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[[File:Pappus Chain Full.svg|thumb|A Pappus chain]]
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In [[geometry]], the '''Pappus chain''' was created by [[Pappus of Alexandria]] in the 3rd century [[AD]].
 
==Construction==
The [[arbelos]] is defined by two circles, ''C''<sub>U</sub> and ''C''<sub>V</sub>, which are tangent at the point '''A''' and where ''C''<sub>U</sub> is enclosed by ''C''<sub>V</sub>.  Let the radii of these two circles be denoted as ''r''<sub>U</sub> and ''r''<sub>V</sub>, respectively, and let their respective centers be the points '''U''' and '''V'''. The Pappus chain consists of the circles in the shaded grey region, which are externally tangent to ''C''<sub>U</sub> (the inner circle) and internally tangent to ''C''<sub>V</sub> (the outer circle).  Let the radius, diameter and center point of the ''n''<sup>th</sup> circle of the Pappus chain be denoted as ''r''<sub>''n''</sub>, ''d''<sub>''n''</sub> and '''P'''<sub>''n''</sub>, respectively.
 
==Properties==
===Centers of the circles===
====Ellipse====
All the centers of the circles in the Pappus chain are located on a common [[ellipse]], for the following reason. The sum of the distances from the ''n''<sup>th</sup> circle of the Pappus chain to the two centers '''U''' and '''V''' of the arbelos circles equals a constant
 
:<math>
\overline{\mathbf{P}_{n}\mathbf{U}} + \overline{\mathbf{P}_{n}\mathbf{V}} =
\left( r_{U} + r_{n} \right) + \left( r_{V} - r_{n} \right) = r_{U} + r_{V}
</math>
 
Thus, the [[Focus (geometry)|foci]] of this ellipse are '''U''' and '''V''', the centers of the two circles that define the arbelos; these points correspond to the midpoints of the line segments '''AB''' and '''AC''', respectively.
 
====Coordinates====
If ''r'' = ''AC''/''AB'', then the center of the ''n''th circle in the chain is:
 
:<math>\left(x_n,y_n\right)=\left(\frac {r(1+r)}{2[n^2(1-r)^2+r]}~,~\frac {nr(1-r)}{n^2(1-r)^2+r}\right)</math>
 
===Radii of the circles===
If ''r'' = ''AC''/''AB'', then the radius of the ''n''th circle in the chain is:
:<math>r_n=\frac {(1-r)r}{2[n^2(1-r)^2+r]}</math>
 
===Circle inversion===
[[File:Pappus Chain Theorem.svg|thumb|right|250px|Under a particular inversion centered on '''A''', the four initial circles of the Pappus chain are transformed into a stack of four equally sized circles, sandwiched between two parallel lines. This accounts for the height formula ''h''<sub>''n''</sub> = ''n'' ''d''<sub>''n''</sub> and the fact that the original points of tangency lie on a common circle.]]
The height ''h''<sub>''n''</sub> of the center of the ''n''<sup>th</sup> circle above the base diameter '''ACB''' equals ''n'' times ''d''<sub>''n''</sub>.<ref>Ogilvy, pp. 54&ndash;55.</ref>  This may be shown by [[Inversion_(geometry)#Circle_inversion|inverting in a circle]] centered on the tangent point '''A'''.  The circle of inversion is chosen to intersect the ''n''<sup>th</sup> circle perpendicularly, so that the ''n''<sup>th</sup> circle is transformed into itself.  The two arbelos circles, ''C''<sub>U</sub> and ''C''<sub>V</sub>, are transformed into parallel lines tangent to and sandwiching the ''n''<sup>th</sup> circle; hence, the other circles of the Pappus chain are transformed into similarly sandwiched circles of the same diameter. The initial circle ''C''<sub>0</sub> and the final circle ''C''<sub>''n''</sub> each contribute ½''d''<sub>''n''</sub> to the height ''h''<sub>''n''</sub>, whereas the circles ''C''<sub>1</sub>&ndash;''C''<sub>''n''&minus;1</sub> each contribute ''d''<sub>''n''</sub>. Adding these contributions together yields the equation ''h''<sub>''n''</sub> = ''n'' ''d''<sub>''n''</sub>.
 
The same inversion can be used to show that the points where the circles of the Pappus chain are tangent to one another lie on a common circle. As noted above, the [[Inversive geometry|inversion]] centered at point '''A''' transforms the arbelos circles ''C''<sub>U</sub> and ''C''<sub>V</sub> into two parallel lines, and the circles of the Pappus chain into a stack of equally sized circles sandwiched between the two parallel lines.  Hence, the points of tangency between the transformed circles lie on a line midway between the two parallel lines. Undoing the inversion in the circle, this line of tangent points is transformed back into a circle.
 
===Steiner chain===
In these properties of having centers on an ellipse and tangencies on a circle, the Pappus chain is analogous to the [[Steiner chain]], in which finitely many circles are tangent to two circles.
 
==References==
{{reflist|1}}
 
==Bibliography==
 
* {{Cite book | author = Ogilvy CS | year = 1990 | title = Excursions in Geometry | publisher = Dover | isbn = 0-486-26530-7| pages = 54&ndash;55 | postscript = <!--None-->}}
 
* {{Cite book | author = [[Leon Bankoff|Bankoff L]] | contribution = How did Pappus do it? | title = The Mathematical Gardner | editor-first = D. A. | editor-last = Klarner | publisher = Prindle, Weber, & Schmidt | location = Boston | year = 1981 | pages = 112–118 | postscript = <!--None-->}}
 
* {{cite book | author = Johnson RA | year = 1960 | title = Advanced Euclidean Geometry: An elementary treatise on the geometry of the triangle and the circle | edition = reprint of 1929 edition by Houghton Miflin | publisher = Dover Publications | location = New York  | isbn = 978-0-486-46237-0 | pages = 116&ndash;117}}
 
* {{cite book | author = Wells D | year = 1991 | title = The Penguin Dictionary of Curious and Interesting Geometry | publisher = Penguin Books | location = New York | isbn = 0-14-011813-6 | pages = 5&ndash;6}}
 
==External links==
*{{mathworld|title=Pappus Chain|urlnamePappusChain|author=Floer van Lamoen and Eric W. Weisstein}}
*{{citeweb|last=Tan|first=Stephen|title=Arbelos|url=http://www.math.ubc.ca/~cass/courses/m308/projects/tan/html/home.html}}
 
[[Category:Arbelos]]
[[Category:Inversive geometry]]

Latest revision as of 10:23, 26 October 2014

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