Search Results (23)

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

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

The goal of this paper is to shed light on the nature of mathematical practice, i.e., on “doing mathematics”. It explores Gödel’s perspective, which offers an approach to understanding mathematics centered on concepts, objects, and structures. The paper has two parts. In the first part, we situate Gödel’s reflections against the backdrop of formalism and Platonism. In the second part, we present the view shaped by Gödel’s ideas that resonates with contemporary discussions in the philosophy of mathematical practice, particularly in its attention to abstraction, generalization, and conceptual discovery, as essential components of mathematical reasoning. We illustrate this view through concrete examples from category theory and geometry. This approach reveals that mathematical practice, far from being merely formal, is a dynamic interplay of intuition, abstraction, structural, and conceptual reasoning. Such a focus underscores the need for developing the theory of concepts along the lines proposed by Gödel to provide a more natural framework for thinking about mathematics. Full article

While Kepler is regarded as a major figure in standard historical accounts of the scientific revolution of early modern Europe, he is typically seen as having one foot in the new scientific culture and one in the old. In some of his work, Kepler appears, along with Galileo, to be on a trajectory towards Newton’s celestial mechanics. In addition to his advocacy of Copernicus’s heliocentrism, he appealed to physical causes in his explanations of the movements of celestial bodies. But other work appears to express a neo-Platonic “metaphysics” or “mysticism”, as most obvious in his embrace of the ancient tradition of the “music of the spheres”. Here I problematize this distinction. The musical features of Kepler’s purported neo-Platonic “metaphysics”, I argue, was also tied to Platonic and neo-Platonic features of the methodology of a tradition of mathematical astronomy that would remain largely untouched by his shift to heliocentrism and that would be essential to his actual scientific practice. Importantly, certain features of the geometric practices he inherited—ones later formalized as “projective geometry”—would also carry those “harmonic” structures expressed in the thesis of the music of the spheres. 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

It is known that the volume of a convex polyhedron can be increased by suitable isometric deformation of its surface resulting in a non-convex shape. Deformation patterns and the associated enclosed volumes of the Platonic polyhedra were theoretically and numerically investigated by a few authors in the past. In this paper, a generic diamond-shaped folding pattern for all Platonic polyhedra is presented, optimised to achieve the maximum enclosed volumes. The numerically obtained volume increases (44.70%, 25.12%, 13.86%, 10.61%, and 4.36% for the regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron, respectively) improve the existing results (44.00%, 24.62%, 13.58%, 9.72%, and 4.27%, respectively). Quasi-rigid inflatable membrane representations of such deformed polyhedra imply a significant change of structural shape due to initial inflation and subsequent compressive stresses transverse to the crease lines. Full article

The depth and extent of Ficino’s reception and use of Proclus has already attracted much scholarly attention. The present paper builds on and tries to enrich these results, focusing specifically on Ficino’s reception of Proclus’ Elements of Physics and Elements of Theology. In the first part I discuss a marginal annotation of Ficino, in which he makes use of arguments about the circular motion of the soul from the Elements of Physics. I provide some clarifications about the annotated text (of Plotinus) and propose one additional possible echo of the Elements of Physics in Ficino’s Platonic Theology and its arguments about the immortality of the soul. The second part of the paper turns to the link between the Elements of Theology and Ficino’s Platonic Theology. Together with some further doctrinal borrowings I suggest that also the structure of the two works bears important affinities. The soul is a central case in point. To ground this claim, I compare specific sections of the two texts. Also, I selectively examine Ficino’s commentary on the Philebus, which is prior to the Platonic Theology and is strongly influenced by the early theorems of the Elements of Theology. Overall, the paper wishes to shed further light on Ficino’s multiform (and not yet fully unveiled) appropriation of Proclus. Full article

The three isomers of the tetraoxa[4]arene derivative, C₂₄H₁₆O₄, which consist of two m-phenylenes and two phenylenes (meta 1, para 2, ortho 3), represent not only intriguing fundamental structures that induce molecular recognition toward non-porous adaptive crystals, but also attractive candidates for crystallographic polymorphism. In this study, we crystallized isomers 2 and 3, in comparison to isomer 1, in order to understand their stable orientations and the corresponding intermolecular interactions in the crystalline state. For example, m-phenylene derivative 1 exhibits polymorphism with both prismatic and block-shaped crystals. Therefore, we prepared p-phenylene derivative 2 and o-phenylene derivative 3, and their structures were fully characterized by SC-XRD, revealing two polymorphs of derivative 2, namely prismatic crystal 2-I and block-shaped crystal 2-II, along with changes to the crystal lattice parameters (2-Ia, 2-Ib, and 2-Ic) based on temperature dependence. In all of its crystal forms, derivative 2 adopts an O-shaped planar structure, where the p-phenylene units face each other. This suggests that the packing mode during the early stages of crystallization, rather than due to any remarkable changes in the molecular structure, directly affects the bulk crystal morphology. On the other hand, derivative 3 adopts a U-shaped vent structure and, to the best of our knowledge, does not form polymorphs. The Platon and Hirshfeld surface analyses indicated that the contributions to the crystal packing were C···C (av. 37.3% for 2-Ia, av. 38.2% for 2-II, and 18.7% for 3), C···H/H···C (av. 37.3% for 2-Ia, av. 38.2% for 2-II, and 18.7% for 3), and O···H/H···O (av. 17.8% for 2-Ia, av. 19.6% for 2-II, and 19.4% for 3), highlighting significant intermolecular CH···π interactions and pseudo-hydrogen bonding forms for derivative 2 and π···π interactions for derivative 3. 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

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

Augustine’s sermons on the Psalms of Ascent, part of the Enarrationes in Psalmos, are a unique entry in the venerable tradition of those writings that aim to help us ascend to a higher reality. These sermons transform the ascent genre by giving, in the place of the Platonic account of ascent, a Christian ascent narrative with a Trinitarian structure. Not just the individual ascends, but the community that is the church, the body of Christ, also ascends. The ascent is up to God, the Idipsum or the Selfsame, the ultimate reality, confessed by the church as God the Father, God the Son, and God the Holy Spirit. Through the grace of the Incarnation, God the Son enables us to ascend, making himself the way of ascent from the humility we must imitate at the beginning of the ascent all the way up to Heaven, where he retains his identity as Idipsum. Meanwhile, the Holy Spirit works in the ascending church to convert our hearts to the love of God and neighbor. I review the Platonic ascent tradition in Plato’s Republic and Plotinus’ Enneads; overview ascent in some of Augustine’s earlier writings; introduce the narrative setting of the sermons on the Psalms of Ascent; and analyze the Trinitarian structure of their ascent narrative. I close with some reflections on the difference between a preached Trinitarianism that encourages ascent and a more academic effort to understand God such as we find in Augustine’s de Trinitate. 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

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