Four circles, each tangent to the other three, are locked into a single relation. Let
curvatureCurvature k = 1 / radius. A bigger k is a tighter circle. The outer wrapping circle bends the opposite way to the ones inside it, so its k is taken as negative. k measure how tightly a circle bends. Then
(k₁ + k₂ + k₃ + k₄)² = 2 (k₁² + k₂² + k₃² + k₄²)
This is Descartes' circle theoremDescartes' circle theorem (1643). For four mutually tangent circles the four curvatures satisfy this one equation. Given three, it fixes the fourth, so you can keep dropping a tangent circle into every gap forever.. Given any three mutually tangent circles it fixes the fourth, so every gap can be filled with a new tangent circle, endlessly: an Apollonian gasket.
From plane to tube
Take that flat gasket and treat its disk as the cross-section of a tube. Sweep the
cross-section once around a large circle and the tube closes into a torus. Every circle
in the gasket traces its own nested tube, so the whole packing becomes a wireframe torus woven
from gaskets: the wrapping circle becomes the torus surface, and the circles inside it become
the nested structure within.
The delay line: local vs global
The torus is cut into slices around the ring. Change a curvature and you can send that
change two ways.
All at once: every slice reshapes together, a
globalMean-field coupling. Every element feels the average of all the others in the same instant, with no travel time. The opposite of a signal that has to propagate from neighbour to neighbour. update where the whole ring feels it in the same instant.
As a wave: each slice sees the change a moment later than the one before it, so the new shape
travels around the ring like a pulse.
That is the difference between global mean-field coupling and local coupling carried neighbour
to neighbour, the same distinction that runs under every page here. One toggle, and you can watch it.
Lineage
The pieces are old and well named: tangent-circle problems from Apollonius of Perga, the
curvature relation from Descartes (1643), rediscovered by Frederick SoddyFrederick Soddy. Chemist and Nobel laureate who set Descartes' theorem to verse in "The Kiss Precise" (1936) and extended it to spheres. Hence "Soddy circles." as the verse
"The Kiss Precise" (1936). The fresh edge is only the framing: an Apollonian gasket revolved into
a torus, with a curvature change made into a travelling wave. Cite the giants; claim the
demonstration, not the discovery.
Apollonius of Perga, Tangencies (c. 200 BCE) · Descartes (1643) · Soddy,
Nature "The Kiss Precise" (1936).
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