Journey 6 of 10 · 6 steps
The Breathing Solid
Fuller’s vector equilibrium
Twelve spheres around one, and the twisting motion that turns one Platonic solid into the next.
In the app each step moves: figures draw themselves, rise from the page, turn and slice, and the narration can be read aloud. Here they are held still, one picture each.
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Step 1 · Sphere Packing
Twelve around one
At most twelve equal spheres can touch a thirteenth. Their centres form the cuboctahedron, which Buckminster Fuller called the vector equilibrium: every edge equals the radius.
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Step 2 · Sphere Packing
From above
Look down a 3-fold axis and the cluster flattens into a hexagonal flower, the same lattice as the Flower of Life.
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Step 3 · The Jitterbug
Equilibrium
Keep only the eight triangles of the vector equilibrium, and let them hinge at their corners.
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Step 4 · The Jitterbug
The jitterbug
Twist the triangles and the whole form contracts: through the icosahedron (at 1 − 1/φ) and down to the octahedron, then back. The triangles never change size.
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Step 5 · Icosahedron
Icosahedron
Halfway through the twist, the twelve corners form a perfect icosahedron.
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Step 6 · Octahedron
Octahedron
Fully contracted, pairs of corners meet and the octahedron remains. Fold it once more and it would collapse into a tetrahedron.