A Module is a collection of methods and constants. The methods in a module may be instance methods or module methods. Instance methods appear as methods in a class when the module is included, module methods do not. Conversely, module methods may be called without creating an encapsulating object, while instance methods may not. See Module#module_function on page 346. In the descriptions that follow, the parameter aSymbol refers to a symbol, which is either a quoted string or a Symbol (such as :name module Mod include Math CONST = 1 def meth end end Mod.type Module Mod.constants ["CONST", "E", "PI"] Mod.instance_methods ["meth"] class methods constants anArray Returns an array of the names of all constants defined in the system. This list includes the names of all modules and classes. p Module.constants.sort[1..5] produces: ["ARGV", "ArgumentError", "Array", "Bignum", "Binding"]

59. Search.cpan.org: Tels / Math-BigInt-Constant-1.05 Tels mathBigInt-constant-1.05. math-BigInt-constant-1.05. This Release, math-BigInt-constant-1.05, Other Releases, math-BigInt-constant-1.04 11 Feb 2002. http://search.cpan.org/~tels/Math-BigInt-Constant-1.05/

Home Authors Recent News ... Feedback in All Modules Distributions Authors Tels > Math-BigInt-Constant-1.05 Math-BigInt-Constant-1.05 This Release Math-BigInt-Constant-1.05 Download Browse 13 Jan 2004 Other Releases Math-BigInt-Constant-1.04 11 Feb 2002 Links CPAN Testers View/Report Bugs Tools CPAN Testers PASS (2) [ View Rating Rate this distribution License Unknown Special Files CHANGES INSTALL LICENSE MANIFEST ... SIGNATURE Modules Math::BigInt::Constant Arbitrary sized constant integers

60. Search.cpan.org: Math::Symbolic::Constant - Constants In Symbolic Calculations Steffen Müller mathSymbolic-0.132 mathSymbolicConstant. Module Version 0.132 Source mathSymbolicConstant - Constants in symbolic calculations. http://search.cpan.org/~smueller/Math-Symbolic-0.132/lib/Math/Symbolic/Constant.

Home Authors Recent News ... Feedback in All Modules Distributions Authors Math-Symbolic-0.132 > Math::Symbolic::Constant Module Version: 0.132 Source Latest Release: Math-Symbolic-0.133 NAME SYNOPSIS DESCRIPTION ... SEE ALSO NAME SYNOPSIS DESCRIPTION This module implements numeric constants for Math::Symbolic trees. EXPORT None by default. METHODS Constructor new Constructor zero Constructor one Constructor euler Constructor pi Method value object , not just every named variable. Method signature Method explicit_signature Method special Method to_string Returns a string representation of the constant. Method term_type Returns the type of the term. (T_CONSTANT) AUTHOR List of contributors: SEE ALSO Math::Symbolic

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