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| [[File:GabrielHorn.png|thumb|3D illustration of Gabriel's Horn.]]
| | Hello from Italy. I'm glad to be here. My first name is Lukas. <br>I live in a small city called Santa Maria Della Versa in western Italy.<br>I was also born in Santa Maria Della Versa 37 years ago. Married in May 2011. I'm working at the post office.<br><br>Also visit my blog post: Oltramare ([http://richmondmailer.com Read Alot more]) |
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| '''Gabriel's Horn''' (also called '''Torricelli's trumpet''') is a [[geometry|geometric]] figure which has [[infinity|infinite]] [[surface area]] but finite [[volume]]. The name refers to the tradition identifying the [[Gabriel (archangel)|Archangel Gabriel]] as the angel who blows the horn to announce [[Judgment Day]], associating the divine, or infinite, with the finite. The properties of this figure were first studied by [[Italy|Italian]] physicist and mathematician [[Evangelista Torricelli]].
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| ==Mathematical definition==
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| [[File:Rectangular hyperbola.svg|thumb|Graph of ''y'' = 1/''x'']]
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| Gabriel's horn is formed by taking the [[Graph of a function|graph]] of <math>y= \frac{1} {x}</math>, with the [[Domain of a function|domain]] <math>x \ge 1</math> (thus avoiding the [[asymptote]] at ''x'' = 0) and [[surface of revolution|rotating]] it in three [[dimension]]s about the x-axis. The discovery was made using [[Cavalieri's principle]] before the invention of [[calculus]], but today calculus can be used to calculate the volume and surface area of the horn between ''x'' = 1 and ''x'' = ''a'', where ''a'' > 1. Using integration (see [[Solid of revolution]] and [[Surface of revolution]] for details), it is possible to find the volume <math>V</math> and the surface area <math>A</math>:
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| :<math>V = \pi \int_{1}^{a} {1 \over x^2}\, \mathrm{d}x = \pi \left( 1 - {1 \over a} \right)</math>
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| :<math>A = 2\pi \int_{1}^{a} {1 \over x} \sqrt{1+{1 \over x^4}} \,\mathrm{d}x \geq 2\pi \int_{1}^{a} {1 \over x} \,\mathrm{d}x = 2\pi \ln a.</math>
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| <math>a</math> can be as large as required, but it can be seen from the equation that the volume of the part of the horn between <math>x = 1</math> and <math>x = a</math> will never exceed <math>\pi</math>; however, it ''will'' get closer and closer to ''<math>\pi</math>'' as <math>a</math> becomes larger. Mathematically, the volume ''approaches ''<math>\pi</math>'' as <math>a</math> approaches infinity''. Using the [[Limit of a function|limit]] notation of calculus, the volume may be expressed as:
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| :<math>\lim_{a \to \infty}\pi \left( 1 - {1 \over a} \right) = \pi.</math>
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| As for the area, the above shows that the area is greater than <math>2\pi</math> times the [[natural logarithm]] of <math>a</math>. There is no [[upper bound]] for the natural logarithm of <math>a</math> as it approaches infinity. That means, in this case, that the horn has an infinite surface area. That is to say;
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| :<math>2 \pi \ln a \rightarrow \infty </math> as <math> a \rightarrow \infty</math>
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| or
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| :<math>\lim_{a \to \infty}2 \pi \ln a = \infty.</math>
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| ==Apparent paradox==
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| When the properties of Gabriel's Horn were discovered, the fact that the rotation of an infinitely large section of the ''x-y'' plane about the ''x''-axis generates an object of finite volume was considered [[paradox]]ical.
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| Actually the section lying in the ''x-y'' plane is the only one which has an infinite area, while any other, parallel to it, has a finite area. The volume, being calculated from the 'weighted sum' of sections, is of course finite.
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| The more obvious approach is to treat the horn as a stack of disks with diminishing radii. As their shape is identical, one is tempted to calculate just the sum of radii which produces the harmonic series that goes to infinity. A more careful consideration shows that one should calculate the sum of their squares.
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| Every disk has a radius r=1/x and an area π.r<sup>2</sup> or π/x<sup>2</sup>. The series 1/x is divergent but for any real ε>0,
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| 1/''x''<sup>1+ε</sup> converges.
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| The apparent paradox formed part of a great dispute over the nature of infinity involving many of the key thinkers of the time including [[Thomas Hobbes]], [[John Wallis]] and [[Galileo Galilei]].<ref>{{cite book|title=Nonplussed!: mathematical proof of implausible ideas|first=Julian|last=Havil|publisher=Princeton University Press|year=2007|isbn=0-691-12056-0|pages=82–91}}</ref>
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| ===Painter's Paradox===
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| Since the Horn has finite volume but infinite surface area, it seems that it could be filled with a finite quantity of paint, and yet that paint would not be sufficient to coat its inner surface – an apparent paradox. In fact, in a theoretical mathematical sense, a finite amount of paint ''can'' coat an infinite area, provided the thickness of the coat becomes vanishingly small "quickly enough" to compensate for the ever-expanding area, which in this case is forced to happen to an inner-surface coat as the horn narrows. However, to coat the outer surface of the horn with a constant thickness of paint, no matter how thin, ''would'' require an infinite amount of paint.<ref>{{cite book|title=Infinity: The Quest to Think the Unthinkable|first=Brian|last=Clegg|publisher=Robinson (Constable & Robinson Ltd)|year=2003|isbn=978-1-84119-650-3|pages=239–242}}</ref> On the other hand, to coat the inner surface, ''requires'' that the paint thickness be vanishingly small, else it will not fit between the sides of the horn as they become infinitely close.
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| Of course, in reality, paint is not infinitely divisible, and at some point the horn would become too narrow for even one molecule to pass.
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| ==Converse==
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| The converse phenomenon of Gabriel's horn – a surface of revolution which has a ''finite'' surface area but an ''infinite'' volume – cannot occur:
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| '''Theorem:'''
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| :Let <math>f: [1,\infty) \to [0,\infty)</math> be a continuously differentiable function.
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| :Write <math>S</math> for the [[solid of revolution]] of the graph <math>y=f(x)</math> about the <math>x</math>-axis.
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| :''If the surface area of <math>S</math> is finite, then so is the volume.''
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| '''Proof:'''
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| :Since the lateral surface area <math>A</math> is finite, note the [[limit superior]]:
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| :<math>
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| \lim_{t \to \infty} \sup_{x \geq t} f(x)^2 ~-~ f(1)^2 = \limsup_{t \to \infty} \int_{1}^{t} (f(x)^2)' \,\mathrm{d}x
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| </math>
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| :<math>
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| \leqslant \int_{1}^{\infty} |(f(x)^2)'| \,\mathrm{d}x = \int_{1}^{\infty} 2 f(x) |f'(x)| \,\mathrm{d}x
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| </math>
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| :<math>
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| \leqslant \int_{1}^{\infty} 2 f(x) \sqrt{1 + f'(x)^2} \,\mathrm{d}x
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| </math>
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| :<math>
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| = {A \over \pi} < \infty.
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| </math>
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| :Therefore, there exists a <math>t_0</math> such that the [[supremum]] <math>\sup\{f(x) \mid x \geq t_0\}</math> is finite.
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| '''Hence,
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| :<math>
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| M = \sup\{f(x) \mid x \geq 1\}</math> must be finite since <math>f</math> is a [[continuous function]], which implies that
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| :<math>f</math> is bounded on the interval <math>[1,\infty)</math>. | |
| '''Finally, note that the volume:'''
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| :<math>
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| V = \int_{1}^{\infty} f(x) \cdot \pi f(x) \,\mathrm{d}x
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| \leqslant \int_{1}^{\infty} {M \over 2} \cdot 2 \pi f(x) \,\mathrm{d}x
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| \leqslant {M \over 2} \cdot \int_{1}^{\infty} 2 \pi f(x) \sqrt{1 + f'(x)^2} \,\mathrm{d}x
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| </math>
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| :<math>
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| = {M \over 2} \cdot A.
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| </math>
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| '''Therefore:'''<br>
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| :''if the area <math>A</math> is finite, then the volume <math>V</math> must also be finite.''
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| == See also ==
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| * [[Hyperbola]]
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| * [[Koch snowflake]]
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| * [[Pseudosphere]]
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| * [[Shape of the Universe]]
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| * [[Surface of revolution]]
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| * [[Zeno's paradoxes]]
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| == Further reading ==
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| * ''Gabriel's Other Possessions'', Melvin Royer, {{DOI|10.1080/10511970.2010.517601}}
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| * ''Gabriel's Wedding Cake'', Julian F. Fleron, http://people.emich.edu/aross15/math121/misc/gabriels-horn-ma044.pdf
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| * ''A Paradoxical Paint Pail'', Mark Lynch, http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-paradoxical-paint-pail
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| * ''Supersolids: Solids Having Finite Volume and Infinite Surfaces'', William P. Love, {{jstor|27966098}}
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| ==References==
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| {{Reflist}}
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| ==External links==
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| *[http://curvebank.calstatela.edu/torricelli/torricelli.htm Information and diagrams about Gabriel's Horn]
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| *[http://planetmath.org/encyclopedia/TorricellisTrumpet.html Torricelli's Trumpet at PlanetMath]
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| *{{MathWorld|title=Gabriel's Horn|urlname=GabrielsHorn}}
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| * [http://demonstrations.wolfram.com/GabrielsHorn/ "Gabriel's Horn"] by John Snyder, the [[Wolfram Demonstrations Project]], 2007.
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| * [http://www.palmbeachstate.edu/honors/Documents/jeansergejoseph.pdf Gabriel's Horn: An Understanding of a Solid with Finite Volume and Infinite Surface Area] by Jean S. Joseph.
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| [[Category:Mathematics paradoxes]]
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| [[Category:Calculus]]
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| [[Category:Gabriel]]
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| [[Category:Surfaces]]
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Hello from Italy. I'm glad to be here. My first name is Lukas.
I live in a small city called Santa Maria Della Versa in western Italy.
I was also born in Santa Maria Della Versa 37 years ago. Married in May 2011. I'm working at the post office.
Also visit my blog post: Oltramare (Read Alot more)