Quaterionic Construction of the W(F_4) Polytopes with Their Dual Polytopes and Branching under the Subgroups B(B_4) and W(B_3)*W(A_1)

4-dimensional $F_{4}$ polytopes and their dual polytopes have been constructed as the orbits of the Coxeter-Weyl group $W(F_{4})$ where the group elements and the vertices of the polytopes are represented by quaternions. Branchings of an arbitrary \textbf{$W(F_{4})$} orbit under the Coxeter groups $W(B_{4}$ and $W(B_{3}) \times W(A_{1})$ have been presented. The role of group theoretical technique and the use of quaternions have been emphasizedComment: 26 pages, 10 figure

Affine Wa(A4), Quaternions, and Decagonal Quasicrystals

We introduce a technique of projection onto the Coxeter plane of an arbitrary higher dimensional lattice described by the affine Coxeter group. The Coxeter plane is determined by the simple roots of the Coxeter graph I2 (h) where h is the Coxeter number of the Coxeter group W(G) which embeds the dihedral group Dh of order 2h as a maximal subgroup. As a simple application we demonstrate projections of the root and weight lattices of A4 onto the Coxeter plane using the strip (canonical) projection method. We show that the crystal spaces of the affine Wa(A4) can be decomposed into two orthogonal spaces whose point groups is the dihedral group D5 which acts in both spaces faithfully. The strip projections of the root and weight lattices can be taken as models for the decagonal quasicrystals. The paper also revises the quaternionic descriptions of the root and weight lattices, described by the affine Coxeter group Wa(A3), which correspond to the face centered cubic (fcc) lattice and body centered cubic (bcc) lattice respectively. Extensions of these lattices to higher dimensions lead to the root and weight lattices of the group Wa(An), n>=4 . We also note that the projection of the Voronoi cell of the root lattice of Wa(A4) describes a framework of nested decagram growing with the power of the golden ratio recently discovered in the Islamic arts.Comment: 26 pages, 17 figure

Quasi Regular Polyhedra and Their Duals with Coxeter Symmetries Represented by Quaternions I

In two series of papers we construct quasi regular polyhedra and their duals which are similar to the Catalan solids. The group elements as well as the vertices of the polyhedra are represented in terms of quaternions. In the present paper we discuss the quasi regular polygons (isogonal and isotoxal polygons) using 2D Coxeter diagrams. In particular, we discuss the isogonal hexagons, octagons and decagons derived from 2D Coxeter diagrams and obtain aperiodic tilings of the plane with the isogonal polygons along with the regular polygons. We point out that one type of aperiodic tiling of the plane with regular and isogonal hexagons may represent a state of graphene where one carbon atom is bound to three neighboring carbons with two single bonds and one double bond. We also show how the plane can be tiled with two tiles; one of them is the isotoxal polygon, dual of the isogonal polygon. A general method is employed for the constructions of the quasi regular prisms and their duals in 3D dimensions with the use of 3D Coxeter diagrams.Comment: 22 pages, 16 figure

Quaternionic Root Systems and Subgroups of the Aut(F4)Aut(F_{4})

Cayley-Dickson doubling procedure is used to construct the root systems of some celebrated Lie algebras in terms of the integer elements of the division algebras of real numbers, complex numbers, quaternions and octonions. Starting with the roots and weights of SU(2) expressed as the real numbers one can construct the root systems of the Lie algebras of SO(4),SP(2)= SO(5),SO(8),SO(9),F_{4} and E_{8} in terms of the discrete elements of the division algebras. The roots themselves display the group structures besides the octonionic roots of E_{8} which form a closed octonion algebra. The automorphism group Aut(F_{4}) of the Dynkin diagram of F_{4} of order 2304, the largest crystallographic group in 4-dimensional Euclidean space, is realized as the direct product of two binary octahedral group of quaternions preserving the quaternionic root system of F_{4}.The Weyl groups of many Lie algebras, such as, G_{2},SO(7),SO(8),SO(9),SU(3)XSU(3) and SP(3)X SU(2) have been constructed as the subgroups of Aut(F_{4}). We have also classified the other non-parabolic subgroups of Aut(F_{4}) which are not Weyl groups. Two subgroups of orders192 with different conjugacy classes occur as maximal subgroups in the finite subgroups of the Lie group $G_{2}$ of orders 12096 and 1344 and proves to be useful in their constructions. The triality of SO(8) manifesting itself as the cyclic symmetry of the quaternionic imaginary units e_{1},e_{2},e_{3} is used to show that SO(7) and SO(9) can be embedded triply symmetric way in SO(8) and F_{4} respectively

Snub 24-Cell Derived from the Coxeter-Weyl Group W(D4)

Snub 24-cell is the unique uniform chiral polytope in four dimensions consisting of 24 icosahedral and 120 tetrahedral cells. The vertices of the 4-dimensional semi-regular polytope snub 24-cell and its symmetry group {(W(D_{4})\mathord{/{\vphantom {(W(D_{4}) C_{2}}}. \kern-\nulldelimiterspace} C_{2}}):S_{3} of order 576 are obtained from the quaternionic representation of the Coxeter-Weyl group \textbf{$W(D_{4}).$}The symmetry group is an extension of the proper subgroup of the Coxeter-Weyl group \textbf{$W(D_{4})$}by the permutation symmetry of the Coxeter-Dynkin diagram \textbf{$D_{4} .$} The 96 vertices of the snub 24-cell are obtained as the orbit of the group when it acts on the vector \textbf{$\Lambda =(\tau, 1, \tau, \tau)$}or\textbf{}on the vector\textbf{$\Lambda =(\sigma, 1, \sigma, \sigma)$}in the Dynkin basis with\textbf{$\tau =\frac{1+\sqrt{5}}{2} {\rm and}\sigma =\frac{1-\sqrt{5}}{2} {\rm .}$} The two different sets represent the mirror images of the snub 24-cell. When two mirror images are combined it leads to a quasi regular 4D polytope invariant under the Coxeter-Weyl group \textbf{$W(F_{4}).$}Each vertex of the new polytope is shared by one cube and three truncated octahedra. Dual of the snub 24 cell is also constructed. Relevance of these structures to the Coxeter groups \textbf{$W(H_{4}){\rm and}W(E_{8})$}has been pointed out.Comment: 15 pages, 8 figure

The Prevalence of Social Science in Gay Rights Cases: The Synergistic Influences of Historical Context, Justificatory Citation, and Dissemination Efforts

Disjunctive legal change is often accompanied by a period of frantic activity as the competing forces of stasis and evolution vie for domination. Nowhere is the battle for legal change likely to be more sharply joined than when the findings of modern science, in their varied and multifarious forms, are pitted directly against prevailing moral or societal precepts. One of the latest incarnations of this trend is the battle over the legal recognition of gay rights. In recent history, the courts have been inundated by gay litigants seeking the rights and protections already afforded other discrete groups within society. In the resulting legal skirmishes, gay individuals are resorting with increasing regularity to the sciences in an effort to overcome the moral opprobrium surrounding homosexuality. The judicial opinions which have resulted from the onslaught of gay litigants have not remained untouched by the scientific information adduced. Rather, as this Article will demonstrate, a disproportionally large number of gay rights opinions contain citations and references to social science information. These judicial opinions have become artifacts of the battle between modern science and existing moral conceptions of homosexuality and provide a discrete microcosm within which to examine science\u27s contribution to legal change. The lessons derived from gay rights cases may help to elucidate other contexts in which science and morality meet head-on

Non-crystallographic reduction of generalized Calogero-Moser models

We apply a recently introduced reduction procedure based on the embedding of non-crystallographic Coxeter groups into crystallographic ones to Calogero–Moser systems. For rational potentials the familiar generalized Calogero Hamiltonian is recovered. For the Hamiltonians of trigonometric, hyperbolic and elliptic types, we obtain novel integrable dynamical systems with a second potential term which is rescaled by the golden ratio. We explicitly show for the simplest of these non-crystallographic models, how the corresponding classical equations of motion can be derived from a Lie algebraic Lax pair based on the larger, crystallographic Coxeter group

Family Unification in Five and Six Dimensions

In family unification models, all three families of quarks and leptons are grouped together into an irreducible representation of a simple gauge group, thus unifying the Standard Model gauge symmetries and a gauged family symmetry. Large orthogonal groups, and the exceptional groups $E_7$ and $E_8$ have been much studied for family unification. The main theoretical difficulty of family unification is the existence of mirror families at the weak scale. It is shown here that family unification without mirror families can be realized in simple five-dimensional and six-dimensional orbifold models similar to those recently proposed for SU(5) and SO(10) grand unification. It is noted that a family unification group that survived to near the weak scale and whose coupling extrapolated to high scales unified with those of the Standard model would be evidence accessible in principle at low energy of the existence of small (Planckian or GUT-scale) extra dimensions.Comment: 13 pages, 2 figures, minor corrections, references adde

Spectrum of the Relativistic Particles in Various Potentials

We extend the notion of Dirac oscillator in two dimensions, to construct a set of potentials. These potentials becomes exactly and quasi-exactly solvable potentials of non-relativistic quantum mechanics when they are transformed into a Schr\"{o}dinger-like equation. For the exactly solvable potentials, eigenvalues are calculated and eigenfunctions are given by confluent hypergeometric functions. It is shown that, our formulation also leads to the study of those potentials in the framework of the supersymmetric quantum mechanics

Notch signaling coordinates ommatidial rotation in the Drosophila eye via transcriptional regulation of the EGF-Receptor ligand Argos

Abstract: In all metazoans, a small number of evolutionarily conserved signaling pathways are reiteratively used during development to orchestrate critical patterning and morphogenetic processes. Among these, Notch (N) signaling is essential for most aspects of tissue patterning where it mediates the communication between adjacent cells to control cell fate specification. In Drosophila, Notch signaling is required for several features of eye development, including the R3/R4 cell fate choice and R7 specification. Here we show that hypomorphic alleles of Notch, belonging to the Nfacet class, reveal a novel phenotype: while photoreceptor specification in the mutant ommatidia is largely normal, defects are observed in ommatidial rotation (OR), a planar cell polarity (PCP)-mediated cell motility process. We demonstrate that during OR Notch signaling is specifically required in the R4 photoreceptor to upregulate the transcription of argos (aos), an inhibitory ligand to the epidermal growth factor receptor (EGFR), to fine-tune the activity of EGFR signaling. Consistently, the loss-of-function defects of Nfacet alleles and EGFR-signaling pathway mutants are largely indistinguishable. A Notch-regulated aos enhancer confers R4 specific expression arguing that aos is directly regulated by Notch signaling in this context via Su(H)-Mam-dependent transcription