Symmetry Reduction Scheme Research Articles

Intrinsic non-flat-foldability of two-tile DDC surfaces composed of glide-reflected irregular quadrilaterals

Functional origami tessellations have certain geometric or physical properties - such as flat-foldability and rigid-foldability - which make them of particular interest for a broad range of applications in science, engineering, and architecture. While some simple variations of certain functional origami tessellations can be designed trivially, a systematic symmetry-reduction scheme is proved to be productive for the computational generation of more complex, non-trivial variations. Such a scheme has been previously applied to the developable double corrugation (DDC) surface, widely known as the Miura-ori, resulting in the development of novel crystalline derivatives, the symmetry groups of which are subgroups of the parent pattern. Computational algorithms can search for and find flat-foldable solutions for a large number of derivatives of the DDC surface, but fail to find solutions for all of them. In this paper, we exploit the symmetry reduction scheme along with classical plane geometry to analytically demonstrate why some crystallographic derivatives of this pattern do not exist. To this end, by applying the local flat-foldability condition at the vertices of different orbits associated with each tessellation, we show that such patterns are never flat-foldable, regardless of the geometric specifications of their constituting quadrilateral facets. In particular, we prove that two-tile DDC surfaces composed of glide-reflected irregular quadrilaterals are intrinsically non-flat-foldable, resulted from geometric incompatibilities between the properties of certain unit cells and the local flat-foldability condition.

Unstable Manifolds of Relative Periodic Orbits in the Symmetry-Reduced State Space of the Kuramoto–Sivashinsky System

Systems such as fluid flows in channels and pipes or the complex Ginzburg-Landau system, defined over periodic domains, exhibit both continuous symmetries, translational and rotational, as well as discrete symmetries under spatial reflections or complex conjugation. The simplest, and very common symmetry of this type is the equivariance of the defining equations under the orthogonal group O(2). We formulate a novel symmetry reduction scheme for such systems by combining the method of slices with invariant polynomial methods, and show how it works by applying it to the Kuramoto-Sivashinsky system in one spatial dimension. As an example, we track a relative periodic orbit through a sequence of bifurcations to the onset of chaos. Within the symmetry-reduced state space we are able to compute and visualize the unstable manifolds of relative periodic orbits, their torus bifurcations, a transition to chaos via torus breakdown, and heteroclinic connections between various relative periodic orbits. It would be very hard to carry through such analysis in the full state space, without a symmetry reduction such as the one we present here.

Computation of invariants of finite abelian groups

We investigate the computation and applications of rational invariants of the linear action of a finite abelian group in the nonmodular case. By diagonalization, such a group action can be described by integer matrices of orders and exponents. We make use of integer linear algebra to compute a minimal generating set of invariants along with the substitution needed to rewrite any invariant in terms of this generating set. In addition, we show how to construct a minimal generating set that consists only of polynomial invariants. As an application, we provide a symmetry reduction scheme for polynomial systems whose solution set is invariant by a finite abelian group action. Finally, we also provide an algorithm to find such symmetries given a polynomial system.

Phase equilibria in the La–Mg–Ge system at 500 °C and crystal structure of the new ternary compounds La11Mg2Ge7 and LaMg3−xGe2

The whole 500°C isothermal section of the La–Mg–Ge ternary system was constructed. The existence and crystal structure of three ternary compounds were confirmed: La2+xMg1−xGe2 (τ2, P4/mbm, tP10–Mo2FeB2, 0≤x≤0.25), La4Mg5Ge6 (τ3, Cmc21, oS60–Gd4Zn5Ge6) and La4Mg7Ge6 (τ4, C2/m, mS34, own structure type). Five novel compounds were identified and structurally characterized: La11Mg2Ge7 (τ1, P42/ncm, tP88-8, own structure type, a=1.21338(5), c=1.57802(6)nm), LaMg3−xGe2 (τ5, P3¯1c, hP34-0.44, own structure type, x=0.407(5), a=0.78408(4), c=1.45257(7)nm), La6Mg23Ge (τ6, Fm3¯m, cF120–Zr6Zn23Si, a=1.46694(6)nm), La4MgGe10−x (τ7, x=0.37(1), C2/m, mS60-1.46, own structure type, a=0.88403(8), b=0.86756(8), c=1.7709(2)nm, β=97.16°(1) and La2MgGe6 (τ8, Cmce, oS72–Ce2(Ga0.1Ge0.9)7, a=0.8989(2), b=0.8517(2), c=2.1064(3)nm). Disordering phenomena were revealed in several La–Mg–Ge phases in terms of partially occupied sites. The crystal structures of La11Mg2Ge7 and LaMg3−xGe2 are discussed in details. The latter is a √3a×√3a×2c superstructure of the LaLi3Sb2 structure type; the symmetry reduction scheme is shown in the Bärnighausen formalism terms.

A symmetry reduction scheme of the Dirac algebra without dimensional defects

In relating the Dirac algebra to homogeneous coordinates of a projective geometry, we present a simple geometric scheme which allows to identify various Lie algebras and Lie groups well-known from classical physics as well as from quantum field theory. We introduce a 1 -point-compactification and quaternionic Mobius transformations, and we use SU* (4) and a symmetry reduction scheme without dimensional defects to identify transformations and particle representations thoroughly. As such, two subsequent nonlinear σ models SU*(4)/U Sp(4) and U Sp(4)/SU(2) × U(1) emerge as well as a possible double coset decomposition of SU*(4) with respect to SU(2) × U(1). Whereas the first model leads to equivalence classes of hyperbolic manifolds and naturally introduces coordinates and velocities, the second coset model leads to a Hermitian symmetric (vector) space (Kahlerian space) of real dimension 6, i.e., to a 3-dimensional complex space with a global symplectic and a local SU(2) × U(1) symmetry which allows to identify the (local) gauge group of electroweak interactions as well as under certain assumptions it admits compact SU(3) transformations as automorphisms of this 3-dimensional (hyper)complex vector space. In the limit of low energies, this geometric SU*(4) scheme naturally yields the (compact) group SU(4) to describe “chiral symmetry” and conserved isospin of hadrons as well as the low-dimensional hadron representations. Last not least, with respect to some of the SU*(4) generators we find a multiplication table which (up to signs) is identical with the octonions represented in the Fano plane.

Abelian BF-theory and spherically symmetric electromagnetism

Three different methods to quantize the spherically symmetric sector of electromagnetism are presented: First, it is shown that this sector is equivalent to Abelian BF-theory in four spacetime dimensions with suitable boundary conditions. This theory, in turn, is quantized by both a reduced phase space quantization and a spin network quantization. Finally, the outcome is compared with the results obtained in the recently proposed general quantum symmetry reduction scheme. In the magnetically uncharged sector, where all three approaches apply, they all lead to the same quantum theory.