Jack S. Avrin
Review: Knots on a Torus: A Model of the Elementary Particles
Symmetry 2012, 4(1), 39-115; doi:10.3390/sym4010039 - published online 9 February 2012
Show/Hide AbstractAbstract: Two knots; just two rudimentary knots, the unknot and the trefoil. That’s all we need to build a model of the elementary particles of physics, one with fermions and bosons, hadrons and leptons, interactions weak and strong and the attributes of spin, isospin, mass, charge, CPT invariance and more. There are no quarks to provide fractional charge, no gluons to sequester them within nucleons and no “colors” to avoid violating Pauli’s principle. Nor do we require the importation of an enigmatic Higgs boson to confer mass upon the particles of our world. All the requisite attributes emerge simply (and relativistically invariant) as a result of particle conformation and occupation in and of spacetime itself, a spacetime endowed with the imprimature of general relativity. Also emerging are some novel tools for systemizing the particle taxonomy as governed by the gauge group SU(2) and the details of particle degeneracy as well as connections to Hopf algebra, Dirac theory, string theory, topological quantum field theory and dark matter. One exception: it is found necessary to invoke the munificent geometry of the icosahedron in order to provide, as per the group “flavor” SU(3), a scaffold upon which to organize the well-known three generations—no more, no less—of the particle family tree.

Toru Ikeda
Article: Symmetries of Spatial Graphs and Rational Twists along Spheres and Tori
Symmetry 2012, 4(1), 26-38; doi:10.3390/sym4010026 - published online 20 January 2012
Show/Hide AbstractAbstract: A symmetry group of a spatial graph Γ in S³ is a finite group consisting of orientation-preserving self-diffeomorphisms of S3 which leave Γ setwise invariant. In this paper, we show that in many cases symmetry groups of Γ which agree on a regular neighborhood of Γ are equivalent up to conjugate by rational twists along incompressible spheres and tori in the exterior of Γ.

Jaime Nava and Vladik Kreinovich
Article: Towards Symmetry-Based Explanation of (Approximate) Shapes of Alpha-Helices and Beta-Sheets (and Beta-Barrels) in Protein Structure
Symmetry 2012, 4(1), 15-25; doi:10.3390/sym4010015 - published online 19 January 2012
Show/Hide AbstractAbstract: Protein structure is invariably connected to protein function. There are two important secondary structure elements: alpha-helices and beta-sheets (which sometimes come in a shape of beta-barrels). The actual shapes of these structures can be complicated, but in the first approximation, they are usually approximated by, correspondingly, cylindrical spirals and planes (and cylinders, for beta-barrels). In this paper, following the ideas pioneered by a renowned mathematician M. Gromov, we use natural symmetries to show that, under reasonable assumptions, these geometric shapes are indeed the best approximating families for secondary structures.

Egon Schulte and Jörg M. Wills
Article: Convex-Faced Combinatorially Regular Polyhedra of Small Genus
Symmetry 2012, 4(1), 1-14; doi:10.3390/sym4010001 - published online 28 December 2011
Show/Hide AbstractAbstract: Combinatorially regular polyhedra are polyhedral realizations (embeddings) in Euclidean 3-space E³ of regular maps on (orientable) closed compact surfaces. They are close analogues of the Platonic solids. A surface of genus g ≥ 2 admits only finitely many regular maps, and generally only a small number of them can be realized as polyhedra with convex faces. When the genus g is small, meaning that g is in the historically motivated range 2 ≤ g ≤ 6, only eight regular maps of genus g are known to have polyhedral realizations, two discovered quite recently. These include spectacular convex-faced polyhedra realizing famous maps of Klein, Fricke, Dyck, and Coxeter. We provide supporting evidence that this list is complete; in other words, we strongly conjecture that in addition to those eight there are no other regular maps of genus g, with 2 ≤ g ≤ 6, admitting realizations as convex-faced polyhedra in E³. For all admissible maps in this range, save Gordan’s map of genus 4, and its dual, we rule out realizability by a polyhedron in E³.

Hiroshi Fukuda , Chiaki Kanomata , Nobuaki Mutoh , Gisaku Nakamura and Doris Schattschneider
Article: Polyominoes and Polyiamonds as Fundamental Domains of Isohedral Tilings with Rotational Symmetry
Symmetry 2011, 3(4), 828-851; doi:10.3390/sym3040828 - published online 12 December 2011
Show/Hide AbstractAbstract: We describe computer algorithms that produce the complete set of isohedral tilings by n-omino or n-iamond tiles in which the tiles are fundamental domains and the tilings have 3-, 4-, or 6-fold rotational symmetry. The symmetry groups of such tilings are of types p3, p31m, p4, p4g, and p6. There are no isohedral tilings with p3m1, p4m, or p6m symmetry groups that have polyominoes or polyiamonds as fundamental domains. We display the algorithms’ output and give enumeration tables for small values of n. This expands earlier works [1,2] and is a companion to [3].

Jose M. P. Carmelo and Maria J. Sampaio
Article: An Application of the Extended Global SO(3) × SO(3) × U(1) Symmetry of the Hubbard Model on a Square Lattice: The Spinon, η-Spinon, and c Fermion Description
Symmetry 2011, 3(4), 780-827; doi:10.3390/sym3040780 - published online 12 December 2011
Show/Hide AbstractAbstract: In this paper we review recent results on the preliminary applications of the new-found extended global SO(3) × SO(3) × U(1) symmetry of the Hubbard model on a bipartite lattice. Our results refer to the particular case of the bipartite square lattice. Specifically, we review a general description for such a model with nearest-neighbor transfer integral t and on-site repulsion U on a square lattice with N²_a >> 1 sites consistent with its extended global symmetry. It refers to three types of elementary objects whose occupancy configurations generate the state representations of the model extended global symmetry. Such objects emerge from a suitable electron-rotated-electron unitary transformation. An application to the spin spectrum of the parent compound La₂CuO₄ is shortly reviewed.