Characterizing the universal rigidity of generic tensegrities

A tensegrity is a structure made from cables, struts, and stiff bars. A d-dimensional tensegrity is universally rigid if it is rigid in any dimension $$d'$$ d ′ with $$d'\ge d$$ d ′ ≥ d . The celebrated super stability condition due to Connelly gives a sufficient condition for a tensegrity to be universally rigid. Gortler and Thurston showed that super stability characterizes universal rigidity when the point configuration is generic and every member is a stiff bar. We extend this result in two directions. We first show that a generic universally rigid tensegrity is super stable. We then extend it to tensegrities with point group symmetry, and show that this characterization still holds as long as a tensegrity is generic modulo symmetry. Our strategy is based on the block-diagonalization technique for symmetric semidefinite programming problems, and our proof relies on the theory of real irreducible representations of finite groups.

Structure of 3He

Using electron scattering data, the diffraction pattern off $$^{3}$$ 3 He shows it to be an equilateral triangle possessing dihedral D$$_{3}$$ 3 point group symmetry (PGS). Previous work showed that $$^{4}$$ 4 He is a 3-base pyramid with C$$_{3v}$$ 3 v PGS. $$^{6}$$ 6 Li is predicted to have C$$_{2v}$$ 2 v PGS. As nuclear $$A \rightarrow $$ A → large, atomic nuclei enter into the ‘protein folding problem’ with many possible groundstate PGS competing for lowest energy.

Classification and Effects of Symmetry of Mechanical Structure and its Application in Design

Symmetry widely exists in natural objects and man-made objects. Mechanical structures, as man-made objects, have the property of symmetry without exception. The existence of symmetry affects the function and performance of mechanical products. Therefore, on the basis of analyzing a large number of examples and referring to the Schoenflies symbol of crystal, the symmetry of mechanical structures is divided into point group symmetry and space group symmetry, and these two types are further subdivided according to the types and spatial positions of the symmetry elements. Then, the general effects of symmetry are summarized according to symmetry types and functions, and several symmetry rules for design are further refined. Finally, after defining the requirements of speed change and technology background, a multispeed device for bicycle shaft drive is proposed by applying symmetry knowledge comprehensively.

Concept System and Application of Point Group Symmetry in Mechanical Structure Design

Symmetry has been widely and deeply researched in basic science, and many mature results have been obtained so far. However, the widespread existence of symmetry in applied science is not in direct proportion to the attention it has received. Through a large number of examples studies, almost all mechanical structures are found to have symmetry, and most of them have the characteristics of point group symmetry. Therefore, the concept of point group symmetry in crystallography was extended to the field of machinery and adjusted according to the mechanical structures. First of all, the classification of mechanical point group symmetry is proposed, and how point group symmetry is applied in machinery is illustrated with examples. Then, the requirements of symmetry are analyzed and compared. Furthermore, the data mining software RapidMiner is used to mine the association rules between requirements and symmetry. Based on the mining results, the four selection principles of point group symmetry are summarized to provide ideas for structure design. Finally, a new type of gear pump with radial force balancing is invented by comprehensively using the mining results and selection principles.

[5,10,15,20-Tetrakis(pentafluorophenyl)porphyrinato]zinc(II) benzene disolvate

Single crystals of the title zinc porphyrinato complex, [Zn(C44H8F20N4)]·2C6H6, were obtained by the solvent evaporation method. The molecular complex exhibits point group symmetry \overline1 with the central ZnII atom located on an inversion centre. The porphyrinato core is approximately planar, and the cation has no other ligating atoms than the four porphyrinato N atoms. π–π interactions between benzene solvent molecules and [Zn(TFPP)] units lead to multilayer packing structures. In addition, intermolecular C—H...F hydrogen bonding is observed between [Zn(TFPP)] molecules.