The restriction in three dimensions

A plane lattice carries rotations of orders one, two, three, four and six, and no others. Space has an extra dimension, fourteen lattice types instead of five, and two hundred and thirty groups instead of seventeen — so the natural expectation is that it carries more rotation orders as well.

It carries exactly the same five. The result is worth working through, because the reason has nothing to do with three-ness and everything to do with what an integer matrix can be.

The argument, unchanged #

The two-dimensional proof runs: a rotation preserving a lattice is an integer matrix in the lattice basis, so its trace is a whole number; the trace of a rotation by $2\pi/n$ in the plane is $2\cos(2\pi/n)$; a whole number between $-2$ and $2$ is one of five values; each gives one $n$.

In space, a rotation by $2\pi/n$ about an axis has matrix $\operatorname{diag}(1) \oplus R_{2\pi/n}$ in a suitable orthogonal frame, and its trace is $1 + 2\cos(2\pi/n)$. Changing to the lattice basis is a similarity transformation, which does not change the trace, and in the lattice basis the matrix is integral. So $1 + 2\cos(2\pi/n)$ is a whole number, hence $2\cos(2\pi/n)$ is, and the same five values are the only possibilities.

The extra dimension contributed a $+1$ to the trace and nothing else. Which is the whole answer, and it is worth naming what carried it: the rotation still acts on a two-dimensional plane, and the third direction is along the axis, where it does nothing. A rotation in three dimensions is a two-dimensional rotation with a spectator direction attached, so it inherits the plane’s constraint exactly.

What forty-eight operations look like #

The cubic lattice is the most symmetric lattice in space, so if extra orders were going to appear anywhere they would appear there. Enumerating its symmetries settles the question by exhaustion.

Every operation preserving a cubic lattice permutes the three axes and may reverse any of them, so it is a signed permutation matrix: one non-zero entry per row and column, each $\pm 1$. There are $3! = 6$ permutations and $2^3 = 8$ sign choices, giving $48$ — a count produced here by generating them rather than by multiplying, and asserted.

Of the forty-eight, half have determinant $+1$ and are proper rotations. The orders occurring among those twenty-four, measured by raising each matrix to successive powers until the identity appears, are:

• order 1 — the identity, once;
• order 2 — nine of them;
• order 3 — eight;
• order 4 — six.

That is $1 + 9 + 8 + 6 = 24$, and there is no six-fold anywhere. Six-fold rotation requires a hexagonal arrangement, in space as in the plane, and a cubic lattice does not have one.

Thirteen axes, and how they are counted #

The rotations sort themselves into axes, and the sorting has a trap in it worth flagging because it is the sort of error a picture would not reveal.

Each non-identity rotation fixes a direction. Grouping the twenty-four rotations by their fixed direction gives thirteen axes: three through the face centres, four through the body diagonals, and six through the edge midpoints. The face axes carry four-fold rotations, the body diagonals three-fold, and the edge axes two-fold.

The trap: the square of a four-fold rotation is a two-fold rotation about the same axis. Counting axes by “which orders occur about them” rather than by “the highest order that occurs” gives sixteen axes rather than thirteen — three face axes counted twice, once as four-fold and once as two-fold. The correct convention is the highest order, and the check on this site asserts thirteen, so the miscount cannot pass quietly.

Thirteen axes, with $4-1$, $3-1$ and $2-1$ non-identity rotations on them respectively: $3 \times 3 + 4 \times 2 + 6 \times 1 = 9 + 8 + 6 = 23$, plus the identity is $24$. The two counts agree, which is the reason to do both.

The other lattices, and where six-fold lives #

The cubic case settles that nothing new appears at the top of the symmetry range, and it leaves a question: where does six-fold rotation live in space, given that a cubic lattice has none?

On the hexagonal lattice, which in three dimensions is a plane hexagonal lattice stacked vertically. Its point symmetry has twenty-four operations, twelve of them proper rotations, and a six-fold axis along the stacking direction. That axis is the only place a six-fold rotation occurs in any three-dimensional lattice, exactly as the hexagonal plane lattice is the only place it occurs in two.

The full list of holohedries — the point symmetry of each lattice type — is short and the counts are worth having beside the plane’s:

Two features of that table are instructive.

The cubic system is the largest and not the one with the highest rotation. Forty-eight operations against the hexagonal system’s twenty-four, and yet its highest rotation order is four. Symmetry is not a single quantity, and “more symmetric” is not a total ordering — a fact the plane hides, since there the largest holohedry and the highest rotation coincide in the hexagonal lattice.

Seven systems, fourteen lattices. Each system has one or more centring variants, and the fourteen come from distributing the centrings among the seven. The distribution is uneven and slightly arbitrary-looking — the orthorhombic system has four variants and the hexagonal one has one — for exactly the reason Frankenheim’s fifteenth lattice was a duplicate: some centrings of some systems are others in different axes.

Where the extra room does show up #

Space is roomier, and the extra room appears in three places — none of them the rotation orders.

More lattices. Five plane lattices become fourteen Bravais lattices, because centring has more options: body-centred, face-centred and base-centred varieties exist where the plane had only one kind of centring. That is a combinatorial expansion of the centring question rather than a new phenomenon.

A new kind of operation. The plane has four motions: translation, rotation, reflection, glide. Space has those and one more — the screw axis, a rotation combined with a translation along its own axis, which has no two-dimensional counterpart because a plane rotation has no axis to translate along. Screw axes are why so many space groups are non-symmorphic, and they are the mechanism behind the helical structures of DNA and of most fibrous proteins.

More reflections. A plane has mirror lines; space has mirror planes and also the rotoinversion operations, rotations combined with inversion through a point. The inversion centre itself is available in the plane as a half turn, but the rotoinversions of order 3, 4 and 6 are genuinely new.

Seventeen becomes two hundred and thirty out of those three expansions, and not out of any new rotation order.

What the machinery here does and does not decide #

This site’s round trip is two-dimensional. Its detector enumerates operations of a plane lattice, its patterns are plane patterns, and none of that reaches into space.

The three-dimensional claims on this page rest on a smaller, self-contained piece of machinery: integer matrices and their orders, computed exactly. The forty-eight signed permutations are generated; each one’s order is found by repeated integer multiplication until the identity appears; determinants are computed to separate proper rotations from improper ones; and the axis count comes from searching for the fixed direction of each rotation among small integer vectors.

Every number on this page comes out of that. What does not come out of it is any statement about the fourteen Bravais lattices or the two hundred and thirty space groups, which are quoted from the literature and are not verified here. The distinction matters: the cubic case is enumerated and the general case is cited, and a reader is entitled to know which is which.

The gate makes the same separation. It re-runs the cubic enumeration on every build and fails if the counts change; it makes no claim about anything larger.

Where the exactness stops #

Two boundaries, and the first is the one this essay exists to mark.

The result is about lattices, not about objects. A crystal may contain a five-fold molecule, a five-fold cluster, or an icosahedral cage, and frequently does. What it may not have is a five-fold operation of the whole structure. The distinction is between local arrangement and global symmetry, and it is the same distinction the plane case turns on.

The most-cited example is the icosahedral virus capsid. Hundreds of virus shells have exact icosahedral symmetry, which includes six five-fold axes, and they crystallise perfectly well; the crystal’s space group simply contains none of the capsid’s five-folds, and the capsid sits at a general position or on whichever of its own axes the crystal happens to share.

The result assumes a lattice, which is where quasicrystals come in. An icosahedral quasicrystal has five-fold and three-fold and two-fold axes and sharp diffraction, and no lattice at all. It is not a counterexample; the theorem’s first line does not apply to it, and describing it properly requires going up to six dimensions, where the restriction permits what three does not.

Reading the result the useful way round #

The statement “only orders 1, 2, 3, 4 and 6” is usually delivered as a prohibition, and it is more useful as a permission with a mechanism attached.

Six-fold and three-fold force a hexagonal lattice. So a structure known to have a three-fold axis is known to have a hexagonal or trigonal cell before anything else is measured, and its cell parameters have two equal axes at $120°$ whether or not the refinement has got there yet.

Four-fold forces a tetragonal or cubic cell, by the same argument at $90°$.

Two-fold forces nothing, which is why monoclinic and orthorhombic structures are so common: a two-fold axis is compatible with every lattice, so a structure with modest symmetry has the fewest constraints on its cell.

Read that way, the restriction is a lookup from a symmetry element to a set of cell constraints, and it is the first step of every structure determination. A crystallographer who has seen a three-fold axis in a diffraction pattern has learnt something about the cell lengths, and reads them off the pattern’s geometry knowing what to expect.

The prohibition and the permission are the same statement, and which one is more useful depends on whether the reader has a structure in hand or a theorem to prove.

The surprising part #

The answer is the same in two dimensions and three, and it is not the same in four. So the restriction is not a fact about physical space and not a fact about crystals; it is a fact about which dimensions are large enough to hold a particular integer matrix.

The clean statement is this: an $n$-fold rotation of a lattice exists in $d$ dimensions exactly when $\varphi(n) \le d$, where $\varphi$ is Euler’s totient — the count of numbers below $n$ with no factor in common with it. For $n = 5$, $\varphi(5) = 4$, so five-fold needs four dimensions. For $n = 3$ and $n = 4$ and $n = 6$, $\varphi$ is $2$, so they fit in the plane. For $n = 7$, $\varphi(7) = 6$.

Reading the table for $d = 2$ and $d = 3$ gives the same five, because $\varphi(n) \le 3$ and $\varphi(n) \le 2$ have the same solutions — $\varphi$ is even for every $n$ above 2, so it is never exactly 3. Three dimensions gives nothing new because three is odd. That is the whole of it, and it is a fact about number theory rather than about space.

The screw axis, which is the genuinely new thing #

Of the three expansions listed above, the screw axis is the one with no plane analogue at all, and it deserves a paragraph on its own because it is where most of the extra two hundred and thirteen groups come from.

A screw axis is a rotation by $2\pi/n$ followed by a translation of $m/n$ of a lattice vector along the rotation axis. Written $n_m$: a $2_1$ axis is a half turn with a half-cell shift, a $6_1$ is a sixth turn with a sixth-cell shift, and so on. Repeating an $n_m$ operation $n$ times gives a pure lattice translation, which is the condition that makes it a symmetry at all.

The plane cannot host one because a plane rotation’s axis is perpendicular to the plane, and translating along it leaves the plane. What the plane has instead is the glide, where the translation runs along a mirror rather than along a rotation axis — so the two operations are analogues in spirit and not in construction.

The consequence for the classification is large. A screw axis constrains its order the same way a rotation does, so no new orders appear; but for each rotation order $n$ there are $n-1$ distinct screw variants, and each gives a different space group. That multiplication is most of the gap between seventeen and two hundred and thirty, and it is why so many space groups are non-symmorphic — no choice of origin removes every translation part at once.

It also has a direct experimental signature. A screw axis removes reflections along the row parallel to it, in exactly the way a glide removes alternate reflections along a row, and reading which are missing is how a screw axis is identified. Nobody has seen one.

Who established it #

The three-dimensional restriction predates its two-dimensional counterpart in the literature, because crystals came first. Haüy’s law of rational indices, from the 1780s, is the empirical version: crystal faces cut the axes in simple whole-number ratios, which is the statement that a lattice is there. Frankenheim, Bravais and Hessel worked out the consequences through the middle of the nineteenth century, and Hessel’s 1830 derivation of the thirty-two crystal classes contains the restriction implicitly.

The totient statement — the one that explains both cases at once and predicts the fourth — was not written down until much later, and comes out of the study of finite subgroups of $GL_n(\mathbb{Z})$ rather than out of crystallography. It is a good example of a result becoming obvious only after being generalised past the case that motivated it.

Where the ladder goes next #

The argument this one extends is the restriction itself, and its geometric partner is the descent construction.

The dimension where the answer changes is four, and what lives there is the periodic structure whose shadow is a quasicrystal.

The classification this restriction makes finite is the seventeen, and its three-dimensional counterpart is the two hundred and thirty.

What the pictures here cannot show. The cube on this page is an axonometric drawing, and every claim about it — forty-eight operations, thirteen axes, no six-fold — comes from the matrices rather than from the projection. A drawing of a cube cannot show that a rotation maps it onto itself, because the drawing distorts the very lengths and angles that would have to be compared.