Theme

The theme: The lattice forbids — page 6

Periodicity is a strong constraint. It rules out five-fold rotations, most rotation orders above six, and a great many patterns that look perfectly reasonable until the arithmetic is done.

The fibres of a cubic, read off a conic. Over the field of 17 elements, the points of the conic x² + xy + y² = 14, which is −β for β = 3. Each point (x, y) with x, y and −x − y all different is one ordering of a set of three roots on which t³ + 3t takes a single value; the 12 such points fall into 2 sets of six. The 6 dashed points lie on the lines where two roots coincide.

Lattices

The third power's shortest vectors lie on a conic

Craig's lattices reach their bound when a polynomial over a prime field takes one value at k points and another at k more, and each shortest vector is such a pair. Counting them for the third power turns out to be counting points on a conic, and the answer is a closed formula: thirty-two orbits at seventeen, as a sixteen-dimensional search had found without explanation. The fourth power is a polynomial in p at exactly the primes that do not split in the field of √−2, and at the others it is not.

How long a second sweep must be, against how far its axis leans. After a full turn about a kept axis, the least arc about a second axis, tilted from the first by the angle on the horizontal axis, that reaches every orbit the first sweep never saw, for 4/mmm about c, mmm about c, 2/m about b at 1.4 Å. Below about twenty degrees of tilt some missing orbits stay blind whatever the arc; just past that threshold the arc is near eighty degrees, and it falls to under twenty at a right angle.

How it is known

The second spindle must lean past one Bragg angle

A crystal turned about an axis its Laue group keeps leaves a hole no amount of turning fills: on the four-fold axis of a tetragonal crystal, one reflection class in nine is never measured. A second axis fills it, but only if it leans far enough, and the threshold is not a rule of thumb. It is the Bragg angle of the outermost reflection lying on the first axis itself, twenty-one degrees at 1.4 ångström. Past it the second sweep shortens quickly, to nineteen degrees at a right angle.

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