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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 $\mathbb{H}^2$ is chiral, the projection of this pattern via the two covering maps gives rise to distinct structures in $\mathbb{E}^3$ . [ABSTRACT FROM AUTHOR]

A broad range of mathematical techniques, ranging from statistics to fuzzy logic, have been used to great advantage in intelligent data analysis. Topology – the fundamental mathematics of shape – has to date been conspicuously absent from this repertoire. This paper shows how topology, properly reformulated for a finite-precision world, can be useful in intelligent data analysis tasks. [ABSTRACT FROM AUTHOR]

Abstract: Crystalline frameworks in 3D Euclidean space can be constructed by projecting tilings of 2D hyperbolic space onto three-periodic minimal surfaces, giving surface reticulations. The technique involves Delaney–Dress tiling theory, group theory, differential and non-Euclidean geometry. Preliminary results of this approach, found at http://epinet.anu.edu.au, are discussed and compared with other approaches. [Copyright &y& Elsevier]