Pappus Chain

In geometry, the Pappus chain is a ring of circles between two tangent circles investigated 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

:$\overline + \overline = \left( r_ + r_ \right) + \left( r_ - r_ \right) = r_ + r_$

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 ''n''th circle in the chain is:

:$\left(x_n,y_n\right)=\left(\frac ~,~\frac \right)$

Radii of the circles

If ''r'' = ''AC''/''AB'', then the radius of the ''n''th circle in the chain is:
:$r_n=\frac$

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.

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

Bibliography

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External links

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*{{cite web, last=Tan, first=Stephen, title=Arbelos, url=http://www.math.ubc.ca/~cass/courses/m308/projects/tan/html/home.html

Arbelos
Inversive geometry
Circle packing