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- Steiner_chain abstract "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.".
- Steiner_chain thumbnail Steiner_chain_12mer.svg?width=300.
- Steiner_chain wikiPageExternalLink ringOfCircles.html.
- Steiner_chain wikiPageExternalLink webstart.
- Steiner_chain wikiPageExternalLink steiner.html.
- Steiner_chain wikiPageID "18866777".
- Steiner_chain wikiPageRevisionID "602458079".
- Steiner_chain hasPhotoCollection Steiner_chain.
- Steiner_chain title "Steiner Chain".
- Steiner_chain urlname "SteinerChain".
- Steiner_chain subject Category:Circles.
- Steiner_chain subject Category:Inversive_geometry.
- Steiner_chain type Abstraction100002137.
- Steiner_chain type Attribute100024264.
- Steiner_chain type Circle113873502.
- Steiner_chain type Circles.
- Steiner_chain type ConicSection113872975.
- Steiner_chain type Ellipse113878306.
- Steiner_chain type Figure113862780.
- Steiner_chain type PlaneFigure113863186.
- Steiner_chain type Shape100027807.
- Steiner_chain comment "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.".
- Steiner_chain label "Cadena de Steiner".
- Steiner_chain label "Catena di Steiner".
- Steiner_chain label "Chaîne de Steiner".
- Steiner_chain label "Corrente de Steiner".
- Steiner_chain label "Steiner chain".
- Steiner_chain label "Steiner-Kette".
- Steiner_chain label "Поризм Штейнера".
- Steiner_chain sameAs Steiner-Kette.
- Steiner_chain sameAs Cadena_de_Steiner.
- Steiner_chain sameAs Chaîne_de_Steiner.
- Steiner_chain sameAs Catena_di_Steiner.
- Steiner_chain sameAs Corrente_de_Steiner.
- Steiner_chain sameAs m.04gkrcn.
- Steiner_chain sameAs Q1368121.
- Steiner_chain sameAs Q1368121.
- Steiner_chain sameAs Steiner_chain.
- Steiner_chain wasDerivedFrom Steiner_chain?oldid=602458079.
- Steiner_chain depiction Steiner_chain_12mer.svg.
- Steiner_chain isPrimaryTopicOf Steiner_chain.
