Search Results (15)

Polyhedral cages (p-cages) are Euclidean geometric structures corresponding to polyhedra with holes. They are a good example of the geometry of some artificial protein cages. In this paper we identify p-cages made out of two families of equivalent polygonal faces, where the face of one family is attached to three other faces while the faces of the other family are attached to three, four, five or six other faces. To restrict ourselves to p-cages with small holes, we consider p-cages where each hole comprises at most four faces. The construction starts from planar graphs made out of two families of equivalent nodes. One can then construct the dual of the solid corresponding to that graph and tile its faces with regular or nearly regular polygons. An energy function is then defined to quantify the amount of irregularity of the p-cages which is then minimised using a simulated annealing algorithm. We have analysed nearly 100,000 possible configurations, ruling out the p-cages made out of faces with deformations exceeding 10%. We then present graphically some of the most interesting geometries. Full article

Polyhedral cages (p-cages) provide a good description of the geometry of some families of artificial protein cages. In this paper we identify p-cages made out of two families of equivalent polygonal faces/protein rings, where each face has at least four neighbours and where the holes are contributed by at most four faces. We start the construction from a planar graph made out of two families of equivalent nodes. We construct the dual of the solid corresponding to that graph, and we tile its faces with regular or nearly regular polygons. We define an energy function describing the amount of irregularity of the p-cages, which we then minimise using a simulated annealing algorithm. We analyse over 600,000 possible geometries but restrict ourselves to p-cages made out of faces with deformations not exceeding 10%. We then present graphically some of the most promising geometries for protein nanocages. Full article

The primary objective of this study is the computation of the matching polynomials of a number of symmetric, semisymmetric, double group graphs, and solids in third and higher dimensions. Such computations of matching polynomials are extremely challenging problems due to the computational and combinatorial complexity of the problem. We also consider a series of recursive graphs possessing symmetries such as D_2h-polyacenes, wheels, and fans. The double group graphs of the Möbius types, which find applications in chemically interesting topologies and stereochemistry, are considered for the matching polynomials. Hence, the present study features a number of vertex- or edge-transitive regular graphs, Archimedean solids, truncated polyhedra, prisms, and 4D and 5D polyhedra. Such polyhedral and Möbius graphs present stereochemically and topologically interesting applications, including in chirality, isomerization reactions, and dynamic stereochemistry. The matching polynomials of these systems are shown to contain interesting combinatorics, including Stirling numbers of both kinds, Lucas polynomials, toroidal tree-rooted map sequences, and Hermite, Laguerre, Chebychev, and other orthogonal polynomials. Full article

Polyhedral cages (p-cages) describe the geometry of some families of artificial protein cages. We identify the p-cages made out of families of equivalent polygonal faces such that the faces of one family have five neighbors and $P_1$ edges, while those of the other family have six neighbors and $P_2$ edges. We restrict ourselves to polyhedral cages where the holes are adjacent to four faces at most. We characterize all p-cages with a deformation of the faces, compared to regular polygons, not exceeding 10%. Full article

A regular map is an abstract generalization of a Platonic solid. It describes a group, a topological cell decomposition of a 2-manifold of type $\{p, q\}$ with only p-gons, such that q of them meet at each vertex in a circular manner, and we have maximal combinatorial symmetry, expressed by the flag transitivity of the symmetry group. On the one hand, we have articles on topological surface embeddings of regular maps by F. Razafindrazaka and K. Polthier, C. Séquin, and J. J. van Wijk.On the other hand, we have articles with polyhedral embeddings of regular maps by J. Bokowski and M. Cuntz, A. Boole Stott, U. Brehm, H. S. M. Coxeter, B. Grünbaum, E. Schulte, and J. M. Wills. The main concern of this partial survey article is to emphasize that all these articles should be seen as contributing to the common body of knowledge in the area of regular map embeddings. This article additionally provides a method for finding symmetrical equivelar polyhedral embeddings of type $\{3, 7\}$ based on symmetrical graph embeddings on convex surfaces. Full article

In the current qualitative study, we explored father (and varying father figures’) ethnoracial and gendered socialization messages toward Latina and Black college women. We conducted six focus group interviews with Black (n = 3 groups) and Latina (n = 3 groups) college women. Guided by Chicana and Black feminist interpretive phenomenological analysis, we identified four clusters which detailed perceived paternal influences in the lives of these college women: (a) paternal caring, (b) gender socialization, (c) value of education, and (d) developing platonic and romantic relationships. Full article

The symmetries of a Riemann surface $\mathsf{\Sigma }\setminus \left\{a_i\right\}$ with n punctures $a_i$ are encoded in its fundamental group ${\pi }_1\left(\mathsf{\Sigma }\right)$. Further structure may be described through representations (homomorphisms) of ${\pi }_1$ over a Lie group G as globalized by the character variety $\mathcal{C}=\mathrm{Hom}({\pi }_1,G)/G$. Guided by our previous work in the context of topological quantum computing (TQC) and genetics, we specialize on the four-punctured Riemann sphere $\mathsf{\Sigma }=S_2^{\left(4\right)}$ and the ‘space-time-spin’ group $G=S{L}_2\left(\mathbb{C}\right)$. In such a situation, $\mathcal{C}$ possesses remarkable properties: (i) a representation is described by a three-dimensional cubic surface $V_{a,b,c,d}(x,y,z)$ with three variables and four parameters; (ii) the automorphisms of the surface satisfy the dynamical (non-linear and transcendental) Painlevé VI equation (or $P_{VI}$); and (iii) there exists a finite set of 1 (Cayley–Picard)+3 (continuous platonic)+45 (icosahedral) solutions of $P_{VI}$. In this paper, we feature the parametric representation of some solutions of $P_{VI}$: (a) solutions corresponding to algebraic surfaces such as the Klein quartic and (b) icosahedral solutions. Applications to the character variety of finitely generated groups $f_p$ encountered in TQC or DNA/RNA sequences are proposed. Full article

Following the discovery of an artificial protein cage with a paradoxical geometry, we extend the concept of homogeneous symmetric congruent equivalent near-miss polyhedral cages, for which all the faces are equivalent, and define bi-homogeneous symmetric polyhedral cages made of two different types of faces, where all the faces of a given type are equivalent. We parametrise the possible connectivity configurations for such cages, analytically derive p-cages that are regular, and numerically compute near-symmetric p-cages made of polygons with 6 to 18 edges and with deformation not exceeding 10%. Full article

Following the experimental discovery of several nearly symmetric protein cages, we define the concept of homogeneous symmetric congruent equivalent near-miss polyhedral cages made out of P-gons. We use group theory to parameterize the possible configurations and we minimize the irregularity of the P-gons numerically to construct all such polyhedral cages for $P=6$ to $P=20$ with deformation of up to 10%. Full article

It is well known that the point group of the root lattice D₆ admits the icosahedral group as a maximal subgroup. The generators of the icosahedral group H₃, its roots, and weights are determined in terms of those of D₆. Platonic and Archimedean solids possessing icosahedral symmetry have been obtained by projections of the sets of lattice vectors of D₆ determined by a pair of integers (m₁, m₂) in most cases, either both even or both odd. Vertices of the Danzer’s ABCK tetrahedra are determined as the fundamental weights of H₃, and it is shown that the inflation of the tiles can be obtained as projections of the lattice vectors characterized by the pair of integers, which are linear combinations of the integers (m₁, m₂) with coefficients from the Fibonacci sequence. Tiling procedure both for the ABCK tetrahedral and the <ABCK> octahedral tilings in 3D space with icosahedral symmetry H₃, and those related transformations in 6D space with D₆ symmetry are specified by determining the rotations and translations in 3D and the corresponding group elements in D₆. The tetrahedron K constitutes the fundamental region of the icosahedral group and generates the rhombic triacontahedron upon the group action. Properties of “K-polyhedron”, “B-polyhedron”, and “C-polyhedron” generated by the icosahedral group have been discussed. Full article

A single qubit may be represented on the Bloch sphere or similarly on the 3-sphere $S^3$ . Our goal is to dress this correspondence by converting the language of universal quantum computing (UQC) to that of 3-manifolds. A magic state and the Pauli group acting on it define a model of UQC as a positive operator-valued measure (POVM) that one recognizes to be a 3-manifold $M^3$ . More precisely, the d-dimensional POVMs defined from subgroups of finite index of the modular group $PSL(2,\mathbb{Z})$ correspond to d-fold $M^3$ - coverings over the trefoil knot. In this paper, we also investigate quantum information on a few ‘universal’ knots and links such as the figure-of-eight knot, the Whitehead link and Borromean rings, making use of the catalog of platonic manifolds available on the software SnapPy. Further connections between POVMs based UQC and $M^3$ ’s obtained from Dehn fillings are explored. Full article

The formation of crystals and symmetry on the atomic scale has persistently attracted scientists through the ages. The structure itself and its subtle dependence on boundary conditions is a reflection of three principles: atomic attraction, repulsion, and the limitations in 3D space. This involves a competition between simplicity and high symmetry on the one hand and necessary structural complexity on the other. This work presents a simple atomistic crystal growth model derived for equivalent atoms and a pair potential. It highlights fundamental concepts, most prominently provided by a maximum number of equilibrium distances in the atom’s local vicinity, to obtain high symmetric structural motifs, among them the Platonic Solids. In this respect, the harmonically balanced interaction during the atomistic nucleation process may be regarded as origin of symmetry. The minimization of total energy is generalized for 3D periodic structures constituting these motifs. In dependence on the pair potential’s short- and long-range characteristics the, by symmetry, rigid lattices relax isotropically within the potential well. The first few coordination shells with lattice-specific fixed distances do not necessarily determine which equilibrium symmetry prevails. A phase diagram calculated on the basis of these few assumptions summarizes stable regions of close-packed fcc and hcp, next to bcc symmetry for predominantly soft short-range and hard long-range interaction. This lattice symmetry, which is evident for alkali metals as well as transition metals of the vanadium and chromium group, cannot be obtained from classical Morse or Lennard-Jones type potentials, but needs the range flexibility within the pair potential. Full article

Deriving its designation from the Greek word for ‘dog’, cynicism is likely the only philosophical ‘interest group’ with a diachronically dependable affinity for various animals—particularly those of the canine kind. While dogs have met with differing value judgments, chiefly along a perceived human–animal divide, it is specifically discourses with cynical affinities that render problematic this transitional field. The Cervantine “coloquio de los perros” has received scholarly attention for its (caninely) picaresque themes, its “cynomorphic” (Ziolkowski) narratological technique, its socio-historically informative accounts relating to Early Modern Europe and the Iberian peninsula, including its ‘zoopoetically’ (Derrida) relevant portrayal of dogs (see e.g., Alves, Beusterien, Martín); nor did the dialog’s mention of cynical snarling go unnoticed. The essay at hand commences with a chapter on questions of method pertaining to ‘animal narration’: with recourse to Montaigne, Descartes, and Derrida, this first part serves to situate the ensuing close readings with respect to the field of Animal Studies. The analysis of the Cervantine texts synergizes thematic and narratological aspects at the discourse historical level; it commences with a brief synopsis of the respective novellas in part 2; Section 3, Section 4 and Section 5 supply a description of the rhetorical modes of crafting plausibility in the framework narrative (“The Deceitful Marriage”), of pertinent (Scriptural) intertexts for the “Colloquy”. Parts 6–7 demonstrate that the choice of canine interlocutors as narrating agencies—and specifically in their capacity as dogs—is discursively motivated: no other animal than this animal, and precisely as animal, would here serve the discursive purpose that is concurrently present with the literal plane; for this dialogic novella partakes of a (predominantly Stoicizing) tradition attempting to resocialize the Cynics, which commences already with the appearance of the Ancient arch-Cynic ‘Diogenes’ on the scene. At the discursive level, a diachronic contextualization evinces that the Cervantine text takes up and outperforms those rhetorical techniques of reintegration by melding Christian, Platonic, Stoicizing elements with such as are reminiscent of Diogenical ones. Reallocating Blumenberg’s reading of a notorious Goethean dictum, this essay submits the formula ‘against the Dog only a dog’ as a concise précis of the Cervantine method at the discursive level, attained to via a decidedly pluralized rhetorical sermocination featuring, at a literal level, specifically canine narrators in a dialogic setting. Full article

We carry out the harmonic analysis on four Platonic spherical three-manifolds with different topologies. Starting out from the homotopies (Everitt 2004), we convert them into deck operations, acting on the simply connected three-sphere as the cover, and obtain the corresponding variety of deck groups. For each topology, the three-sphere is tiled into copies of a fundamental domain under the corresponding deck group. We employ the point symmetry of each Platonic manifold to construct its fundamental domain as a spherical orbifold. While the three-sphere supports an orthonormal complete basis for harmonic analysis formed by Wigner polynomials, a given spherical orbifold leads to a selection of a specific subbasis. The resulting selection rules find applications in cosmic topology, probed by the cosmic microwave background. Full article

Combinatorially regular polyhedra are polyhedral realizations (embeddings) in Euclidean 3-space E³ of regular maps on (orientable) closed compact surfaces. They are close analogues of the Platonic solids. A surface of genus g ≥ 2 admits only finitely many regular maps, and generally only a small number of them can be realized as polyhedra with convex faces. When the genus g is small, meaning that g is in the historically motivated range 2 ≤ g ≤ 6, only eight regular maps of genus g are known to have polyhedral realizations, two discovered quite recently. These include spectacular convex-faced polyhedra realizing famous maps of Klein, Fricke, Dyck, and Coxeter. We provide supporting evidence that this list is complete; in other words, we strongly conjecture that in addition to those eight there are no other regular maps of genus g, with 2 ≤ g ≤ 6, admitting realizations as convex-faced polyhedra in E³. For all admissible maps in this range, save Gordan’s map of genus 4, and its dual, we rule out realizability by a polyhedron in E³. Full article