Pappus chain

The Pappus chain ( also Pappus chain) is in the geometry of an infinite sequence of mutually contacting circles in a Arbelos. It is named after the Greek geometer Pappus of Alexandria, who examined her for the first time in the 3rd century.

Construction

A Arbelos is formed by the three semicircles about, and ( crescent- shaped figure in the upper figure). The incircle of Arbelos with the center of the first circle of the Pappus chain, the other ( with the centers, & c f ) are obtained by concatenation of circles on the respective preceding circle in the chain, the large semicircle and touch one of the two smaller semicircles. In the figure, to which it relates and the more text that is on the left semicircle, the chain could be continued as well to the right ( the semi-circle over contact).

One can consider the Pappus chain in a mirrored at Arbelos, then the added to the circle Arbelos semicircle that does not affect all sections of the chain (in this case the above ) to a link in the chain.

Properties

One Number the circles of the Pappus chain as follows: The for loop added Arbelos - semicircle is assigned the number 0, the Arbelos - inscribed circle the number 1 and f s (corresponding to the indexing of the circle center points in the figure above ), the radii of these circles denote by you. The radii of the two small semicircles are Arbelos and.

• The circle with the number has the radius

• The center of the circle having the number is the distance from the Arbelos baseline. The figure at right illustrates this for the circle with the number 3

• The centers of the circles of the Pappus chain lie on an ellipse ( dashed lines in the figure above ). The focal points of the ellipse are the midpoints of the lines and.

• The points where the circles of the Pappus chain touch each other, located on a circle.