Pappus chain

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"_U and "C"_V, which are tangent at the point A and where "C"_U is enclosed by "C"_V. Let the radii of these two circles be denoted as "r"_U and "r"_V, 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"_U (the inner circle) and internally tangent to "C"_V (the outer circle). Let the radius, diameter and center point of the "n"^th circle of the Pappus chain be denoted as "r"_"n", "d"_"n" and P_"n", 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"^th circle of the Pappus chain to the two centers U and V of the arbelos circles equals a constant

:$overline\{mathbf\{P\}\_\{n\}mathbf\{U\; +\; overline\{mathbf\{P\}\_\{n\}mathbf\{V\; =\; left(\; r\_\{U\}\; +\; r\_\{n\}\; ight)\; +\; left(\; r\_\{V\}\; -\; r\_\{n\}\; ight)\; =\; r\_\{U\}\; +\; r\_\{V\}$

Thus, the 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 of the "n"th circle in the chain is:

:$left(x\_n,y\_n\; ight)=left(frac\; \{r(1+r)\}\{2\; [n^2(1-r)^2+r]\; \}~,~frac\; \{nr(1-r)\}\{n^2(1-r)^2+r\}\; ight)$

Radii of the circles

If "r" = "AC"/"AB", then the radius of the of the "n"th circle in the chain is::$r\_n=frac\; \{(1-r)r\}\{2\; [n^2(1-r)^2+r]\; \}$

Circle inversion

The height "h"_"n" of the center of the "n"^th circle above the base diameter ACB equals "n" times "d"_"n". [Ogilvy, pp. 54–55.] This may be shown by inverting in a circle centered on the tangent point A. The circle of inversion is chosen to intersect the "n"^th circle perpendicularly, so that the "n"^th circle is transformed into itself. The two arbelos circles, "C"_U and "C"_V, are transformed into parallel lines tangent to and sandwiching the "n"^th circle; hence, the other circles of the Pappus chain are transformed into similarly sandwiched circles of the same diameter. The initial circle "C"₀ and the final circle "C"_"n" each contribute ½"d"_"n" to the height "h"_"n", whereas the circles "C"₁–"C"_"n"−1 each contribute "d"_"n". Adding these contributions together yields the equation "h"_"n" = "n" "d"_"n".

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 inversion centered at point A transforms the arbelos circles "C"_U and "C"_V 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.

teiner 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

Bibliography

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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