Recent advances in AI and data science are transforming scientific workflows and even scientific discovery methods. Materials research and development requires automated methods to predict material structures that have the potential to achieve desired properties. Mathematics can provide useful concepts and tools to advance materials design. Theoretical descriptions of graphs and discrete geometric analysis have long contributed, but recent trends suggest broader and deeper applications. This article presents some of the successful collaborations and discusses future developments in mathematical frameworks.

Borate minerals have a great structural diversity of bonding modes of FBUs（fundamental building units）by Bφ₃ triangular and Bφ₄ tetrahedral coordination groups（φ＝O，OH）. This paper introduces the description methods for bonding modes of FBUs in the borate minerals: one is to describe the B-B graphs, and the other is to use descriptors. This paper also introduces the thermal decomposition process of borax Na₂［B₄O₅（OH）₄］･8H₂O and structural development of the FBU’s bonding modes in the borax during the thermal decomposition. With increasing temperature, borax dehydrates（dehydroxylates）and transforms to anhydrous Na₂B₄O₇ through tincalconite Na₂［B₄O₅（OH）₄］･3H₂O. Anhydrous Na₂B₄O₇ first occurs as γ-Na₂B₄O₇ phase, then changes to α-Na₂B₄O₇ phase. In the thermal decomposition process from borax to α-Na₂B₄O₇ phase, bonding modes of the occurring phase are exactly the same as some of the bonding modes in the previous phase.

Conway’s topograph is a graph derived of the lattice-basis reduction theory, and provides an approachable visualization of the theory. The author used the topograph to visualize, simplify, and speedup the algorithms of the powder indexing software CONOGRAPH, and then applied the same ideas to ab-initio indexing of electron backscatter diffraction (EBSD) images. The purpose of this paper is to illustrate how the topograph and the reduction theory can be used in multiple ways in computational crystallography, which has been rarely covered in the course of mathematical crystallography, unlike the use of space groups.

The strong isotropic lattices, honeycomb, K₄, and diamond lattices, can be transformed into the spin frustration lattices, kagome, hyper-kagome, and pyrochlore lattices, respectively, by the line graph relation. This means that the strong isotropic lattices possess“hidden”electronic properties and phenomena. Here, I introduce several examples：the spin frustration in a honeycomb MOF, Cu₃（L）₂（L＝HHTP and THQ）, and a molecule-based K₄, Rb₃［（–）-NDI-Δ］₂. I also describe the exotic band structure of a molecule-based honeycomb lattice, Rb₃（p-TT）, and the unusual phase transitions in a molecule-based diamond, bpBDTDA, which are considered to result from bond frustration.

As a simple consequence of the ice rule, we propose the balance principle of ice, which states that the numbers of inward and outward hydrogen bonds across any closed surface inside ice satisfying the ice rule are equal. The equilibrium principle clearly explains the effects of polarization and defects in ice. It can also be inferred that the phase transition between hydrogen-ordered ice and hydrogen-disordered ice differs from the transition between the normal ordered and disordered phases.