1 unit
Sphere packing · generation 5 990 spheres GPU-instanced exact Soddy–Gossett packing

Starts at generation 5. GPU instancing reaches generation 9: 126,090 spheres.

Of the outer diameter · balanced at 44.95%. Snaps to four equal seeds.

The cyan circle stays tangent to the gold reference circle while radius and curvature vary inversely.

r = 0 ⇒ k = ∞  ·  r = 1 ⇒ k = 1  ·  r = ∞ ⇒ k = 0

Three circles determine the fourth tangent circle.

Match the dashed target: k = 2.

Packing galleryexact tangent seeds
Rendering

Authored MaGE treatment · adaptive quality

Color represents

Radius maps size across the palette. Generation counts reflections. Curvature is |k| = 1/r.

Palette
Independent overlays
Recursive calculationReady

Each construction dims unrelated circles, highlights the three tangent parents, substitutes their curvatures, then draws the one forced circle.

Descartes' circle theorem (k₁+k₂+k₃+k₄)² = 2(k₁²+k₂²+k₃²+k₄²) Select a circle to connect the equation to its construction.