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Concept

Kissing number

The largest number of equal non-overlapping balls that can all touch one central ball of the same size. It is 6 in the plane, 12 in space and 24 in four dimensions, and only a handful of other dimensions have known values.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Two hundred and forty roots, built and counted. An eight by eight grid whose upper cells stand for the pairs of coordinates a root can use, beside bars counting how many roots stand at each angle from a given root: 1, 56, 126, 56 and 1.

Two hundred and forty directions

The quaternions have twenty-four units and they are the vertices of the most symmetric object in four dimensions. Eight dimensions has two hundred and forty of them, and the quaternions turn out to be how they are built — twice over, with a hundred and ninety-two left to explain.

algebra · Quaternions The most rotations that can be kept a given angle apart. Linear programming bound on the number of rotations pairwise at least θ apart, θ from 54° to 180°, log scale. Groups: four-group 4 at 180° (bound 4.00); tetrahedron 12 at 120° (bound 12.00); cube 24 at 90° (bound 28.94); icosahedron 60 at 72° (bound 60.00).

Sixty rotations as far apart as rotations can be

The sixty rotations of the icosahedron are each at least 72 degrees from every other. A single polynomial, with coefficients that are all positive and values that are nowhere positive where it matters, proves that no sixty rotations do better and no sixty-one do as well. The same calculation proves the tetrahedron's twelve best at 120 degrees, and leaves the cube's twenty-four short of what it allows.

algebra · Quaternions

Named alongside it

The objects these essays reach for when they reach for this one.

QuaternionSymmetry groupChebyshev polynomialDualityExceptional objectGolden ratioLatticeLinear programmingRoot systemRotationUnit group

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