short communications
The intrinsic group–subgroup structures of the Diamond and Gyroid minimal surfaces in their conventional unit cells
^{a}Niels Bohr Institute, University of Copenhagen, Denmark, ^{b}Research School of Physics, Australian National University, Australia, and ^{c}School of Chemistry, University of Sydney, Australia
^{*}Correspondence email: mcpe@nbi.ku.dk
The intrinsic, hyperbolic crystallography of the Diamond and Gyroid minimal surfaces in their conventional unit cells is introduced and analysed. Tables are constructed of symmetry subgroups commensurate with the translational symmetries of the surfaces as well as group–subgroup lattice graphs.
Keywords: minimal surfaces; hyperbolic geometry; symmetry groups; subgroup lattices; conventional unit cells.
1. Introduction
The Primitive, Diamond (Schwarz, 1890) and Gyroid (Schoen, 1970) minimal surfaces are well known structures in the context of materials science, where they emerge in simulations of and experiments on a variety of systems ranging from butterfly wing scales (Dolan et al., 2015) and biological detergent systems (Mezzenga et al., 2019) to bulk polymer phases (Castelletto & Hamley, 2004).
Their crystallographic properties are well documented (Sadoc & Charvolin, 1989; Robins et al., 2004; Hyde et al., 2014) though only for the primitive of the sidepreserving translation group. However, many applications such as molecular simulations (Kirkensgaard et al., 2014), field theory based approaches (Welch et al., 2019) or meshing algorithms (Pellé & Teillaud, 2014) rely on mutually orthogonal lattice vectors of equal length, corresponding to the conventional unit cells of these surfaces. The results from such computations are therefore better analysed in the conventional unitcell settings (Fig. 1).
Here, we extend previous results on the intrinsic (hyperbolic) crystallography of these surfaces (Robins et al., 2004) to these settings; we introduce canonical versions of their conventional in the universal covering space, , along with the group theory needed to construct the related translational subgroups. We derive the group–subgroup structure of the symmetries commensurate with the surfaces in these settings. We label our groups by their orbifold symbols which are explained at length in the literature (Thurston, 1980; Conway et al., 2008; Hyde et al., 2014).
For completeness, we include the Primitive surface in our calculations and tables but stress that these are identical to the ones presented earlier (Robins et al., 2004), as the primitive and conventional unit cells of the Primitive surface coincide. We note that recent changes to GAP's enumeration algorithms mean that our ordering differs slightly from the previous report.
2. Preliminaries and method
The intrinsic symmetry group of these three minimal surfaces can be labelled with the orbifold symbol . Similarly, upon compactification by pairwise identifying sides of the unit cells in Fig. 1, the resulting surfaces are the (genus3) tritorus, the (genus9) enneatorus and the (genus5) pentatorus, respectively. Hence, the Conway orbifold symbols , and describe the related translational groups.
We list the translations needed to construct the translational subgroups corresponding to the domains in Fig. 1 in the supporting information along with additional information and data on these groups.
Next, we construct and analyse all symmetry groups of the surfaces by the procedure described by Robins et al. (2004) using the methods outlined below along with the data presented in the supporting information. First, all subgroups for a given surface (represented via the quotient group labelled , where is the appropriate translational subgroup) are computed using GAP and KBMag (Epstein et al., 1991; The GAP Group, 2021). Then, for each its generators are identified and the associated cosets are calculated. From these cosets, the Delaney–Dress (Dress, 1987; DelgadoFriedrichs, 2003) and orbifold symbols of the are computed. The suite of resulting groups describe all possible intrinsic symmetry groups of the parent minimal surfaces whose translations are those of their conventional unit cells.
3. Results and discussion
The results of our enumeration can be found in Table 1. Expanded tables containing detailed information on each can be found in the supporting information alongside group–subgroup lattice graphs outlining the structure of our three quotient groups.

We note that the translational domains shown in Fig. 1 and the corresponding translations listed in the supporting information are not unique. One can represent the pentatorus as e.g. a 20gon rather than the outlined 30gon, and accordingly the enneatorus can be represented as a 36gon rather than our 60gon. However, we retain the sixfold symmetry around the origin for easier subsequent analysis.
Changing translational symmetries of the primitive unit cells to those of the conventional unit cells admits additional symmetry groups on the surfaces. Fig. 2 shows an example of a trivalent net on the Gyroid which respects the translations of the conventional, but not the primitive, of that surface. The specific decoration and its embeddings in and on the Gyroid were derived as outlined elsewhere (Pedersen & Hyde, 2018; Hyde & Pedersen, 2021). Whereas the primitive cells admit 131 distinct groups, the enlarged unit cells of the Diamond and Gyroid surfaces include 463 and 234 groups, respectively, each associated with a threedimensional More information on these groups can be found in the supporting information and at https://gitlab.com/mcpe/tpmsgroups.
 Figure 2 No. 222 in the table for the Gyroid in the supporting information. The net – or rather its symmetry – is not commensurate with the primitive yet can be embedded in via the conventional Right: the same net shown in the universal covering space, . Crystallographic information on the canonical embedding (DelgadoFriedrichs & O'Keeffe, 2003 
Supporting information
Supporting information and data on the relevant groups. DOI: https://doi.org/10.1107/S2053273321012936/ae5110sup1.pdf
Funding information
MCP thanks the Villum Foundation for financial support through grant No. 22833.
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