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Soddy's hexlet
In geometry, Soddy's hexlet is a chain of six spheres (shown in grey in Figure 1), each of which is tangent to both of its neighbors and also to three mutually tangent given spheres. In Figure 1, these three spheres are shown as an outer circumscribing sphere (blue), and two spheres (not shown) above and below the plane the centers of the hexlet spheres lie on. In addition, the hexlet spheres are tangent to a fourth sphere (red in Figure 1), which is not tangent to the three others.
According to a theorem published by Frederick Soddy in 1937, it is always possible to find a hexlet for any choice of mutually tangent spheres ''A'', ''B'' and ''C''. Indeed, there is an infinite family of hexlets related by rotation and scaling of the hexlet spheres (Figure 1); in this, Soddy's hexlet is the spherical analog of a Steiner chain of six circles. Consistent with Steiner chains, the centers of the hexlet spheres lie in a single plane, on an ellipse. Soddy's hexlet was also discovered independently in Japan, as shown by Sangaku tablets from 1822 in the Kanagawa prefecture.
==Definition==
Soddy's hexlet is a chain of six spheres, labeled ''S''1–''S''6, each of which is tangent to three given spheres, ''A'', ''B'' and ''C'', that are themselves mutually tangent at three distinct points. (For consistency throughout the article, the hexlet spheres will always be depicted in grey, spheres ''A'' and ''B'' in green, and sphere ''C'' in blue.) The hexlet spheres are also tangent to a fourth fixed sphere ''D'' (always shown in red) that is not tangent to the three others, ''A'', ''B'' and ''C''.
Each sphere of Soddy's hexlet is also tangent to its neighbors in the chain; for example, sphere ''S''4 is tangent to ''S''3 and ''S''5. The chain is closed, meaning that every sphere in the chain has two tangent neighbors; in particular, the initial and final spheres, ''S''1 and ''S''6, are tangent to one another.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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