Search Results (10)

We characterize computer science as an interplay between two modes of reasoning: the Aristotelian (procedural) method and the Platonic (declarative) approach. We contend that Aristotelian, step-by-step thinking dominates in computer programming, while Platonic, static reasoning plays a more prominent role in computational complexity. Various frameworks elegantly blend both Aristotelian and Platonic reasoning. A key example explored in this paper concerns nondeterministic polynomial time Turing machines. Beyond this interplay, we emphasize the growing importance of the ‘computing by observing’ paradigm, which posits that a single derivation tree—generated with a string-rewriting system—can yield multiple interpretations depending on the choice of the observer. Advocates of this paradigm formalize the Aristotelian activities of rewriting and observing within automata theory through a Platonic lens. This approach raises a fundamental question: How do these Aristotelian activities re-emerge when the paradigm is formulated in propositional logic? By addressing this issue, we develop a novel simulation method for nondeterministic Turing machines, particularly those bounded by polynomial time, improving upon the standard textbook approach. 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

A topological interlocking assembly (TIA) is an assembly of blocks together with a non-empty subset of blocks called the frame such that every non-empty set of blocks is kinematically constrained and can therefore not be removed from the assembly without causing intersections between blocks of the assembly. TIA provides a wide range of real-world applications, from modular construction in architectural design to potential solutions for sound insulation. Various methods to construct TIA have been proposed in the literature. In this paper, the approach of constructing TIA by applying the Escher trick to tilings of orientable surfaces is discussed. First, the strengths of this approach are highlighted for planar tilings, and the Escher trick is then exploited to construct a planar TIA that is based on the truncated square tiling, which is a semi-regular tiling of the Euclidean plane. Next, the Escher-Like approach is modified to construct TIAs that are based on arbitrary orientable surfaces. Finally, the capabilities of this modified construction method are demonstrated by constructing TIAs that are based on the unit sphere, the truncated icosahedron, and the deltoidal hexecontahedron. Full article

This paper reviews current numerical developments for atmospheric modelling. Numerical atmospheric modelling now looks back to a history of about 70 years after the first successful numerical prediction. Currently, we face new challenges, such as variable and adaptive resolution and ultra-highly resolving global models of 1 km grid length. Large eddy simulation (LES), special applications like the numerical prediction of pollution and atmospheric contaminants belong to the current challenges of numerical developments. While pollution prediction is a standard part of numerical modelling in case of accidents, models currently being developed aim at modelling pollution at all scales from the global to the micro scale. The methods discussed in this paper are spectral elements and other versions of Local-Galerkin (L-Galerkin) methods. Classic numerical methods are also included in the presentation. For example, the rather popular second-order Arakawa C-grid method can be shown to result as a special case of an L-Galerkin method using low-order basis functions. Therefore, developments for Galerkin methods also apply to this classic C-grid method, and this is included in this paper. The new generation of highly parallel computers requires new numerical methods, as some of the classic methods are not well suited for a high degree of parallel computing. It will be shown that some numerical inaccuracies need to be resolved and this indicates a potential for improved results by going to a new generation of numerical methods. The methods considered here are mostly derived from basis functions. Such methods are known under the names of Galerkin, spectral, spectral element, finite element or L-Galerkin methods. Some of these new methods are already used in realistic models. The spectral method, though highly used in the 1990s, is currently replaced by the mentioned local L-Galerkin methods. All methods presented in this review have been tested in idealized numerical situations, the so-called toy models. Waypoints on the way to realistic models and their mathematical problems will be pointed out. Practical problems of informatics will be highlighted. Numerical error traps of some current numerical approaches will be pointed out. These are errors not occurring with highly idealized toy models. Such errors appear when the test situation becomes more realistic. For example, many tests are for regular resolution and results can become worse when the grid becomes irregular. On the sphere no regular grids exist, except for the five derived from Platonic solids. Practical problems beyond mathematics on the way to realistic applications will also be considered. A rather interesting and convenient development is the general availability of computer power. For example, the computational power available on a normal personal computer is comparable to that of a supercomputer in 2005. This means that interesting developments, such as the small sphere atmosphere with a resolution of 1 km and a spherical circumference between 180 and 360 km are available to the normal owner of a personal computer (PC). Besides the mathematical problems of new approaches, we will also consider the informatics challenges of using the new generation of models on mainframe computers and PCs. 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

An algebraic approach to the design of resource-efficient carbon-reinforced concrete structures is presented. Interdisciplinary research in the fields of mathematics and algebra on the one hand and civil engineering and concrete structures on the other can lead to fruitful interactions and can contribute to the development of resource-efficient and sustainable concrete structures. Textile-reinforced concrete (TRC) using non-crimp fabric carbon reinforcement enables very thin and lightweight constructions and thus requires new construction strategies and new manufacturing methods. Algebraic methods applied to topological interlocking contribute to modular, reusable, and hence resource-efficient TRC structures. A modular approach to construct new interlocking blocks by combining different Platonic and Archimedean solids is presented. In particular, the design of blocks that can be decomposed into various n-prisms is the focus of this paper. It is demonstrated that the resulting blocks are highly versatile and offer numerous possibilities for the creation of interlocking assemblies, and a rigorous proof of the interlocking property is outlined. Full article

Eric Lord has observed: “In nature, helical structures arise when identical structural subunits combine sequentially, the orientational and translational relation between each unit and its predecessor remaining constant.” This paper proves Lord’s observation. Constant-time algorithms are given for the segmented helix generated from the intrinsic properties of a stacked object and its conjoining rule. Standard results from screw theory and previous work are combined with corollaries of Lord’s observation to allow calculations of segmented helices from either transformation matrices or four known consecutive points. The construction of these from the intrinsic properties of the rule for conjoining repeated subunits of arbitrary shape is provided, allowing the complete parameters describing the unique segmented helix generated by arbitrary stackings to be easily calculated. Free/Libre open-source interactive software and a website which performs this computation for arbitrary prisms along with interactive 3D visualization is provided. We prove that any subunit can produce a toroid-like helix or a maximally-extended helix, forming a continuous spectrum based on joint-face normal twist. This software, website and paper, taken together, compute, render, and catalog an exhaustive “zoo” of 28 uniquely-shaped platonic helices, such as the Boerdijk–Coxeter tetrahelix and various species of helices formed from dodecahedra. Full article

The concept of Platonism is extended by introducing the new concept of a “Platonic computer” which is incorporated in metacomputics. The theoretical framework of metacomputics postulates that such a Platonic computer exists within the realm of ‘forms’ and is made by, of, with, and from metaconsciousness. Metaconsciousness is defined as the “power to conceive, to perceive, and to be self-aware” and is the formless, contentless infinite potentiality. The infinite potentiality of metaconsciousness is expressed as specific actualities via Platonic computation. This means that an abstract entity is the conscious state of being a specific actuality and is the processing output of the Platonic computer. As such, the physical computer made of silicon is but a shadow or poor imitation of the Platonic computer. Nevertheless, by programing the physical computer it is possible to simulate the generation of abstract entities as the processing output of the Platonic computer. 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 icosahedron and the dodecahedron have the same graph structures as their algebraic conjugates, the great dodecahedron and the great stellated dodecahedron. All four polyhedra are equilateral and have planar faces—thus “EP”—and display icosahedral symmetry. However, the latter two (star polyhedra) are non-convex and “pathological” because of intersecting faces. Approaching the problem analytically, we sought alternate EP-embeddings for Platonic and Archimedean solids. We prove that the number of equations—E edge length equations (enforcing equilaterality) and $2E-3F$ face (torsion) equations (enforcing planarity)—and of variables ( $3V-6$ ) are equal. Therefore, solutions of the equations up to equivalence generally leave no degrees of freedom. As a result, in general there is a finite (but very large) number of solutions. Unfortunately, even with state-of-the-art computer algebra, the resulting systems of equations are generally too complicated to completely solve within reasonable time. We therefore added an additional constraint, symmetry, specifically requiring solutions to display (at least) tetrahedral symmetry. We found 77 non-classical embeddings, seven without intersecting faces—two, four and one, respectively, for the (graphs of the) dodecahedron, the icosidodecahedron and the rhombicosidodecahedron. Full article