Pappos chain

The Pappos chain in the Arbelos

The pappos chain (also pappus chain ) is geometrically an infinite series of touching circles in an arbelos . It is named after the Greek geometer Pappos of Alexandria , who first examined it in the 3rd century .

construction

Pappos chain in the mirrored Arbelos

An arbelos is formed by the three semicircles above , and (sickle-shaped figure in the figure above ). The inscribed circle of the Arbelos with the center is the first circle of the Pappos chain, the others (with the centers , etc.) result from a series of circles that form the previous circle of the chain, the large semicircle above and one of the two smaller semicircles touch. In the illustration, to which the following text also refers, this is the left semicircle above , the chain could just as easily have been continued to the right ( touching the semicircle above ). ${\displaystyle AB}$ ${\displaystyle AC}$ ${\displaystyle CB}$ ${\displaystyle P_{1}}$ ${\displaystyle P_{2}}$ ${\displaystyle P_{3}}$ ${\displaystyle AB}$ ${\displaystyle AC}$ ${\displaystyle CB}$

You can also look at the Pappos chain in a mirrored Arbelos, then the Arbelos semicircle, which does not touch all the circles of the chain (here the one above ), becomes a link of the chain. ${\displaystyle AB}$ ${\displaystyle CB}$

properties

Distance of circle 3 from the Arbelos baseline

The circles of the Pappos chain are numbered as follows: The Arbelos semicircle added to the circle above is given the number 0, the Arbelos inscribed circle the number 1 u. s. f. (according to the indexing of the circle centers in the figure above), the radii of these circles are denoted by . Let the radii of the two small Arbelos semicircles be and . ${\displaystyle CB}$ ${\displaystyle P_{n}}$ ${\displaystyle r_{n}}$ ${\displaystyle a={\overline {CB}}/2=r_{0}}$ ${\displaystyle b={\overline {AC}}/2}$

• The circle with the number has the radius . ${\displaystyle n}$
  ${\displaystyle r_{n}={\frac {ab(a+b)}{n^{2}a^{2}+b(a+b)}}}$
• The center of the circle with the number is the distance from the Arbelos baseline . The figure on the right illustrates this for the circle with the number 3. ${\displaystyle P_{n}}$ ${\displaystyle n}$ ${\displaystyle 2nr_{n}}$ ${\displaystyle AB}$
• The centers of the circles of the Pappos chain lie on an ellipse (dashed in the figure above). The focal points of this ellipse are the centers of the lines and . ${\displaystyle AB}$ ${\displaystyle AC}$
• The points at which the circles of the Pappos chain touch each other lie on a circle.

See also

• Steiner chain
• Book of Lemmas

Web links

Commons : Pappus chain  - collection of images

• Floor van Lamoen, Eric W. Weisstein: Pappus Chain . In: MathWorld (English).
• Jürgen Köller: Pappus chain . Math tinkering.
• Walter Fendt: The Pappos chain . (HTML5 app). HTML5 apps for math .
• Bob Allanson: Pappus's Arbelos (animated Java applet, English)