Unusual Permutation Groups in Negative Curvature Carbon and Boron Nitride Structures

The concept of symmetry point groups for regular polyhedra can be generalized to special permutation groups to describe negative curvature polygonal networks that can be expanded to possible carbon and boron nitride structures through leapfrog transformations, which triple the number of vertices. Thus a D surface with 24 hep-tagons and 56 hexagons in the unit cell can be generated by a leapfrog transformation from the Klein figure consisting only of the 24 heptagons. The permutational symmetry of the Klein figure can be described by the simple PSL(2,7) (or heptakisoctahedral) group of order 168 with the conjugacy class structure E + 24C7 + 24C73 + 56C3 + 21C2 + 42C4. Analogous methods can be used to generate a D surface with 12 octagons and 32 hexagons by a leapfrog transformation from the Dyck figure consisting only of the 12 octagons. The permutational symmetry of the Dyck figure can be described by a group of order 96 and the conjugacy class structure E + 24S8 + 6C4 + 3C42 + 32C3 + 12C2 + 18S4. This group is not a simple group since it has a normal subgroup chain leading to the trivial group C1 through subgroups of order 48 and 16 not related to the octahedral or tetrahedral groups

Regular Polytopes, Root Lattices, and Quasicrystals

The icosahedral quasicrystals of five-fold symmetry in two, three, and four dimensions are related to the corresponding regular polytopes exhibiting five-fold symmetry, namely the regular pentagon (H2 reflection group), the regular icosahedron {3,5} (H3 reflection group), and the regular four-dimensional polytope {3,3,5} (H4 reflection group). These quasicrystals exhibiting five-fold symmetry can be generated by projections from indecomposable root lattices with twice the number of dimensions, namely A4→H2, D6→H3, E8→H4. Because of the subgroup relationships H2 ⊂ H3 ⊂ H4, study of the projection E8→H4 provides information on all of the possible icosahedral quasicrystals. Similar projections from other indecomposable root lattices can generate quasicrystals of other symmetries. Four-dimensional root lattices are sufficient for projections to two-dimensional quasicrystals of eight-fold and twelve-fold symmetries. However, root lattices of at least six-dimensions (e.g., the A6 lattice) are required to generate twodimensional quasicrystals of seven-fold symmetry. This might account for the absence of seven-fold symmetry in experimentally observed quasicrystals

Unusual Permutation Groups in Negative Curvature Carbon and Boron Nitride Structures

The concept of symmetry point groups for regular polyhedra can be generalized to special permutation groups to describe negative curvature polygonal networks that can be expanded to possible carbon and boron nitride structures through leapfrog transformations, which triple the number of vertices. Thus a D surface with 24 hep-tagons and 56 hexagons in the unit cell can be generated by a leapfrog transformation from the Klein figure consisting only of the 24 heptagons. The permutational symmetry of the Klein figure can be described by the simple PSL(2,7) (or heptakisoctahedral) group of order 168 with the conjugacy class structure E + 24C7 + 24C73 + 56C3 + 21C2 + 42C4. Analogous methods can be used to generate a D surface with 12 octagons and 32 hexagons by a leapfrog transformation from the Dyck figure consisting only of the 12 octagons. The permutational symmetry of the Dyck figure can be described by a group of order 96 and the conjugacy class structure E + 24S8 + 6C4 + 3C42 + 32C3 + 12C2 + 18S4. This group is not a simple group since it has a normal subgroup chain leading to the trivial group C1 through subgroups of order 48 and 16 not related to the octahedral or tetrahedral groups

Crystallographic and Quasicrystallographic Lattices from the Finite Groups of Quaternions

Quaternions are ordered quadruples of four numbers subject to specified rules of addition and multiplication, which can represent points in four-dimensional (4D) space and which form finite groups under multiplication isomorphic to polyhedral groups. Projection of the 8 quaternions of the dihedral group D2h, with only two-fold symmetry, into 3D space provides a basis for crystal lattices up to orthorhombic symmetry (a "* b "* c). Addition of three-fold symmetry to D2h gives the tetrahedral group Td with 24 quaternions, whose projection into 3D space provides a basis for more symmetrical crystal lattices including the cubic lattice (a = b = c). Addition of five fold symmetry to Td gives the icosahedral group Ih with 120 quaternions, whose projection into 3D space introduces the --J5 irrationality and thus cannot provide the basis for a 3D crystal lattice. However, this projection of Ih can provide a basis for a 6D lattice which can be divided into two orthogonal 3D subspaces, one representing rational coordinates and the other representing COOI\u27- dinates containing the --J5 irrationality similar to some standard models for icosahedral quasicrystals

Riemann Surfaces as Descriptors for Symmetrical Negative Curvature Carbon and Boron Nitride Structures

Leapfrog transformations starting with the genus 3 Klein and Dyck tessellations consisting of 24 heptagons and 12 octagons, respectively, can generate possible highly symmetrical structures for allo-tropes of carbon and the isosteric boron nitride, (BN)x. The Klein tessellation, alternatively described as a platonic {3,7} tessellation, corresponds to the Riemann surface for the multi-valued function w = 7√(z(z-1)2), which can also described by the homogeneous quartic polynomial ξ3n + η3ω + ω3ξ = 0. The symmetry of this polynomial is related to the heptakisoctahedral automorphism group of the Klein tessellation of order 168. Similarly the Dyck or {3,8} tessellation can be described by a Riemann surface which corresponds to the homogeneous Fermat quartic polynomial ξ4 + η4 + ω4 = 0. The symmetry of the Fermat quartic relates to the automorphism group of the Dyck tessellation of order 96

Robotic-Movement Payload Lifter and Manipulator

A payload lifter/manipulator module includes a rotatable joint supporting spreader arms angularly spaced with respect to one another. A rigid arm is fixedly coupled to the joint and extends out therefrom to a tip. A tension arm has a first end and a second end with the first end being fixedly coupled to the tip of the rigid arm. The tension arm incorporates pivots along the length thereof. Each pivot can be engaged by or disengaged from the outboard end of a spreader arm based on a position of the spreader arm. A hoist, positioned remotely with respect to the module and coupled to the second end of the tension arm, controls the position of the spreader arms to thereby control the position of the rigid arm's tip. Payload lifter/manipulator assemblies can be constructed with one or more of the modules

Development of a Tendon-Actuated Lightweight In-Space MANipulator (TALISMAN)

An invention of a new and novel space robotic manipulator is described. By using a combination of lightweight truss links, a novel hinge joint, tendon-articulation and passive tension stiffening, this new robotic manipulator architecture achieves compact packaging, high strength, stiffness and dexterity while being very lightweight compared to conventional manipulators. The manipulator is also very modular; easy to scale for different reach, load and stiffness requirements; enabling customization for a diverse set of applications. Novel features of the new manipulator concept are described as well as some of the approaches to implement these design features. Two diverse applications are presented to show the versatility of the concept. First generation prototype hardware was designed, manufactured and has been assembled into a working manipulator that is being used to refine and extend development efforts