## A foundation for the theory of symmetry of ordered nanostructures |

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algebraic constructions algebraic geometry algebraic group atoms automorphisms axes axis base Bernal bundle cell centre coherent boundaries cohomology coincides column configuration connected coordinates corresponding crystal crystalline crystallographic cubic defined Desargues configuration determined diamond diffeomorphic dimensional discrete dodecahedron DS cluster edges elements embedded Euclidean space fiber fibre space field finite projective plane functions geometrical structural complexes Gosset group G helicoids hexacycles hexagonal Hopf fibration icosahedral icosahedron ideal incidence graph incidence signs incidence table infinite integer invariant isomorphic Lie algebra Lie group linear manifolds minimal nanoparticle nanostructures nanostructures with coherent number of vertices obtained octahedron Petri polygon phase point group polyhedra polyhedron polynomials polytope 240 possible quasicrystal reflection group represents ring rod substructures root lattice root system rotation semi-simple semi-simple algebras smooth sphere straight lines subconfiguration subgroup subsystems subtable symmetry tetrahedra three-dimensional topological torus transformations transition union variety vector space vertex