Four dimensions adds a sixteenth and a twenty-fourth

A stretch buys an eighth and a twelfth of a turn found that a lattice in the plane, allowed to land on part of itself rather than on all of itself, can be turned by two fractions of a turn that no rotation of a lattice can make. The stretch hides in a square root: the trace of the map sees the stretch once and the determinant sees it twice, and squaring the cosine cancels it. In space the same trick fails, because every turn in space keeps a line, and the line must be stretched by the same factor as the plane across it.

Four dimensions has no such line. A rotation there can turn two perpendicular planes at once and leave no direction fixed, so whatever stopped space from keeping the eighth does not apply. And four dimensions is already roomier for rotations alone, since five-fold rotation becomes legal there, along with the ten-fold, the eight-fold and the twelve-fold. The question is what a stretch adds on top of that.

It adds exactly two orders, sixteen and twenty-four. They come from the plane’s eighth and twelfth by a substitution that works in every even dimension. A ceiling that holds in every dimension allows four more, and an argument about fields keeps all four out.

A ceiling in every dimension #

A similarity σ of a lattice in $n$ dimensions is $r$ times an orthogonal map $R$, and in a basis of the lattice it is a matrix of integers. Its determinant is therefore an integer, and it is also $\pm r^n$, since an orthogonal map has determinant $\pm1$. So $r^n$ is a whole number, whatever $r$ is. Then

$$R^n = \frac{\sigma^n}{r^n}$$

is a matrix of rational numbers. If $R$ has finite order $m$, then $R^n$ has finite order $m/\gcd(m, n)$, and a rational matrix of finite order preserves a lattice, the sum of the images of the integer lattice under its powers, so it is conjugate to an integer matrix of the same order. The order of $R^n$ must be an order an integer matrix can have in $n$ dimensions. Call it the power rule.

In the plane, the power rule asks the square of the turn to have order $1$, $2$, $3$, $4$ or $6$, so the turn itself has order $1$, $2$, $3$, $4$, $6$, $8$ or $12$. That is exactly the plane’s answer, reached by a different road, so in two dimensions the ceiling is the floor as well. In space the rule asks the cube of the turn to have one of those five orders, which allows $9$, $12$ and $18$ besides, and the argument about the kept line rules out all three. The rule is a ceiling there and not reached.

In four dimensions the orders an integer matrix can have are those whose least block sum of cyclotomic factors has degree at most four, $1$, $2$, $3$, $4$, $5$, $6$, $8$, $10$ and $12$, as the degrees that name the restriction and the earlier counting set out. The power rule allows any $m$ for which $m/\gcd(m, 4)$ is one of these, and that is fifteen orders:

$$1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 8,\ 10,\ 12,\ 16,\ 20,\ 24,\ 32,\ 40,\ 48.$$

The nine at the start are rotations already, with no stretch. The six at the end are the candidates.

The plane’s eighth, with every power doubled #

The plane’s least eighth is the integer matrix with characteristic polynomial $x^2 - 2x + 2$. Its eigenvalues are $1 \pm i$, of modulus $\sqrt2$ and argument 45°. Replace $x$ by $x^2$ and the polynomial becomes

$$x^4 - 2x^2 + 2,$$

whose roots are the square roots of $1 \pm i$. They have modulus $\sqrt[4]{2}$ and arguments 22.5°, 157.5°, 202.5° and 337.5°. All four lie on one circle, so the companion matrix of the polynomial is a similarity of some lattice in four dimensions, and the arguments are multiples of a sixteenth of a turn. The polynomial is irreducible, since 2 divides every coefficient below the top and 4 does not divide the last, which is Eisenstein’s criterion, so the matrix is a single block and does not come apart into two plane similarities. Its turn has order sixteen.

The same substitution in the plane’s least twelfth, $x^2 - 3x + 3$, gives $x^4 - 3x^2 + 3$, Eisenstein at three, with roots of modulus $\sqrt[4]{3}$ at 15°, 165°, 195° and 345°. That is a twenty-fourth of a turn.

What these maps do to the lattice is worth saying in the language three rotations of order twelve used for rotations in four dimensions. A turn in four dimensions turns two perpendicular planes, each by its own angle. The sixteenth turns one plane by 22.5° and the other by 157.5°, which is seven sixteenths, and the order of the whole is the least common multiple of the two, sixteen. A rotation could not turn either plane by a sixteenth, since the sixteenth roots of unity need eight dimensions. A similarity can, because its eigenvalues are not roots of unity but roots of unity times $\sqrt[4]{2}$, and those can have degree four where the roots of unity themselves have degree eight.

The census in four dimensions #

Whether anything else occurs is a finite search with one trick in it. A single block of a four-dimensional similarity is a companion matrix of a quartic $x^4 + ax^3 + bx^2 + cx + d$ whose four roots all have the same modulus $r$. The map $\lambda \mapsto r^2/\lambda$ then sends each root to its complex conjugate, which is again a root, so the polynomial equals its own reversal up to scale, and comparing coefficients gives

$$c^2 = a^2\,|d|.$$

That one equation throws out almost every quartic before any arithmetic is spent on it. The search took $d$ from $-16$ to $16$, $a$ from $-8$ to $8$ and $c$ from $-40$ to $40$, kept the 14,112 quartics that pass the filter with $|b|$ within the bound the modulus allows, and raised each companion matrix to successive powers in exact integers until one was a positive multiple of the identity or the sixtieth was reached. Seventy were similarities. Block sums of smaller blocks with a common stretch, a line and a three-dimensional block or two planes, gave 112 more.

The orders that occur are $1$, $2$, $3$, $4$, $5$, $6$, $8$, $10$, $12$, $16$ and $24$. The nine rotation orders all appear with stretch one, as they must, and the least block for each is a product of cyclotomic polynomials, the integer rotation. Sixteen and twenty-four appear and nothing else does. Of the six candidates the power rule left open, two are realised and four are absent: twenty, thirty-two, forty and forty-eight.

A search in a box cannot prove an absence, and these four need a proof. It is short, and it is the same argument for each.

A square root of two eighths #

The sixteenth has a description that makes it less exotic than its order suggests. Square its companion matrix and the eigenvalues square with it: $1 + i$ twice and $1 - i$ twice. So $\sigma^2$ is a similarity of order eight that turns two perpendicular planes, each by 45°, and stretches both by $\sqrt2$. In a suitable basis it is the plane’s eighth written down twice, once on each plane. The sixteenth is a square root of two copies of the plane’s eighth, and the twenty-fourth is a square root of two copies of its twelfth.

That is the general shape of the Eisenstein family too. The polynomial in $x^k$ has a $k$-th power that is $k$ copies of the plane’s block, so the lattice in $2k$ dimensions is a stack of $k$ square or hexagonal lattices, and the similarity cycles through them while turning each a $k$-th of the way to the plane’s own turn. A rotation can cycle through planes in the same way. The four-dimensional rotation that permutes the coordinates cyclically does exactly that, and its order is only four. What the similarity adds is that the cycle can close on a turn and a stretch instead of on the identity, so the $k$ steps divide the plane’s 45° rather than a whole turn.

Two familiar cases sit on either side of this. A rotation of order five in four dimensions, the one the crystallographic restriction forbids in the plane, does not decompose into planes turned by plane-legal angles at all: it turns one plane by a fifth and the other by two fifths, and neither is a fraction any plane lattice allows. And the inflation of a quasicrystal, which which inflation factors exist studied as an eigenvalue of an integer matrix, is a similarity of a four-dimensional module with a stretch like $1 + \sqrt2$ and no turn at all. The sixteenth is the third kind: a turn the plane cannot make, built entirely from turns the plane can.

What a field keeps out #

Suppose a single block of degree four has a turn of order $m$, and let λ be one of its eigenvalues, $r$ times a primitive $m$-th root of unity ζ. The power rule’s step, read on eigenvalues, says that $\lambda^4 = r^4\zeta^4$ with $r^4$ a whole number. So $\lambda^4$ generates the same field as $\zeta^4$, the field of roots of unity of order $m/\gcd(m, 4)$.

For the four candidates that field has degree four already. Twenty gives fifth roots of unity, thirty-two eighth roots, forty tenth roots and forty-eight twelfth roots, and each of those fields has degree four over the rationals. But λ itself has degree four, since it is a root of the block’s irreducible quartic, and its field contains the field of $\lambda^4$. Two fields of the same degree, one inside the other, are the same field. So λ lies in the field of $\zeta^4$, and so does its complex conjugate, and so does their ratio:

$$\frac{\lambda}{\bar\lambda} = \zeta^2.$$

That is a root of unity of order $m/\gcd(m, 2)$: ten, sixteen, twenty and twenty-four. The field of eighth roots of unity contains roots of unity of order eight and no higher, so it cannot contain a sixteenth. The fields of tenth and twelfth roots cannot contain a twentieth or a twenty-fourth. Thirty-two, forty and forty-eight fall at this step.

Twenty survives it, since a tenth root of unity does lie in the field of fifth roots, and falls one step later. In a field like that one, closed under complex conjugation, the product $\lambda\bar\lambda = r^2$ is fixed by every automorphism once all the eigenvalues share one modulus, so $r^2$ is rational. Then $\lambda^2 = r^2\zeta^2$, and since $\zeta^2$ is minus a square of a fifth root of unity, λ can lie in the field only if $-r^2$ is a square there. That would put the square root of a negative rational into the field of fifth roots of unity, whose only quadratic subfield is $\mathbb{Q}(\sqrt5)$, which is real. So twenty is impossible as a single block. As a block sum it would need a five-fold part, which takes four dimensions alone, and a four-fold part besides, which leaves no room.

For sixteen and twenty-four the argument has nothing to hold on to, and that is why they exist. Their fourth powers are $i$ and a sixth root of unity, which generate fields of degree two, so a degree-four λ is not forced into them, and room is left over for the stretch. The two Eisenstein quartics use exactly that room.

The ceiling, and where it is reached #

The power rule is sharp in the plane and not in space or in four dimensions. The figure sets out what is known of it from two dimensions to six.

In two dimensions every order the rule allows is found. In three, three are ruled out, by the argument about the kept line. In four, two are found and four ruled out, by the argument about fields. In five and six dimensions the rule allows many more: fifteen, twenty, twenty-five, thirty, forty, fifty and sixty in five, and in six twenty-eight orders up to sixty, eleven of them beyond what a rotation there can do. A search there would need quintic and sextic blocks with the same filter, and none has been run. The field argument would decide each candidate as it decided the four here, but has not been run on them.

What can be said in every even dimension is a construction. The polynomials

$$x^{2k} - 2x^k + 2 \qquad \text{and} \qquad x^{2k} - 3x^k + 3$$

are irreducible, by Eisenstein’s criterion at two and at three, and their roots are the $k$-th roots of the plane’s eigenvalues $1 \pm i$ and $\frac{3 \pm i\sqrt3}{2}$. All lie on one circle, so each companion matrix is a similarity of a lattice in $2k$ dimensions, turning by an $8k$-th and a $12k$-th of a turn. For $k$ from one to four the exact powers confirm orders $8$, $12$, $16$, $24$, $24$, $36$, $32$ and $48$, the last two in eight dimensions.

So a stretch lets the order of a turn grow in step with the dimension. A rotation’s largest order grows with the totient function, which is to say in jumps that follow the primes: six in the plane and in space, twelve in four and five dimensions, thirty in six and seven, sixty in eight. The Eisenstein similarity grows by exactly twelve every two dimensions. It is ahead by a factor of two in the plane and in four dimensions, slightly ahead at six, thirty-six against the rotation’s thirty, and behind at eight, where the rotation reaches sixty. Neither dominates the other. The point is the kind of growth: the stretch converts a question about cyclotomic fields, where degrees grow like the totient, into a question about radicals of them, where the degree can be spent on the radical instead.

What the census rests on #

The ceiling is a theorem, two lines long. The exclusion of twenty, thirty-two, forty and forty-eight is a theorem, a paragraph long. The existence of sixteen and twenty-four is a pair of explicit integer matrices whose powers were computed exactly. The search between them is a check that nothing was missed in the theorems, and it could have failed: a block of order twenty, from a quartic outside the theory’s expectations, would have shown up in the box.

The search depends on one filter, $c^2 = a^2|d|$. It is necessary rather than sufficient, and it was not trusted further than that. Every quartic it passed was tested by raising the companion matrix to successive powers in arbitrary-precision integers and comparing the result with a multiple of the identity entry by entry. The block sums were formed only from blocks with stretches equal to within rounding, and each sum was then tested the same way, so an accidental equality of stretches could at most propose a sum, never certify one.

The refused claim is the one the power rule invites, that its ceiling is the answer. It is the answer in the plane, which is what makes it tempting. In four dimensions it lets in twenty, and nothing in a search’s failure to find twenty proves anything. The proof that refuses it is the one above: the fifth roots of unity have no room for the square root of a negative number.

What a lattice in four dimensions cannot show #

The figures draw eigenvalues on circles and tables of orders, and no figure here draws a lattice. A four-dimensional lattice turned by a sixteenth and stretched by $\sqrt[4]{2}$ can be projected onto a plane, but the projection of a similarity is not a similarity, and the picture would show a cloud of points with no visible relation to its image. The eigenvalues are the faithful picture, since they are the turn and the stretch and nothing else.

Nor does anything here say which lattices carry these similarities, the question the plane answered with its fields. A block of order sixteen makes a lattice a module over the ring generated by $\sqrt[4]{2}$ times a sixteenth root of unity, and the lattices it preserves form a family of forms, as a five-fold lattice has one free number found for rotations of order five. The family for the sixteenth has not been mapped.

Whose results these are #

That the determinant of a similarity is the $n$-th power of its stretch, and that its turn’s $n$-th power is therefore rational, are elementary and have no single source. Similar sublattices in higher dimensions were studied by Conway, Rains and Sloane, whose 1999 paper treats which stretches occur for the root lattices, and by Baake, Moody and others for the modules behind quasicrystals, where a similarity is an inflation. The census of turn orders by dimension, the four exclusions and the Eisenstein family are worked out here. They have probably been noticed before, but no source for them is known here.

Still open: five and six dimensions, and the lattices behind the sixteenth #

Two questions are left, one finite and one not. The finite one is the census in five and six dimensions. The power rule leaves forty-odd candidates there, the Eisenstein family realises a few, and the field argument, which needed only the degree of the field of a power of the turn, should decide most of the rest without a search.

The other is what the four-dimensional lattices with a sixteenth look like. The plane answered the corresponding question completely: a lattice turns by an eighth exactly when it holds a square lattice. In four dimensions the ring behind the sixteenth is not a ring of integers of a cyclotomic field but of a field generated by a radical, and whether its lattices include anything familiar, a root lattice turned and stretched onto itself, has not been asked. The root lattice $D_4$ is the obvious first place to look, since it carries the largest group of rotations any lattice in four dimensions has, and no lattice carries all three twelves found how selective even that group is: it holds only one of the three kinds of operation of order twelve.