Under what circumstances can a regular n-gon be made such that all its points are lattice points, for some lattice?

Here, I do not mean to restrict only to square lattices, but to any discrete (i.e., points cannot be found arbitrarily close to each other) subset of the plane closed under integer weight affine combinations. [That is, a discrete torsor for some subgroup of the vectors of the plane.]

These things are sometimes called crystallographic.

In general, the symmetry group of a regular n-gon will be crystallographic just in case the n-th cyclotomic polynomial has degree ≤ 2, which will happen precisely for n = 1, 2, 3, 4, or 6.

The proof is like so:

The lattice in question can be thought of as the subring of the complex numbers generated by any primitive nth root of unity. From this point of view, the question of discreteness is the question as to whether this subset of the complex numbers contains arbitrarily small nonzero values.

Note that if our subring contains arbitrarily small nonzero values, then they can be multiplied by their complex conjugate (also in the subring) to yield arbitrarily small positive values.

So the question is really as to the discreteness of the real components here; that is, the discreteness of the additive group generated by cosines of multiples of \(360°/n\).

If all these cosines are rational, then this group is discrete. Otherwise, it is not (by taking suitable linear combinations of an irrational cosine and \(\cos(0) = 1\)). As \(\cos(mx)\) is an integer polynomial of \(\cos(x)\) for any integer \(m\), it suffices to know whether \(\cos(360°/n)\) is rational.

This is equivalently the question of whether \(2 \cos(360°/n)\) is rational, and as \(2 \cos(360°/n) = r + r^{-1}\) for the rotation \(r\) by \(360°/n\), we see that it is an algebraic integer (that is, a root of a monic polynomial; \(r\) is an algebraic integer, since \(r^n - 1 = 0\), and so is \(r^{-1}\) in the same way, and any sum or product of algebraic integers is an algebraic integer; a more explicit argument is at footnote 1). Thus, \(2 \cos(360°/n)\) can only be rational if it is actually an integer. Since cosine ranges from -1 to 1, we can conclude that this is only rational when \(\cos(360°/n) = 0\), \(\pm 1\), or \(\pm 1/2\). This happens precisely at n = 1, 2, 3, 4, or 6.

Instead of invoking the rational root theorem, we could note that our subring contains arbitrarily small nonzero values just in case it contains any nonzero value of size < 1, as such a value can be multiplied by itself repeatedly to become arbitrarily small. If \(2 \cos(360°/n)\) were not a whole number, we could subtract from it a suitable multiple of 1 to find a value of size strictly between 0 and 1 within our subring. Thus, as above, we get discreteness just in case \(2 \cos(360°/n)\) is a whole number.

Argued yet another way, for n > 2, the algebraic degree of \(2 \cos(360°/n)\) (in the sense of the degree of the corresponding field extension of the rationals) must be half the degree of r, by considering how roots bundle with their complex conjugate. This is \(\phi(n)\), and thus we obtain rationality/discreteness just when \(\phi(n) = 2\), where \(\phi\) is the totient function. Again, this happens precisely at n = 3, 4, or 6 (plus the n = 1 and n = 2 cases which correspond to \(\phi(n) = 1\)).

Yet alternatively, we again note that our subring contains arbitrarily small nonzero values just in case it contains any nonzero value of size < 1. We thus can rule out discreteness for n > 6 by noting that in these cases, \(\abs{r - 1} < 1\) (as \(\abs{r - 1} = 2 \sin(180°/n) < 2 \sin(30°) = 1\)). And we can rule out discreteness for n = 5 by noting that in this case, \(\abs{r + r^{-1}} < 1\) (as \(\abs{r + r^{-1}} = 2 \cos(360°/n) = 2 \cos(72°) < 2 \cos(60°) = 1\)). While the discrete lattices for n = 1, 2, 3, 4, or 6 are readily understood (the integers for n = 1 or 2, the Gaussian integers for n = 4, and the Eisenstein integers for n = 3 or 6).

(Basically, out of \(\sin(x/2)\) and \(\cos(x)\), when \(0 < x < 90°\), one or the other will be \(< 1/2\), except when both are equal to \(1/2\) at \(x = 60°\). Thus, for n > 4, we can rule out all n other than n = 6.)

This completes the proof. We obtain not just crystallographic restriction but also by the same argument that the only rational cosines and sines of rational angles (in the sense of rational multiples of full revolutions) are at 0, \(\pm 1\), or \(\pm 1/2\).

A similar argument also shows that the only rational tangents or cotangents of rational angles are 0 or \(\pm 1\). We could conclude this fact about rational (co)tangents also by considering the properties of the Gaussian integers as a unique factorization domain.

Old proof that I left unfinished but that there must be something to:

Take a regular n-gon within a lattice.

Its central point may not be a lattice point, but if we pick an arbitrary lattice point, we can scale everything up by a factor of n in terms of its displacement from this point, yielding a new regular n-gon whose center is a lattice point.

This n-gon’s vertices can be considered as lattice vectors relative to its center. Relative to any one of these vectors, the other vectors are given by rotations corresponding to the complex numbers which are n-th roots of unity.

Since all of these vectors are lattice vectors, so are arbitrary integer linear combinations of them. Thus, the entire ring of integer linear combinations of n-th roots of unity must form a lattice.

So now the matter reduces to checking conditions under which this occurs. This amounts to checking conditions under which this ring is “2-dimensional” over the integers, which amounts to looking at the degree of the n-th cyclotomic polynomial.

(TODO: I save the details for later, when there’s a point to my writing this.)

The key lemma to see is that every discrete subgroup of 2d real vector space is isomorphic to the additive group Z^n for some n <= 2 (whichever n characterizes the dimension of its convex, i.e. real, closure). Aka, every non-empty lattice has a “fundamental pair of periods”. However, the ring generated by n-th roots of unity is isomorphic to Z^(ϕ(n)), with ϕ(n) as the degree of the n-th cyclotomic polynomial.

Footnotes:

  1. Consider the polynomial functions recursively defined via \(P_{n +2}(x) = x P_{n + 1}(x) - P_n(x)\), with \(P_0(x) = 2\) and \(P_1(x) = x\). (These are closely related to the Chebyshev polynomials). We have that \(P_n\) is a monic polynomial of degree \(n\), and also we can see inductively that \(P_n(x + 1/x) = x^n + 1/x^n\), which is to say, \(P_n(2 \cos(x)) = 2 \cos(nx)\). Thus, \(P_n(2 \cos(360°/n)) = 2\), which makes \(2 \cos(360°/n)\) a root of a monic polynomial, which by the Rational Root Theorem implies that if \(2 \cos(360°/n)\) is rational, it must be an integer. ↩