Conjugate Chains
- Conjugate Steiner chains with ''n'' = 4
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Steiner chain with the two given circles shown in red and blue.
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Same set of circles, but with a different choice of given circles.
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Same set of circles, but with yet another choice of given circles.
If a Steiner chain has an even number of circles, then any two diametrically opposite circles in the chain can be taken as the two given circles of a new Steiner chain to which the original circles belong. If the original Steiner chain has n circles in m wraps, and the new chain has p circles in q wraps, then the equation holds
A simple example occurs for Steiner chains of four circles (n = 4) and one wrap (m = 1). In this case, the given circles and the Steiner-chain circles are equivalent in that both types of circles are tangent to four others; more generally, Steiner-chain circles are tangent to four circles, but the two given circles are tangent to n circles. In this case, any pair of opposite members of the Steiner chain may be selected as the given circles of another Steiner chain that involves the original given circles. Since m = p = 1 and n = q = 4, Steiner's equation is satisfied:
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Famous quotes containing the word chains:
“‘Rise like Lions after slumber
In unvanquishable number,
Shake your chains to earth like dew
Which in sleep had fallen on you—
Ye are many—they are few.”
—Percy Bysshe Shelley (1792–1822)