How many spheres can touch one sphere? Newton said twelve, Gregory said thirteen
In May 1694, Isaac Newton and the astronomer David Gregory sat in Cambridge arguing about stars of equal brightness spread around a central one, and the conversation turned into a geometry problem that would outlive both of them. Take a billiard ball. How many identical balls can touch it at once? Newton was certain the answer was twelve. Gregory thought a thirteenth could be squeezed in. Neither man could prove it, and the question — the kissing number of three-dimensional space — stayed open for the next 259 years.
What makes this worth a fight is that it is genuinely close. Each neighbour hides a patch of the central ball spanning 30° in every direction — about 6.70% of its surface. Twelve neighbours hide 80.38% and leave 19.62% bare, nearly three times what a thirteenth sphere would need. Counting area alone, fourteen spheres would fit. The obstruction is not a shortage of room; it is the shape of the room, and that is a far harder thing to reason about.
Put twelve spheres in the cannonball arrangement — centres at the vertices of a cuboctahedron — and the whole assembly rattles. Every neighbour can slide across the central sphere, and its neighbours can shuffle aside to let it pass. In the plane a ring of six circles is welded solid; here, nothing is. You can even rotate the twelve continuously into the vertices of an icosahedron, a motion Buckminster Fuller called the jitterbug, and in that position no two neighbours touch each other at all — even the closest pair has 3.4° of daylight between them, and you still cannot fit another sphere.
Try it: Shake the arrangement hard and watch the amber free area slosh across the central sphere. It always totals 19.62%, and the largest single empty cap stays stubbornly short of the 60° a thirteenth sphere demands — the cannonball arrangement's deepest hole is 45°, and shaking only redistributes the shortfall. Then switch to the icosahedron: the contact count drops from 24 pairs to zero, the spheres float free of one another, and the answer is still twelve. Press Try to add a 13th and watch where it ends up.
Nobody has ever disputed the flat case. Six coins fit around a coin, and their contact points cut the rim into six arcs of exactly 60° that tile it with nothing left over. There is no slack to argue about, no configuration to check: 6 × 60° = 360°, and the answer falls out of arithmetic. Move up one dimension and the same accounting gives 80.38% coverage with a fifth of the surface spare. The gap between those two pictures is the entire difficulty — and it is why Newton could be right without being able to say why.
Turn the dial and the whole character of the problem changes. On the left, six neighbours consume the rim exactly and the count is forced. On the right, twelve neighbours leave a fifth of the surface bare, and that leftover is what kept the question open for 259 years.
Try it: Turn the dial from the flat picture to the round one and watch the slack row jump from 0.00% to 19.62%. The right-hand panel is an equal-area map of the central sphere, so the amber really is one-fifth of what you are looking at. Press Try a 13th sphere to see the red cap it needs sitting in the deepest hole the arrangement has — and still spilling over the edges. Schütte and van der Waerden finally closed that gap with a proof in 1953.
One dimension up, the answer is 24, and it comes from an object with no three-dimensional analogue whatsoever. Take every permutation of (±1, ±1, 0, 0): twenty-four points, each √2 from the origin, each √2 from its nearest neighbours. Spheres of radius √2⁄2 at all of them touch a central sphere and just miss one another. Those points are the minimal vectors of the D₄ lattice and the vertices of the 24-cell — the one regular polytope that exists in four dimensions and nowhere else, and the only regular self-dual polytope whose faces are not simplices. Oleg Musin proved in 2003 that 24 cannot be beaten, fifty years after the three-dimensional case fell.
Try it: Set the xw slider moving and leave yz at zero for a simple turn, then add the second rotation for a double rotation — something no three-dimensional object can do, since it needs two independent planes to spin in. Vertices swelling toward you are rotating into the fourth dimension, not growing. Switch on the 24 spheres to see the kissing configuration itself.
Dimensions 8 and 24 are the strange ones. In 1979, Odlyzko and Sloane — and independently Levenshtein — used linear programming on spherical codes to prove that 240 spheres in dimension 8 and 196,560 in dimension 24 are the exact maxima. Those proofs landed thirty-seven years before the corresponding packing problems were settled, which is backwards from what you would expect: the kissing number is a question about one sphere's neighbourhood, and that local rigidity is what makes it tractable. Move off those six dimensions and the honest answer stops being a number and becomes a range.
The Newton–Gregory question, settled in 1953.
Hover or tap a column. Six dimensions — 1, 2, 3, 4, 8 and 24 — have an exact answer; everywhere else the honest statement is a range. Dimension 19 is the widest of them: somewhere between 11,948 and 31,066 spheres can touch one, and the lower end of that range moved as recently as 2026.
Try it: Hover across the dimensions and watch the amber bands open up. Dimension 5 is off by four spheres; dimension 12 could be anywhere from 840 to 1,355; dimension 19 spans a factor of 2.6, and its lower end was pushed up as recently as 2026. Every one of those bands is a live question.