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The Shape Sphere Skeleton: K2,3
Verified
Claim: The symmetry stratification of the N=3 shape sphere is the complete
bipartite graph K2,3: exactly 5 vertices (2 equilateral poles + 3 binary collisions)
connected by isosceles meridians. This structure is forced by the S3 permutation action.
Drag to rotate · Scroll to zoom · Gold = equilateral poles · Red = collision points
The Five Vertices
The Kendall shape sphere for three particles has exactly five distinguished points:
- L+ — equilateral triangle (positive orientation)
- L− — equilateral triangle (negative orientation)
- B12 — particles 1 and 2 collide
- B23 — particles 2 and 3 collide
- B31 — particles 3 and 1 collide
Why K2,3
The bipartition is forced: the two equilateral poles have full S3 symmetry (the
“hubs”), while each collision point has S2 symmetry (the “leaves”).
Every isosceles triangle lies on a great circle connecting one hub to one leaf.
Since there are 2 hubs and 3 leaves, and every hub-leaf pair is connected by such a
meridian, the 1-skeleton is the complete bipartite graph K2,3.
This is not a choice of coordinates or a modelling decision. It is the combinatorial
structure forced by the permutation group acting on three distinguishable particles
in Euclidean space, quotiented by translations, rotations, and scale.
Why this matters for Barbour’s programme:
The shape sphere is the arena for everything Julian builds on. K2,3 is its skeleton —
the minimal graph you cannot deform away. The collision points carry the monopole topology
(spin-statistics), the equilateral poles are the minimum-complexity configurations,
and the meridians are the isosceles transition paths between them. Every structural claim
about N=3 shape dynamics passes through these five points.