A note on the two symmetry-preserving covering maps of the gyroid minimal surface

Our study of the gyroid minimal surface has revealed that there are two distinct covering maps from the hyperbolic plane onto the surface that respect its intrinsic symmetries. We show that if a decoration of <InlineEquation ID="Equ1"> <EquationSource Format="TEX">$\mathbb{H}^2$</EquationSource> </InlineEquation> is chiral, the projection of this pattern via the two covering maps gives rise to distinct structures in <InlineEquation ID="Equ2"> <EquationSource Format="TEX">$\mathbb{E}^3$</EquationSource> </InlineEquation>. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2005