# Steiner Chain

In geometry, a Steiner chain is a set of n circles, all of which are tangent to two given non-intersecting circles (blue and red in Figure 1), where n is finite and each circle in the chain is tangent to the previous and next circles in the chain. In the usual closed Steiner chains, the first and last (nth) circles are also tangent to each other; by contrast, in open Steiner chains, they need not be. The given circles α and β do not intersect, but otherwise are unconstrained; the smaller circle may lie completely inside or outside of the larger circle. In these cases, the centers of Steiner-chain circles lie on an ellipse or a hyperbola, respectively.

Steiner chains are named after Jakob Steiner, who defined them in the 19th century and discovered many of their properties. A fundamental result is Steiner's porism, which states:

If at least one closed Steiner chain of n circles exists for two given circles α and β, then there is an infinite number of closed Steiner chains of n circles; and any circle tangent to α and β in the same way is a member of such a chain.

"Tangent in the same way" means that the arbitrary circle is internally or externally tangent in the same way as a circle of the original Steiner chain. A porism is a type of theorem relating to the number of solutions and the conditions on it. Porisms often describe a geometrical figure that cannot exist unless a condition is met, but otherwise may exist in infinite number; another example is Poncelet's porism.

The method of circle inversion is helpful in treating Steiner chains. Since it preserves tangencies, angles and circles, inversion transforms one Steiner chain into another of the same number of circles. One particular choice of inversion transforms the given circles α and β into concentric circles; in this case, all the circles of the Steiner chain have the same size and can "roll" around in the annulus between the circles similar to ball bearings. This standard configuration allows several properties of Steiner chains to be derived, e.g., its points of tangencies always lie on a circle. Several generalizations of Steiner chains exist, most notably Soddy's hexlet and Pappus chains.

### Other articles related to "steiner chain, chain":

Pappus Chain - Properties - Steiner Chain
... and tangencies on a circle, the Pappus chain is analogous to the Steiner chain, in which finitely many circles are tangent to two circles ...
Steiner Chain - Generalizations
... The simplest generalization of a Steiner chain is to allow the given circles to touch or intersect one another ... In the former case, this corresponds to a Pappus chain, which has an infinite number of circles ... is a three-dimensional generalization of a Steiner chain of six circles ...

### Famous quotes containing the words chain and/or steiner:

Loyalty to petrified opinions never yet broke a chain or freed a human soul in this world—and never will.
Mark Twain [Samuel Langhorne Clemens] (1835–1910)

The violent illiteracies of the graffiti, the clenched silence of the adolescent, the nonsense cries from the stage-happening, are resolutely strategic. The insurgent and the freak-out have broken off discourse with a cultural system which they despise as a cruel, antiquated fraud. They will not bandy words with it. Accept, even momentarily, the conventions of literate linguistic exchange, and you are caught in the net of the old values, of the grammars that can condescend or enslave.
—George Steiner (b. 1929)