Arithmetic derivative

File:Pappus Chain Full.svg

A 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

$\overline{{\mathbf{P}}_n\mathbf{U}}+\overline{{\mathbf{P}}_n\mathbf{V}}=(r_U+r_n)+(r_V-r_n)=r_U+r_V$

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.

If r = AC/AB, then the center of the nth circle in the chain is:

$(x_n,y_n)=(\frac{r(1+r)}{2[n^2(1-r{)}^2+r]},\frac{nr(1-r)}{n^2(1-r{)}^2+r})$

Radii of the circles

If r = AC/AB, then the radius of the nth circle in the chain is:

$r_n=\frac{(1-r)r}{2[n^2(1-r{)}^2+r]}$

Circle inversion

File:Pappus Chain Theorem.svg

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_n = n d_n and the fact that the original points of tangency lie on a common circle.

The height h_n of the center of the n^th circle above the base diameter ACB equals n times d_n. 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 C₀ and the final circle C_n each contribute ½d_n to the height h_n, whereas the circles C₁–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

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Bibliography

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

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