engleski

Crystallographic Origami

Symmetry is both an intuitive and an abstract mathematical notion (an isometric transformation acting on an Euclidean space). The ability to notice symmetries acting on an three-dimensional object is very important for understanding (not only) crystal structures. Many of the abstract properties, particularly the fact that symmetries of an object form a mathematical structure called a group, can be discovered by observing objects possessing different kinds of symmetries. It is well known that crystal structures can be grouped into 32 crystal classes, which correspond to 32 symmetry groups called point groups. Polyhedra models have been used for a long time in teaching symmetry issues, and nets for constructing them are easily obtainable. Such cardboard models have the general disadvantage that they are prone to be easily destroyed, and their construction can be quite untidy. Modular origami is a well-known technique of making paper models from several units folded from rectangular paper sheets and assembled into the final model, without gluing or cutting, so the model is (more or less) easily disassembled and reassembled. There are several books and web-pages devoted to constructions of modular origami polyhedra. Most known models belong to the cubic holohedral (hexoctahedral) class and the non-crystallographic full icosahedral group. Modular origami polyhedra models are mostly developed from an aesthetic viewpoint, so there are many decorative versions of a cube, a regular tetrahedron etc., but symmetry issues are rarely discussed, and to the author’s knowledge there is no systematisation of the modular origami polyhedra according to symmetry. The aim of this presentation was to develop a systematic approach to the application of modular origami polyhedra in teaching and using crystallographic point groups. Several models from various classes were presented, assembled and their symmetry properties discussed.

crystallographic point groups; crystal class models; teaching group theory

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