Coincidences in 4 dimensions

The coincidence site lattices (CSLs) of prominent 4-dimensional lattices are considered. CSLs in 3 dimensions have been used for decades to describe grain boundaries in crystals. Quasicrystals suggest to also look at CSLs in dimensions $d>3$. Here, we discuss the CSLs of the root lattice $A_4$ and the hypercubic lattices, which are of particular interest both from the mathematical and the crystallographic viewpoint. Quaternion algebras are used to derive their coincidence rotations and the CSLs. We make use of the fact that the CSLs can be linked to certain ideals and compute their indices, their multiplicities and encapsulate all this in generating functions in terms of Dirichlet series. In addition, we sketch how these results can be generalised for 4–dimensional $\Z$–modules by discussing the icosian ring.

💡 Research Summary

The paper investigates coincidence site lattices (CSLs) in four‑dimensional Euclidean space, focusing on two archetypal lattices: the root lattice A₄ and the hypercubic lattices (ℤ⁴ and its densest sublattice D₄). While CSLs have long been used to describe grain boundaries in three‑dimensional crystals, the discovery of quasicrystals motivates extending the concept to dimensions d > 3. The authors adopt quaternion algebra as the central tool because the rotation group SO(4) factorises as SU(2) ⊗ SU(2), allowing any rotation to be represented by a pair of unit quaternions (q₁, q₂). A rotation is a coincidence rotation for a lattice L if q₁ L q₂⁻¹ ⊂ L; this condition translates into an inclusion of certain quaternionic ideals.

For the A₄ lattice the authors exploit its relationship with the icosian ring ℐ, a maximal order in the rational quaternion algebra. ℐ consists of “icosian” quaternions that encode the icosahedral symmetry underlying A₄. By considering left and right ideals of ℐ generated by a quaternion pair (q₁, q₂), the corresponding CSL is identified with the intersection of these ideals with ℐ itself. The index of a CSL is shown to be the norm N(I) of the associated ideal I, which coincides with the absolute value of the determinant of (R − I), where R is the rotation matrix. The multiplicity a(m) – the number of distinct CSLs of index m – is derived from the factorisation properties of the norm in ℐ. The authors give explicit formulas for a(m) in terms of the prime decomposition of m, distinguishing primes that split, ramify, or remain inert in the quaternionic order.

For the hypercubic case, the analysis proceeds similarly but uses the fact that D₄ can be identified with the set of Hurwitz quaternions of even norm. A coincidence rotation is then a unit quaternion q such that q D₄ q⁻¹ ∩ D₄ yields a sublattice of finite index. Again the index equals the norm of the quaternion, and the multiplicity function is expressed through the arithmetic of the Hurwitz order. The paper provides closed‑form Dirichlet series generating functions \