Soddy's hexlet

Soddy's hexlet is a ring of six mutually tangent spheres, each tangent to the same two fixed spheres. Its configuration is constrained by a relation among the spheres' curvatures.

Soddy's formula: A relation among the signed curvatures of mutually tangent spheres in three-dimensional space. It governs the curvatures in a hexlet and helps explain the ring's closure.
Tangency: Contact between geometric objects at a shared point or along a shared boundary without crossing. Every neighboring pair in the hexlet meets at a point.
Frederick Soddy: An English chemist who received the 1921 Nobel Prize in Chemistry for research on radioactive substances and isotopes. Soddy described the sphere configuration and its curvature relation in a 1936 poem.
Coaxial spheres: Spheres whose centers lie on a common line. The two fixed spheres of a hexlet are commonly arranged coaxially.
Oriented sphere: A sphere together with a choice of orientation, allowing its curvature to be treated as signed. Signed curvature distinguishes the two fixed spheres when applying Soddy's formula.
Curvature: A measure of how rapidly a curve or surface departs from being straight or flat. Sphere curvature, taken as reciprocal radius, is the quantity used in Soddy's relation.
The Kiss Precise: A 1936 poem by Frederick Soddy presenting a formula for curvatures of mutually tangent circles and spheres. The poem popularized the mathematical relation associated with Soddy's name.
Chain of tangent spheres: A finite sequence of spheres in which each sphere touches its neighbors. The six moving spheres form a closed tangent chain between the fixed spheres.
Sphere packing: An arrangement of non-overlapping spheres in space, often with specified contacts between spheres. A hexlet is a tightly constrained local configuration of tangent spheres.
Three-dimensional geometry: The branch of geometry that studies points, lines, surfaces, and solids in three-dimensional space. The hexlet is a spatial configuration of spheres rather than a planar circle arrangement.

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Six circles theoremCompared with: Its six-step closure resembles the theorem’s cyclic arrangement but uses sphere tangency.