Electromagnetically induced grating: Difference between revisions

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[[Image:Salinon.svg|thumb|right|350px|The salinon (red) and the circle (blue) have the same area.]]


The '''salinon''' (meaning "salt-cellar" in Greek) is a [[geometrical figure]] that consists of four [[semicircle]]s. It was first introduced in the ''[[Book of Lemmas]]'', a work attributed to [[Archimedes]].<ref name=WOLF>
{{citeweb
| title="Salinon." From MathWorld--A Wolfram Web Resource
| last=Weisstein
| first=Eric W.
| url=http://mathworld.wolfram.com/Salinon.html
| accessdate=2008-04-14
}}</ref>


==Construction==
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Let ''O'' be the origin on a [[Cartesian plane]]. Let ''A'', ''D'', ''E'', and ''B'' be four points on a line, in that order, with ''O'' bisecting line ''AB''. Let ''AD'' = ''EB''. Semicircles are drawn above line ''AB'' with [[diameter]]s ''AB'', ''AD'', and ''EB'', and another semicircle is drawn below with diameter ''DE''. A salinon is the figure bounded by these four semicircles.<ref>
{{Cite book
| last=Nelsen
| first=Roger B.
| chapter=Proof Without Words: The Area of a Salinon
| title=Mathematics Magazine
| page=130
| year=2002
| url=http://www.lclark.edu/~mathsci/salinon.pdf
| postscript=<!--None-->
}}</ref>
 
==Properties==
===Area===
Archimedes introduced the salinon in his ''Book of Lemmas'' by applying Book II, Proposition 10 of [[Euclid's Elements|Euclid's ''Elements'']]. Archimedes noted that "the area of the figure bounded by the circumferences of all the semicircles [is] equal to the area of the circle on CF as diameter."<ref name=CUT>
{{cite web
|url=http://www.cut-the-knot.org/proofs/Lemma.shtml
|title=Salinon: From Archimedes' Book of Lemmas from Interactive Mathematics Miscellany and Puzzles
|accessdate=2008-04-15
|last=Bogomolny
|first=A.
|work=from Interactive Mathematics Miscellany and Puzzles
}}</ref>
 
Namely, the area of the salinon is:
:<math>A=\frac{1}{4}\pi\left(r_1+r_2\right)^2.</math><ref name=WOLF/>
 
====Proof====
Let the radius of the [[midpoint]] of ''AD'' and ''EB'' be denoted as ''G'' and ''H'', respectively. Therefore, ''AG''&nbsp;=&nbsp;''GD''&nbsp;=&nbsp;''EH''&nbsp;=&nbsp;''HB''&nbsp;=&nbsp;''r''<sub>1</sub>. Because ''DO'', ''OF'', and ''OE'' are all radii to the same semicircle, ''DO''&nbsp;=&nbsp;''OF''&nbsp;=&nbsp;''OE''&nbsp;=&nbsp;''r''<sub>2</sub>. By segment addition, ''AG''&nbsp;+&nbsp;''GD''&nbsp;+&nbsp;''DO''&nbsp;=&nbsp;''OE''&nbsp;+&nbsp;''EH''&nbsp;+&nbsp;''HB''&nbsp;=&nbsp;2''r''<sub>1</sub>&nbsp;+&nbsp;''r''<sub>2</sub>. Since ''AB'' is the diameter of the salinon, ''CF'' is the line of symmetry. Because they all are radii of the same semicircle, ''AO''&nbsp;=&nbsp;''BO''&nbsp;=&nbsp;''CO''&nbsp;=&nbsp;2''r''<sub>1</sub>&nbsp;+&nbsp;''r''<sub>2</sub>.
 
Let ''P'' be the center of the large circle. Because ''CO''&nbsp;=&nbsp;2''r''<sub>1</sub>&nbsp;+&nbsp;''r''<sub>2</sub> and ''OF''&nbsp;=&nbsp;''r''<sub>2</sub>, ''CF''&nbsp;=&nbsp;2''r''<sub>1</sub>&nbsp;+&nbsp;2''r''<sub>2</sub>. Therefore, the radius of the circle is ''r''<sub>1</sub>&nbsp;+&nbsp;''r''<sub>2</sub>. The area of the circle&nbsp;=&nbsp;π(''r''<sub>1</sub>&nbsp;+&nbsp;''r''<sub>2</sub>)<sup>2</sup>.
 
Let ''x'' = ''r''<sub>1</sub> and ''y'' = ''r''<sub>2</sub>. The area of the semicircle with diameter ''AB'' is:
 
:<math>AB=\frac{1}{2}\pi\left(2x+y\right)^2</math>.
 
The area of the semicircle with diameter ''DE'' is:
 
:<math>DE=\frac{1}{2}\pi y^2</math>
 
The area of each of the semicircles with diameters ''AD'' and ''EB'' is
 
:<math>AD=EB=\frac{1}{2}\pi x^2</math>
 
Therefore, the area of the salinon is:
 
:<math>\frac{1}{2}\pi\left(\left(2x+y\right)^2-2x^2+y^2\right)=\frac{1}{2}\pi\left(2x^2+4xy+2y^2\right)=\pi\left(x^2+2xy+y^2\right)=\pi\left(x+y\right)^2=\pi\left(r_1+r_2\right)^2</math>
 
Q.E.D.<ref>
{{citeweb
|url=http://jwilson.coe.uga.edu/emt668/EMAT6680.2000/Umberger/EMAT6690smu/Essay4smu/Essay4smu.html
|last=Umberger
|first=Shannon
|title=Essay # 4 - The Arbelos and the Salinon
|accessdate=2008-04-18
}}</ref>
 
===Arbelos===
 
Should points ''D'' and ''E'' converge with ''O'', it would form an [[arbelos]], another one of Archimedes' creations, with [[symmetry]] along the [[y-axis]].<ref name=CUT/>
 
==References==
{{reflist}}
 
[[Category:Geometric shapes]]
[[Category:Archimedes]]

Latest revision as of 23:25, 17 August 2014


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