Crystallography & Symmetry
patterns derived from their groups
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The coincidence relations of the cubic lattice up to Σ25

The coincidence relations of the cubic lattice up to Σ25
The coincidence relations of the cubic lattice up to Σ25. The rotations that bring a cubic lattice into coincidence with itself, up to index 25: 17 distinct relations across 12 indices, each found by enumerating integer quaternions and each index computed as the size of a sublattice rather than from the usual formula — the two are then required to agree. 5 of the 12 indices carry more than one relation, which is why the tables write 13a and 13b. Every index is odd. That is not a feature of this range: a rational orthogonal matrix written in lowest terms has an odd denominator, so the factors of two always cancel.

The rotations that bring a cubic lattice into coincidence with itself, up to index 25: 17 distinct relations across 12 indices, each found by enumerating integer quaternions and each index computed as the size of a sublattice rather than from the usual formula — the two are then required to agree. 5 of the 12 indices carry more than one relation, which is why the tables write 13a and 13b. Every index is odd. That is not a feature of this range: a rational orthogonal matrix written in lowest terms has an odd denominator, so the factors of two always cancel.

5 essays call sigma-series. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about. Every one of this site's 393 essays names its parameters at the call site, which the standard pass of 2026-08-09 established and param-floor holds.

Where it is called

Changing this generator changes every one of these figures.

Σ5: two square lattices at 36.87°. Two square lattices, one turned through 36.87° about a shared point. At this angle one point in 5 lands exactly on a point of the other lattice — 29 of the 149 drawn — and those shared points are themselves a lattice, the coincidence site lattice, of index 5. The angle comes from tan(θ/2) = 1/3, and Σ is the odd part of 3² + 1² = 10. Nothing here is measured: whether a point is shared is decided by an integer congruence. Symmetry at work

Turn a lattice against itself and almost nothing lines up

Two copies of one lattice rotated about a shared point share that point and, at almost every angle, no other. At a discrete set of angles they share a whole sublattice — one point in three, or five, or seven — and a grain boundary built on such an orientation costs a fraction of what a general one costs, because a fraction of the atoms are already where both sides want them.

 Every coincidence index is odd. The rotations that bring a cubic lattice into coincidence with itself, up to index 25: 17 distinct relations across 12 indices, each found by enumerating integer quaternions and each index computed as the size of a sublattice rather than from the usual formula — the two are then required to agree. 5 of the 12 indices carry more than one relation, which is why the tables write 13a and 13b. Every index is odd. That is not a feature of this range: a rational orthogonal matrix written in lowest terms has an odd denominator, so the factors of two always cancel. Symmetry at work

Every coincidence index is odd, and in the plane most of them do not exist

The indices at which two copies of a cubic lattice share points are 3, 5, 7, 9, 11 and every odd number after them. There is a two-line proof that no even index can occur. Ask the same question about a square lattice and the answer is a different list entirely, governed by which numbers are sums of two squares — so Σ3, which is the commonest boundary in every metal, has no plane analogue at all.

 6°: a boundary with a dislocation every 9.5 cells. Two crystals of the same lattice, each turned by half of 6 degrees in opposite senses, meeting on the dashed line. Almost everywhere along it the atoms of one side face the atoms of the other at very nearly the right distance — the boundary is good crystal — and at the marked places the misfit has accumulated to a whole lattice vector and an extra half-plane has to be inserted. Those are the edge dislocations, and they are 9.5 cells apart against the 9.6 that Frank's formula gives. Symmetry at work

A small angle is a row of dislocations

Turn one crystal a degree against another and the coincidence arithmetic says they share almost nothing. The boundary between them is nevertheless nearly perfect crystal, and both statements are true: the misfit stays small for a long way and then, all at once, needs an extra half-plane.

 Σ5: three lattices in one picture. Two copies of the square lattice turned by 36.87 degrees against one another — one drawn pale, one drawn in the second colour — with the points they share ringed. The fine dots are the lattice generated by both together, the DSC lattice, which contains each crystal with index 5 exactly as the coincidences sit inside each crystal with index 5. Three lattices nested at the same index, and the middle one is the crystal. Symmetry at work

The dislocations a boundary allows

Two crystals meeting at a coincidence angle share one lattice and generate another. The second is where a boundary's own defects live, its shortest vector is one over the square root of the index, and a dislocation's energy is the square of that.

 No two cubic grains are more than sixty-three degrees apart. For each proper class: how many rotations describe one misorientation, the largest disorientation there is, and the mean over uniformly random orientations. The maximum is found by sampling and then climbing locally, so it is a lower bound that has stopped moving rather than a solved value — and it lands on the numbers the literature records. Symmetry at work

The angle two grains differ by

A crystal's axes are not labelled, so a relative orientation between two grains has as many descriptions as the symmetry allows — five hundred and seventy-six of them for a cubic crystal — and their rotation angles run from a few degrees to more than a hundred and seventy. The honest answer is the smallest, and its largest possible value is a number: no two cubic grains are more than sixty-three degrees apart, whatever anybody does to them.

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