Polyhedra, complexes, nets and symmetry

Skeletal polyhedra and polygonal complexes in ordinary Euclidean 3-space are finite or infinite 3-periodic structures with interesting geometric, combinatorial and algebraic properties. They can be viewed as finite or infinite 3-periodic graphs (nets) equipped with additional structure imposed by the faces, allowed to be skew, zigzag or helical. A polyhedron or complex isregularif its geometric symmetry group is transitive on the flags (incident vertex–edge–face triples). There are 48 regular polyhedra (18 finite polyhedra and 30 infinite apeirohedra), as well as 25 regular polygonal complexes, all infinite, which are not polyhedra. Their edge graphs are nets well known to crystallographers and they are identified explicitly. There are also six infinite families ofchiralapeirohedra, which have two orbits on the flags such that adjacent flags lie in different orbits.

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Extended geometric symmetry group of HAAH-type molecules

10.1117/12.131128 ◽

1992 ◽

Author(s):

Alexander V. Burenin

Keyword(s):

Symmetry Group ◽

Geometric Symmetry

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DOUBLY PERIODIC TEXTILE STRUCTURES

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216509007646 ◽

2009 ◽

Vol 18 (12) ◽

pp. 1597-1622 ◽

Cited By ~ 13

Author(s):

HUGH R. MORTON ◽

SERGEI GRISHANOV

Keyword(s):

Periodic Structures ◽

Original Structure ◽

Closed Curves ◽

Polynomial Invariants ◽

Topological Features ◽

Algebraic Properties ◽

Full Symmetry ◽

Thickened Torus ◽

Traditional Structures ◽

Woven Textile

Knitted and woven textile structures are examples of doubly periodic structures in a thickened plane made out of intertwining strands of yarn. Factoring out the group of translation symmetries of such a structure gives rise to a link diagram in a thickened torus, as in [2]. Such a diagram on a standard torus in S3 is converted into a classical link by including two auxiliary components which form the cores of the complementary solid tori. The resulting link, called a kernel for the structure, is determined by a choice of generators u, v for the group of symmetries. A normalized form of the multi-variable Alexander polynomial of a kernel is used to provide polynomial invariants of the original structure which are essentially independent of the choice of generators u and v. It gives immediate information about the existence of closed curves and other topological features in the original textile structure. Because of its natural algebraic properties under coverings we can recover the polynomial for kernels based on a proper subgroup from the polynomial derived from the full symmetry group of the structure. This enables two structures to be compared at similar scales, even when one has a much smaller minimal repeating cell than the other. Examples of simple traditional structures are given, and their Alexander data polynomials are presented to illustrate the techniques and results.

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