Search Results (25)

Vibration band gap structures are advanced materials for vibration wave mitigation from metamaterials to phononic crystals from simple geometrical manipulations. Here, we present geometrical structures, made from platonic solids, that are capable of providing multi-passband frequency ranges with face symmetry in each unit cell. We fabricated the metamaterial structures using stereolithography, after which we experimentally characterized band gaps through impulse vibration testing. Experimental results have shown that the band gaps can be changed for different types of platonic structures along with the loading direction. This provided a comparison between axial and two bending direction band gaps, revealing ranges where the structures behave in either a “fluid-like” or an “optical-like” manner. Dodecahedron unit cells have exhibited the most promising results, when compared with reduced relative densities and a number of stacking unit cells. We utilized the coherence function during signal processing analysis, which provided strong predictions for the band gap frequency ranges. Full article

This manuscript examines the rotational dynamics of rigid bodies in five-dimensional Euclidean space. This results in ten coupled nonlinear differential equations for angular velocities. Restricting rotations along certain axes reduces the 5D equations to sets of 4D Euler equations, which collapse to the classical 3D Euler equations. This demonstrates consistency with established mechanics. For bodies with equal principal moments of inertia (e.g., hyperspheres and Platonic solids), the rotation velocities remain constant over time. In cases with six equal and four distinct inertia moments, the solutions exhibit harmonic oscillations with frequencies determined by the initial conditions. Rotations are stable when the body spins around an axis with the largest or smallest principal moment of inertia, thus extending classical stability criteria into higher dimensions. This study defines a 5D angular momentum operator and derives commutation relations, thereby generalizing the familiar 3D and 4D cases. Additionally, it discusses the role of Pauli matrices in 5D and the implications for spin as an intrinsic property. While mathematically consistent, the hypothesis of a fifth spatial dimension is ultimately rejected since it contradicts experimental evidence. This work is valuable mainly as a theoretical framework for understanding spin and symmetry. This paper extends Euler’s equations to five dimensions (5D), demonstrates their reduction to four dimensions (4D) and three dimensions (3D), provides closed-form and oscillatory solutions under specific inertia conditions, analyzes stability, and explores quantum mechanical implications. Ultimately, it concludes that 5D space is not physically viable. Full article

Structures composed of classical dipoles in higher-dimensional space present a unique opportunity to venture beyond the conventional paradigm of few-body or cluster physics. In this work, we consider the six convex regular polychora that exist in an Euclidean four-dimensional space as a theoretical benchmark for hte investigation of dipolar systems in higher dimensions. The structures under consideration represent the four-dimensional counterparts of the well-known Platonic solids in three-dimensions. A dipole is placed in each vertex of the structure and is allowed to interact with the rest of the system via the usual dipole–dipole interaction generalized to the higher dimension. We use numerical tools to minimize the total interaction energy of the systems and observe that all six structures represent dipole clusters with a zero net dipole moment. The minimum energy is achieved for dipoles arranging themselves with orientations whose angles are commensurate or irrational fractions of the number $\pi$. Full article

Shapes and topologies of lattice materials have been extensively studied, yet very few studies have dealt with shapes inspired by ancient mathematicians, such as the Platonic solids discovered by Plato in 360 BC or the mathematical behavior of the unexplored “semi-regular” solids of Pacioli (1445–1517). Using the finite element analysis method, the buckling and post-buckling behavior of Platonic and Paciolian cells subjected to a compressive load were analyzed. In these solids, the energy absorbed per unit mass is an increasing function with the number of faces, similar to porosity, which reaches a maximum value for solids comprised of 90–100 surfaces. 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

The packing and assembly of Platonic solids have fascinated mathematicians for ages. Recently, this fundamental geometrical problem has also attracted the attention of physicists, chemists, and engineers. This growing interest is due to the rapid advancements in various related fields, ranging from the formation of colloidal crystals and the design of metal–organic frameworks to the development of ultra-lightweight metamaterials, which are closely tied to the fast-evolving 3D printing technology. Numerous reports have focused on the assembly of Platonic polyhedra, particularly tetrahedra, for which an optimal packing strategy remains unidentified to this day. However, less attention has been given to the dodecahedron and its networks. This work introduces a new type of framework, designed from regular dodecahedra combined with icosahedron-based binders. The relatively simple design protocol employed here results in a remarkable variety of intriguing networks, which could be potentially useful in fields such as architecture, regenerative medicine, or aeronautics. Additionally, the dodecahedral networks presented in this study led to the discovery of intriguing structures resembling distorted graphene sheets. These structures exhibit features characteristic of both graphene and diamond. Full article

Recent advancements in geospatial technologies have significantly expanded the volume and diversity of geospatial data, unlocking new and innovative applications that require novel Geographic Information Systems (GIS). (Discrete) Global Grid Systems (DGGSs) have emerged as a promising solution to further enhance modern geospatial capabilities. Current DGGSs employ a simple, low-resolution polyhedral approximation of the Earth for efficient operations, but require a projection between the Earth’s surface and the polyhedral faces. Equal-area DGGSs are desirable for their low distortion, but they fall short of this promise due to the inefficiency of equal-area projections. On the other hand, efficiency-first DGGSs need to better address distortion. We introduce a novel mesh-based DGGS (MBD) which generalizes efficient operations over watertight triangular meshes with spherical topology. Unlike traditional approaches that rely on Platonic or Catalan solids, our mesh-based method leverages high-resolution spherical meshes to offer greater flexibility and accuracy. MBD allows high-resolution polyhedra (HRP) to be used as the base polyhedron of a DGGS, significantly reducing distortion. To address the operational challenges, we introduce a new hash encoding method and an efficient barycentric indexing method (BIM). MBD extends Atlas of Connectivity Maps to the BIM to provide efficient spatial and hierarchical traversal. We introduce several new base polyhedra with lower areal and angular distortion, and we experimentally validate their properties and demonstrate their efficiency. Our experimentation shows that we achieve constant-time operations for high-resolution MBD, and we recommend polyhedra to be used as the base polyhedron for low-distortion DGGSs, compact faces, and efficient operations. 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

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

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

This paper proposes a new approach to relate the effective thermal conductivity of open-cell solid foams to their porosity. It is based on a recently published approach for estimating the dielectric permittivity of isotropic porous media. A comprehensive assessment was performed comparing the proposed mixing relation with published experimental data for thermal conductivity and with numerical data from state-of-the-art relations. The mixing relation for the estimation of thermal conductivities based on dodecahedrons as building blocks shows good agreement with experimental data over a wide range of porosity. Full article

This paper focuses on the dynamics of the eight tridimensional principal slices of the tricomplex Mandelbrot set: the Tetrabrot, the Arrowheadbrot, the Mousebrot, the Turtlebrot, the Hourglassbrot, the Metabrot, the Airbrot (octahedron), and the Firebrot (tetrahedron). In particular, we establish a geometrical classification of these 3D slices using the properties of some specific sets that correspond to projections of the bicomplex Mandelbrot set on various two-dimensional vector subspaces, and we prove that the Firebrot is a regular tetrahedron. Finally, we construct the so-called “Stella octangula” as a tricomplex dynamical system composed of the union of the Firebrot and its dual, and after defining the idempotent 3D slices of ${\mathcal{M}}_3$, we show that one of them corresponds to a third Platonic solid: the cube. Full article

Open-cell solid foams are rigid skeletons that are permeable to fluids, and they are used as direct heaters or thermal dissipaters in many industrial applications. Using susceptors, such as dielectric materials, for the skeleton and exposing them to microwaves is an efficient way of heating them. The heating performance depends on the permittivity of the skeleton. However, generating a rigorous description of the effective permittivity is challenging and requires an appropriate consideration of the complex skeletal foam morphology. In this study, we propose that Platonic solids act as building elements of the open-cell skeletal structures, which explains their effective permittivity. The new, simplistic geometrical relation thus derived is used along with electromagnetic wave propagation calculations of models that represent real foams to obtain a geometrical, parameter-free relation, which is based only on foam porosity and the material’s permittivity. The derived relation facilitates an efficient and reliable estimation of the effective permittivity of open-cell foams over a large range of porosity. Full article