DOI: 10.36890/iejg.1940104 ISSN: 1307-5624

Euclidean and Hyperbolic Polyhedral Crystallography via Computer Visualisation

János Katona, Emil Molnar, Pavel Pech, Istvan Prok, Jenő Szirmai, Arnasli Yahya
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According to the actuality and importance of the topic, we thought it would be useful to provide an overview of our former, partly joint contributions:How to visualise non-Euclidean geometries in the computer screen? Following W.~Thurston, we consider $8$ three-dimensional homogeneous Riemannian geometries$$\mathbf{E}^3,~\mathbf{S}^3,~\mathbf{H}^2,~\mathbf{S}^2 \times \mathbf{R},~\mathbf{H}^2 \times \mathbf{R},~\widetilde{\mathrm{SL}_2\mathbf{R}},~\mathrm{NIL}, \mathrm{SOL}.$$Among them the Euclidean $\mathbf{E}^3$ and the Bolyai-Lobachevsky hyperbolic $\mathbf{H}^3$ geometry seem to be the most important, as "absolute" geometry (by J\'{a}nos Bolyai),where the crystallography as common basis of material sciences can play new roles for the near future. Our pictures will illustrate these geometries,the first one for remembering our usual experiences, the second one as new phenomena, related to the analogous previous ones.The common basis will be the classical real projective space geometry $\mathcal{P}^3$ that would be good also for the other Thurston geometries as shown in our former papers, but we have to restrict ourselves here. Moreover, didactical reasons lead us to start with plane geometry $\mathcal{P}^2$ on Euclidean basis.Of course, to be concise, we use the concept and machinery of real vector spaces $\mathbf{V}^3$, then $\mathbf{V}^4$ and their dual form spaces $\boldsymbol{V}_3$ resp. $\boldsymbol{V}_4$ for points resp. (hyper)planes. Thus for orthogonality, we introduce (hyper)plane $\rightarrow$ point polarity and (equivalently) scalar product in each in different manner, to distinguish $\mathbf{E}^3$ and $\mathbf{H}^3$.Our main tools will be the Coxeter-type projective coordinate simplices in $\mathcal{P}^3$ through $\mathbf{V}^4$ and $\boldsymbol{V}_4$, where we can compute to the later coordinategeometry, then to project onto an appropriate picture plane, so onto the computer screen. The titles of chapters you see later on for further information. At the end, we also mention higher-dimensional geometry as our new initiative by geometric (Grassmann-Clifford or shortly (G-C) algebra). Pictures and their captures are essential for helping the kind readers at first glance.

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