Ideal Polyhedra Research Articles

Construction of rhombic triacontahedron discrete global grid systems

ABSTRACT Discrete Global Grid System (DGGS) is a new multi-resolution geospatial data modeling and processing scheme for the digital earth. The icosahedron is commonly regarded as an ideal polyhedron for constructing DGGSs with small distortions; however, the shape of its face is triangular, making it difficult to incorporate the matrix structure used for geospatial data storage and parallel computing. To overcome this limitation, this study utilizes the rhombic triacontahedron (RT) as the basic polyhedron to construct DGGSs. An equal-area projection between the surface of RT and the sphere is developed and used to design a grid-generation algorithm for the aperture 4 hexagonal DGGS based on RT. Compared with the equal-area DGGS based on the icosahedron, the proposed scheme results in smaller angular projection distortions, with the mean and standard deviation decreasing by 41.6% and 30.9%, respectively. The grid cells of the RT DGGS also achieve more optimized geometric characteristics in shape compactness, length deviation, and angle deviation than those in the icosahedron DGGS. Additionally, the cross-surface computation efficiency provides advantages in code conversion to latitude and longitude and proximity queries. Furthermore, the use of RT offers a new and better framework within the context of DGGS research and application.

Influence of the Different Types of Auxiliary Noncarboxylate Organic Ligands on the Topologies and Magnetic Relaxation Behavior of Zn-Dy Heterometallic Single Molecule Magnets.

In this work, we first synthesized a Zn-Dy complex, [Zn6Dy2(L)6(tea)2(CH3OH)2]·6CH3OH·8H2O (H2L = N-3-methoxysalicylidene-2-amino-3-hydroxypyridine, teaH3 = triethanolamine, 1), by employing H2L, anhydrous ZnCl2, and Dy(NO3)3·5H2O reacting with auxiliary ligand teaH3 in the mixture of CH3OH and DMF. When teaH3 and the solvent CH3OH in the reaction system of 1 were replaced by the auxiliary ligand 2,6-pyridinedimethanol (pdmH2) and the solvent MeCN, another Zn-Dy complex, [Zn4Dy4(L)6(pdm)2(pdmH)4]·10CH3CN·5H2O (2), was obtained. For 1, its crystal structure can be viewed as a dimer of two Zn3DyIII units. However, for 2, four DyIII form a zigzag arrangement, and each of its terminals linked two ZnII ions. Interestingly, although the structural topologies of 1 and 2 are different, the coordination geometries of DyIII in 1 and 2 are all triangular dodecahedron (TDD-8). The difference is that the continuous shape measure (CShM) values of DyIII in 1 are larger than the corresponding values in 2. Magnetic investigation revealed that the diluted sample 1@Y exhibits two magnetic relaxation processes, while 2 only exhibits a single relaxation process. Ab initio calculations indicated that, in the crystal lattice of 1, two complexes exhibiting slightly different CShM values of DyIII result in the double relaxation behavior of 1@Y. However, for 2, one of two DyIII fragments possesses a fast quantum tunneling of magnetization (QTM), resulting in its magnetic process presented at T < 1.8 K, so 2 exhibits single relaxation behavior. More importantly, the theoretical calculations also clearly indicated that the weak ligation at equatorial sites of DyIII in 1 and 2 ensure 1@Y and 2 possess SMM behavior, although the coordination geometry of DyIII (TDD-8) in 1 and 2 severely deviates from the ideal polyhedron and its axial symmetry is low.

Combinatorial Ricci flows for ideal circle patterns

This paper considers Chow-Luo's combinatorial Ricci flows in the setting of ideal circle patterns. We establish several equivalent conditions regarding long time behaviors of these flows. As consequences, Rivin's and Bobenko-Springborn's theorems are obtained. Moreover, this approach suggests a computational method to find the desired ideal circle patterns (and ideal hyperbolic polyhedra).

Ideal polyhedral surfaces in Fuchsian manifolds

Let S_{g,n} be a surface of genus g > 1 with n>0 punctures equipped with a complete hyperbolic cusp metric. Then it can be uniquely realized as the boundary metric of an ideal Fuchsian polyhedron. In the present paper we give a new variational proof of this result. We also give an alternative proof of the existence and uniqueness of a hyperbolic polyhedral metric with prescribed curvature in a given conformal class.

Parameterization of the crystal structure of garnet in terms of symmetry-adapted basis-vectors of the ideal tetrahedron and octahedron: Application to the pressure-dependence of the crystal structure of Y3Al5O12 between 0 and 126 GPa

Abstract Non-linear optimisation of the anion coordinates in garnet-structured compounds (A3B2C3O12) with space group I a 3 ¯ d has been used to define a universal prototypic crystal structure in which the tetrahedral site (CO4) and the octahedral site (BO6) exhibit close to the ideal point group symmetries of 4 ¯ 3 m and m 3 ¯ m respectively. Distortions in actual garnet-structured compounds are then described in terms of condensed normal modes of the two isolated, ideal polyhedra that are consistent with the site point group symmetry within the space group I a 3 ¯ d . The methodology is applied to the pressure-dependence of the crystal structure of Y3Al5O12 from 0 GPa to 126 GPa that have been determined from first-principles calculations. The structural evolution of Y3Al5O12 with pressure is dominated by changes in the AlivO4 site which rotates and becomes significantly more regular as pressure increases. The coupled response of the octahedral site is more complex as it responds to the changes in the tetrahedral site. The evolution of the crystal structure of YAG with pressure is briefly compared to the experimentally determined pressure-dependence of the crystal structure of pyrope, Mg3Al2Si3O12.

The Growth Rates of Ideal Coxeter Polyhedra in Hyperbolic 3-Space

In [6], Kellerhals and Perren conjectured that the growth rates of the reflection groups given by hyperbolic Coxeter polyhedra are always Perron numbers. We prove that this conjecture is always true for the case of ideal Coxeter polyhedra in H 3 . We also find out the ideal Coxeter polyhedron in H 3 with the smallest growth rate. Finally, we show that there are correlations between the volumes and the growth rates of ideal Coxeter polyhedra in H 3 in many cases.

Cooling rate dependence and local structure in aluminum monatomic metallic glass

The local atomic structure in aluminium monatomic metallic glass is studied using molecular dynamics simulations combined with the embedded atom method (EAM). We have used a variety of analytical methods to characterise the atomic configurations of our system: the Pair Distribution Function (PDF), the Common Neighbour Analysis (CNA) and the Voronoi Tessellation Analysis. CNA was used to investigate the order change from liquid to amorphous phases, recognising that the amount of icosahedral clusters increases with the decrease of temperature. The Voronoi analysis revealed that the icosahedral-like polyhedral are the predominant ones. It has been observed that the PDF function shows a splitting in the second peak, which cannot be attributed to the only ideal icosahedral polyhedron 〈0, 0, 12, 0〉, but also to the formation of other Voronoi polyhedra 〈0, 1, 10, 2〉 . Further, the PDFs were then integrated giving the cumulative coordination number in order to compute the fractal dimension (df).

Discrete conformal maps and ideal hyperbolic polyhedra

We establish a connection between two previously unrelated topics: a particular discrete version of conformal geometry for triangulated surfaces, and the geometry of ideal polyhedra in hyperbolic three-space. Two triangulated surfaces are considered discretely conformally equivalent if the edge lengths are related by scale factors associated with the vertices. This simple definition leads to a surprisingly rich theory featuring M\"obius invariance, the definition of discrete conformal maps as circumcircle preserving piecewise projective maps, and two variational principles. We show how literally the same theory can be reinterpreted to addresses the problem of constructing an ideal hyperbolic polyhedron with prescribed intrinsic metric. This synthesis enables us to derive a companion theory of discrete conformal maps for hyperbolic triangulations. It also shows how the definitions of discrete conformality considered here are closely related to the established definition of discrete conformality in terms of circle packings.

Volume and geometry of homogeneously adequate knots

We bound the hyperbolic volumes of a large class of knots and links, called homogeneously adequate knots and links, in terms of their diagrams. To do so, we use the decomposition of these links into ideal polyhedra, developed by Futer, Kalfagianni and Purcell. We identify essential product disks in these polyhedra.

Stereochemistry of complexes with double and triple metal-ligand bonds: a continuous shape measures analysis.

To each coordination polyhedron we can associate a normalized coordination polyhedron that retains the angular orientation of the central atom-ligand bonds but has all the vertices at the same distance from the center. The use of shape measures of these normalized coordination polyhedra provides a simple and efficient way of discriminating angular and bond distance distortions from an ideal polyhedron. In this paper we explore the applications of such an approach to analyses of several stereochemical problems. Among others, we discuss how to discern the off-center displacement of the metal from metal-ligand bond shortening distortions in families of square planar biscarbene and octahedral dioxo complexes. The normalized polyhedron approach is also shown to be very useful to understand stereochemical trends with the help of shape maps, minimal distortion pathways, and ligand association/dissociation pathways, illustrated by the Berry and anti Berry distortions of triple-bonded [X≡ML4] complexes, the square pyramidal geometries of Mo coordination polyhedra in oxido-reductases, the coordination geometries of actinyl complexes, and the tetrahedricity of heavy atom-substituted carbon centers.

Rank gradient in co-final towers of certain Kleinian groups

We prove that if the fundamental group of an orientable finite volume hyperbolic 3-manifold has finite index in the reflection group of a right-angled ideal polyhedron in \mathbb{H}^3 then it has a co-final tower of finite sheeted covers with positive rank gradient. The manifolds we consider are also known to have co-final towers of covers with zero rank gradient.

Canonical triangulations of Dehn fillings

Every cusped, finite-volume hyperbolic three-manifold has a canonical decomposition into ideal polyhedra. We study the canonical decomposition of the hyperbolic manifold obtained by filling some (but not all) of the cusps with solid tori: in a broad range of cases, generic in an appropriate sense, this decomposition can be predicted from that of the unfilled manifold. As an example, we treat all hyperbolic fillings on one cusp of the Whitehead link complement.

On the boundary of a special class of hyperbolic two-dimensional simplicial complexes

A class of hyperbolic two-dimensional complexes X is defined. It is shown that the limit set of the action of π1(X) on the universal covering \({\widetilde{X}}\) of X is equal to the visual boundary \({\partial \widetilde{X}}\) of and that \({\partial \widetilde{X}}\) is path connected and locally path connected. A kind of Sierpinski set is described which is homeomorphic to the visual boundary of certain ideal polyhedra.

Synthesis and structure of bismuth-containing complexes [Ph3PMe] 2 + [BiI5]2− and [Ph3PMe] 2 + [BiI5 · C5H5N]2− · C5H5N

The complexes [Ph3PMe]2+[BiI5]2− (I) and [Ph3PMe]2+ [BiI5 · C 5H5N]2− · C 5H5N (II) were synthesized by the reaction of bismuth triiodide with triphenylmethylphosphonium iodide, and their structures were determined by X-ray diffraction analysis. The P atom in cation I has slightly distorted tetragonal coordination polyhedron (the CPC angles 109.42(4)° and 109.52(4)°, the bond lengths P-CPh 1.779(2), P-CMe 1.793(1) A. The Bi atom in the anion of complex I has an ideal trigonal-bipyramidal coordination polyhedron (Bi-Ieq 3.0031(4), Bi-Ieq 3.0485(5) A). The crystal of complex II consists of the anions [BiI5 · C 5H5N]2−, solvated pyridine molecules, and two types of crystallographically independent tetrahedral triphenylmethylphosphonium cations (the angles CPC 106.9(1)°–111.7(1)°, the distances P-CPh 1.785(3)–1.792(3), P-CMe 1.793(3), 1.786(3) A). The Bi atoms in the anion of complex II have a distorted octahedral coordination polyhedron (Bi-I 3.0878(4)–3.1240(3), Bi-N(1) 2.628(3) A).

A variational principle for weighted Delaunay triangulations and hyperideal polyhedra

We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with non-intersecting site-circles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations may be interpreted as images of hyperbolic polyhedra with one vertex on and the remaining vertices beyond the infinite boundary of hyperbolic space. Thus, the main theorem states necessary and sufficient conditions for the existence and uniqueness of such polyhedra with prescribed combinatorial type and dihedral angles. More generally, we consider weighted Delaunay triangulations in piecewise flat surfaces, allowing cone singularities with prescribed cone angles in the vertices. The material presented here extends work by Rivin on Delaunay triangulations and ideal polyhedra.

On the isometries of ideal polyhedra

On montre que siX est un polyedre ideal complet fini satisfaisant la condition CAT(−1) locale, alors toute isometrie deX homotope a l’identite est l’identite, pourvu queπ1(X) ne soit pas elementaire. On montre aussi que le groupe des isometies deX est fini.

On an Elementary Proof of Rivin's Characterization of Convex Ideal Hyperbolic Polyhedra by their Dihedral Angles

On an Elementary Proof of Rivin's Characterization of Convex Ideal Hyperbolic Polyhedra by their Dihedral Angles

On an Elementary Proof of Rivin's Characterization of Convex Ideal Hyperbolic Polyhedra by their Dihedral Angles

In 1832, Jakob Steiner (Systematische Entwicklung der Abhangigkeit geometrischer Gestalten von einander, Reimer (Berlin)) asked for a characterization of those planar graphs which are combinatorially equivalent to polyhedra inscribed in the sphere. This question was answered in the 1990s by Igor Rivin (Ann. of Math. 143(1996), 51–70), as a byproduct of his classification of ideal polyhedra in hyperbolic 3-space. Rivin also proposed a more direct approach to these results in Rivin (Ann. of Math. 139 (1994), 553–580). In this paper, we prove a combinatorial result (Theorem 6) which enables one to complete the program of Rivin (Ann. Math. 139(1994), 553–580).

The Laplacian and Dirac operators on critical planar graphs

On a periodic planar graph whose edge weights satisfy a certain simple geometric condition, the discrete Laplacian and d-bar operators have the property that their determinants and inverses only depend on the local geometry of the graph. We obtain explicit expressions for the logarithms of the (normalized) determinants, as well as the inverses of these operators. We relate the logarithm of the determinants to the volume plus mean curvature of an associated hyperbolic ideal polyhedron. In the associated dimer and spanning tree models, for which the determinants of d-bar and the Laplacian respectively play the role of the partition function, this allows us to compute the entropy and correlations in terms of the local geometry. In addition, we define a continuous family of special discrete analytic functions, which, via convolutions gives a general process for constructing discrete analytic functions and discrete harmonic functions on critical planar graphs.

ON ARCHIMEDEAN LINK COMPLEMENTS

We study a subclass of alternating links for which the complete hyperbolic metric can be realised directly by pairwise identification of faces of two ideal hyperbolic polyhedra. Our main result is a characterization of these links: essentially, the corresponding polyhedra are exactly the Archimedean solids with trivalent vertices. Furthermore, we show that the only knots which arise are the two dodecahedral knots, and the figure eight knot.