Archimedean Solids Research Articles

The signaling dimension of two-dimensional and polytopic systems

The signaling dimension of any given physical system represents its classical simulation cost, that is, the minimum dimension of a classical system capable of reproducing all the input/output correlations of the given system. The signaling dimension landscape is vastly unexplored; the only non-trivial systems whose signaling dimension is known -- other than quantum systems -- are the octahedron and the composition of two squares.}{Building on previous results by Matsumoto, Kimura, and Frenkel, our first result consists of deriving bounds on the signaling dimension of any system as a function of its Minkowski measure of asymmetry. We use such bounds to prove that the signaling dimension of any two-dimensional system (i.e. with two-dimensional set of admissible states, such as polygons and the real qubit) is two if and only if such a set is centrally symmetric, and three otherwise, thus conclusively settling the problem of the signaling dimension for such systems.} {Guided by the relevance of symmetries in the two dimensional case, we propose a branch and bound division-free algorithm for the exact computation of the symmetries of any given polytope, in polynomial time in the number of vertices and in factorial time in the dimension of the space. Our second result then consist of providing an algorithm for the exact computation of the signaling dimension of any given system, that outperforms previous proposals by exploiting the aforementioned bounds to improve its pruning techniques and incorporating as a subroutine the aforementioned symmetries-finding algorithm. We apply our algorithm to compute the exact value of the signaling dimension for all rational Platonic, Archimedean, and Catalan solids, and for the class of hyper-octahedral systems up to dimension five.

Archimedean solids in the fifteenth and sixteenth centuries

Several artists, artisans, and mathematicians described fascinating solid bodies in the fifteenth and sixteenth centuries. The knowledge they developed on the subject was so progressive that it is considered a milestone in the history of polyhedra. In the first part of this study we analyze, from a chronological and comparative perspective, the consistent studies developed between 1460 and 1583 on those that came to be recognized as Archimedean Solids. The authors who engaged in such studies were Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Augustin Hirschvogel, an Anonymous Author who accomplished remarkable studies between 1538 and 1556, Wentzel Jamnitzer, Daniele Barbaro, Lorenz Stöer, Rafael Bombelli, and Simon Stevin. In the second part, we discuss how the revolutionary method of describing solid bodies with planar nets contributed to the rediscovery of the Archimedean Solids. We also present our interpretation of some of the studies by the Anonymous Author and our conclusions on his identity and influence on other authors.

Overlapping edge unfoldings for convex regular-faced polyhedra

Herein, we discuss the existence of overlapping edge unfoldings for convex regular-faced polyhedra. Horiyama and Shoji showed that there are no overlapping edge unfoldings for all platonic solids and five of the Archimedean solids. The remaining five Archimedean solids were found to have edge unfoldings that overlap. In this study, we propose a method called rotational unfolding to find an overlapping edge unfolding of a polyhedron. We show that all the edge unfoldings of an icosidodecahedron, a rhombitruncated cuboctahedron, an n-gonal Archimedean prism (3≤n≤23), an m-gonal Archimedean antiprism (3≤m≤11), and 48 types of Johnson solids do not overlap. Our algorithm finds overlapping edge unfoldings for the snub cube, and 44 types of Johnson solids. We present analytic proof that an overlapping edge unfolding exists in an n-gonal Archimedean prism (n≥24), and an m-gonal Archimedean antiprism (m≥12). Our results prove the existence of overlapping edge unfoldings for convex regular-faced polyhedra.

Stiffness, strength, anisotropy, and buckling of lattices derived from TPMS and Platonic and Archimedean solids

Lattice metamaterials have gained considerable attention due to their distinctive topological structures and multifunctional properties. In this work, the effect of topology, loading conditions, and relative density on the effective mechanical properties of various novel lattice architectures is investigated numerically and experimentally. Thirteen strut-based lattices derived from triply periodic minimal surfaces (five lattices) as well as Platonic (three lattices) and Archimedean (five lattices) solids are considered for the first time, and their anisotropic mechanical properties, including uniaxial, shear, and bulk moduli and strengths as well as their total stiffness, buckling strengths, Poisson’s ratio, and anisotropy are investigated as a function of a wide range of relative densities (0.1% to 37%). Finite element analysis is employed to capture the full effective behavior of these lattices using periodic boundary conditions. Bifurcation analysis is performed to predict the threshold relative density governing their buckling vs yielding deformation behavior. Selected lattice structures of various relative densities are 3D printed using polymer selective laser sintering additive manufacturing technique and tested under quasi-static uniaxial compression where the experimental and numerical results are compared. The numerical results indicate that the deformation behavior can be altered between stretching and bending dominated mode of deformation as function of loading. Archimedean lattices are shown to outperform a wide range of strut-based lattices. This work opens the doors for more investigations of the multifunctional properties of these novel types of lattices and their engineering applications. Furthermore, the generated comprehensive data are useful in optimizing latticed structures using topology optimization techniques.

Cosmological domain walls from the breaking of S4 flavor symmetry

In this work, we delve into the often overlooked cosmological implications of the spontaneous breaking of the non-Abelian discrete groups, specifically focusing on the formation of domain walls in the case of S4 flavor symmetry. In particular, we investigate three interesting breaking patterns of the S4 group and study the structure of the domain walls in the broken phase for three possible residual symmetries. The presentation of domain walls in the case of multiple vacua is usually complicated, which therefore implies that most of the analyses only approximate their presentation. Here, we propose a subtle way to represent the S4 domain wall networks by presenting the vacua in each breaking pattern as vectors with their components corresponding to their coordinates in the flavon space. Then, through the properties of the obtained vectors, we find that the domain wall networks can be represented by Platonic or Archimedean solids where the vertices represent the degenerate vacua while the edges correspond to the domain walls that separate them. As an illustration, we propose a type-II seesaw model that is based on the S4 flavor symmetry, and study its phenomenological implications on the neutrino sector. To solve the domain wall problem within this toy model, we consider an approach based on high-dimensional effective operators induced by gravity that explicitly break the structure of the induced vacua favoring one vacuum over the others. Published by the American Physical Society 2024

The Group of Transformation Which Preserving Distance on Cuboctahedron and Truncated Octahedron Space

Minkowski geometry is a non-Euclidean geometry in a finite number of dimensions. In a Minkowski geometry the unit ball is a symmetric, convex closed set instead of the usual sphere in Euclidean space. In [14], it is shown that there are some geometries which unit spheres are cuboctahedron and truncated octahedron-which are Archimedean solids-, they are also Minkowski geometries. In geometry determining the group of isometries of a space with a metric is a fundamental problem. In this article we show that the group of isometries of the 3−dimensional spaces covered by CO − metric and TO − metric are the semi-direct product of Oh and T(3), where octahedral group Oh is the (Euclidean) symmetry group of the octahedron and T(3) is the group of all translations of the 3−dimensional space.

The Isometry Group of Tetrakis Hexahedron and Disdyakis Dodecahedron Spaces

Polyhedra are used in several fields by mathematicians and other scientists. It is easy to think of examples from architecture. Polyhedra have been used to scientifically explain the world around us. In the early days of study, polyhedra included only convex polyhedra. Since the ancient Greeks, many thinkers have worked on convex polyhedra. There are only five regular convex polyhedra known as Platonic solids, thirteen semi-regular convex polyhedra known as Archimedean solids, and thirteen irregular convex polyhedra which are duals of the Archimedean solids and known as Catalan solids. In this study, we show that the isometry group of the threedimensional analytic space formed by the metrics of the Tetrakis hexahedron and the Disdyakis dodecahedron is the semi-direct product of Oh and T(3), where the octahedral group Oh is the (Euclidean) symmetry group of the octahedron and T(3) is the group of all translations of the three-dimensional space.

Modular Construction of Topological Interlocking Blocks—An Algebraic Approach for Resource-Efficient Carbon-Reinforced Concrete Structures

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.

Competition between frustration and spin dimensionality in the classical antiferromagnetic $n$-vector model with arbitrary $n$

A new method to characterize the strength of magnetic frustration is proposed by calculating the minimum dimensionality of the absolute ground states of the classical nearest-neighbor antiferromagnetic nn-vector model with arbitrary nn. Platonic solids in three and four dimensions and Archimedean solids have lowest-energy configurations in a number of spin dimensions equal to their real-space dimensionality. Fullerene molecules and geodesic icosahedra can produce ground states in as many as five spin dimensions. Frustration is also characterized by the maximum value of the round-state energy when the exchange interactions are allowed to vary.

Platonic and Archimedean Solids As a Basis for the Structure of Self-Accommodation Complexes of Martensite Crystals in Alloys with Shape Memory Effects

Platonic and Archimedean Solids As a Basis for the Structure of Self-Accommodation Complexes of Martensite Crystals in Alloys with Shape Memory Effects

An algorithmic approach to Rupert’s problem

A polyhedron P ⊂ R 3 \mathbf {P} \subset \mathbb {R}^3 has Rupert’s property if a hole can be cut into it, such that a copy of P \mathbf {P} can pass through this hole. There are several works investigating this property for some specific polyhedra: for example, it is known that all 5 Platonic and 9 out of the 13 Archimedean solids admit Rupert’s property. A commonly believed conjecture states that every convex polyhedron is Rupert. We prove that Rupert’s problem is algorithmically decidable for polyhedra with algebraic coordinates. We also design a probabilistic algorithm which can efficiently prove that a given polyhedron is Rupert. Using this algorithm we not only confirm this property for the known Platonic and Archimedean solids, but also prove it for one of the remaining Archimedean polyhedra and many others. Moreover, we significantly improve on almost all known Nieuwland numbers and finally conjecture, based on statistical evidence, that the Rhombicosidodecahedron is in fact not Rupert.

Platonic and Archimedean solids in discrete metal-containing clusters.

Metal-containing clusters have attracted increasing attention over the past 2-3 decades. This intense interest can be attributed to the fact that these discrete metal aggregates, whose atomically precise structures are resolved by single-crystal X-ray diffraction (SCXRD), often possess intriguing geometrical features (high symmetry, aesthetically pleasing shapes and architectures) and fascinating physical properties, providing invaluable opportunities for the intersection of different disciplines including chemistry, physics, mathematical geometry and materials science. In this review, we attempt to reinterpret and connect these fascinating clusters from the perspective of Platonic and Archimedean solid characteristics, focusing on highly symmetrical and complex metal-containing (metal = Al, Ti, V, Mo, W, U, Mn, Fe, Co, Ni, Pd, Pt, Cu, Ag, Au, lanthanoids (Ln), and actinoids) high-nuclearity clusters, including metal-oxo/hydroxide/chalcogenide clusters and metal clusters (with metal-metal binding) protected by surface organic ligands, such as thiolate, phosphine, alkynyl, carbonyl and nitrogen/oxygen donor ligands. Furthermore, we present the symmetrical beauty of metal cluster structures and the geometrical similarity of different types of clusters and provide a large number of examples to show how to accurately describe the metal clusters from the perspective of highly symmetrical polyhedra. Finally, knowledge and further insights into the design and synthesis of unknown metal clusters are put forward by summarizing these "star" molecules.

Deployable polyhedral mechanisms with radially reciprocating motion based on novel basic units and an additive-then-subtractive design strategy

This paper is to deal with the mechanism design problem for constructing deployable polyhedral mechanisms on the basis of novel basic units with an additive-then-subtractive design strategy. The aimed deployable polyhedral mechanisms can be, but are not limited to, Platonic and Archimedean solids. The basic units involved in the type synthesis process refer to one-degree-of-freedom regular polygonal prisms, whose deployable performance has been already proven to be excellent. To show the procedure of the type synthesis method, the case study on a deployable regular-tetrahedron mechanism is presented as a design example. The mobility properties of the proposed deployable regular-tetrahedron mechanism are then confirmed with the combination of screw theory and line geometry. In addition, the kinematic model is presented to study the deployable performance of deployable polyhedral mechanisms. By adopting the proposed type synthesis method, novel deployable Platonic and Archimedean polyhedral mechanisms are proposed with wider-margin radially reciprocating motion. This work not only provides the design and analysis method but also contributes new types of mechanisms, hoping to have a reference for the research of deployable polyhedral mechanisms.

Water-stable porous Al24 Archimedean solids for removal of trace iodine

In this paper, we report a unique type of core-shell crystalline material that combines an inorganic zeolitic cage structure with a macrocyclic host arrangement and that can remove trace levels of iodine from water effectively. These unique assemblies are made up of an inorganic Archimedean truncatedhexahedron (tcu) polyhedron in the kernel which possesses six calixarene-like shell cavities. The cages have good adaptability to guests and can be assembled into a series of supramolecular structures in the crystalline state with different lattice pore shapes. Due to the unique core-shell porous structures, the compounds are not only stable in organic solvents but also in water. The characteristics of the cages enable rapid iodine capture from low concentration aqueous I2/KI solutions (down to 4 ppm concentration). We have studied the detailed process and mechanism of iodine capture and aggregation at the molecular level. The facile synthesis, considerable adsorption capacity, recyclability, and β- and γ-radiation resistance of the cages should make these materials suitable for the extraction of iodine from aqueous effluent streams (most obviously, radioactive iodide produced by atomic power generation).

Two-point resistances in Archimedean resistor networks

The rise of novel materials such as graphene compounds or carbon nanotubes recently revived the interest on the role of symmetries upon electrical properties. Among possible solids, symmetrical polytopes, widely realized in natural and artificial matter, deserve particular attention. When equal resistors are attached between neighbouring nodes of Perfect (Platonic) solids (PS) along the edges, the two-point resistance between any pair of nodes was established in the 1990s by van Steenwijk using an elegant and efficient method. Using van Steenwijk’s method, Moddy and Aravind subsequently derived the resistance values for two Archimedean and two Catalan solids. Here, with the same method, we derive the exact expression for the two-point resistance between any two nodes for resistor networks made of equal resistors placed along the edges of the thirteen Archimedean solids (AS). While the calculation remains elementary for the simplest AS, a dedicated method using computer assistance was developed for the Snub Cube, the Snub Dodecahedron and the three AS with three different rotation symmetries (the Great Rhombicuboctahedron, the Small Rhombicosidodecahedron and the Great Rhombicosidodecahedron). The largest resistance value (191/90) is obtained between the two opposite nodes of the Truncated Dodecahedron, and the second largest (42815/21114) is obtained between the two opposite nodes of the Great Rhombicosidodecahedron. The smallest resistance value (14137/38016) is obtained between some of the neighbouring nodes of the Snub Cube, whereas the resistance between neighbouring nodes of the Icosahedron (11/30) is slightly smaller. Some general symmetry relations between two-point resistances in AS networks are also derived, as well as some relations with two-point resistances in PS networks. The complete set of exact two-point resistance values for PS and AS networks can be used to check numerical codes or to evaluate the capacity of regular solids to provide appropriate models for given experimental situations.

Optimal Finite Homogeneous Sphere Approximation

The two-dimensional sphere can’t be approximated by finite homogeneous spaces. We describe the optimal approximation and its distance from the sphere. We compare this distance to the distance achieved by all Platonic and Archimedean solids.

Polyhedra, Tiles and Graphene Defects: The Case of Tetraoctite

: This mini review concerns hydrocarbons, (CH)n, and carbon allotropes, Cn, and their relationships with regular solids and regular surfaces, respectively. Platonic and Archimedean solids and surfaces related to carbon allotropes are described as an introduction. An overview is then provided on how Stone-Wales defects lead to a series of structures: pentaheptite, Haeckelite, net C, net W, planar C4, biphenylene, graphyne, graphdiyne, and tetraoctite. This last compound is discussed in detail together with its relation to the Mills-Nixon effect on cyclooctatetraene (COT).

Terpyridine-Based 3D Discrete Metallosupramolecular Architectures.

Terpyridine (tpy)-based 3D discrete metallosupramolecular architectures, which are often inspired by polyhedral geometry and the biological structures found in nature, have drawn significant attention from the community of metallosupramolecular chemistry. Because of the linear tpy-M(II)-tpy connectivity, the creation of sophisticated 3D metallosupramolecules based on tpy remains a formidable synthetic challenge. Nevertheless, with recent advancement in ligand design and self-assembly, diverse 3D metallosupramolecular polyhedrons, such as Platonic solids, Archimedean solids, prims as well as Johnson solids, have been constructed and their potential applications have been explored. This review summarizes the progress on tpy-based discrete 3D metallosupramolecules, aiming to shed more light on the design and construction of novel discrete architectures with molecular-level precision through coordination-driven self-assembly.

A Multicenter Metal–Organic Framework for Quantitative Detection of Multicomponent Organic Mixtures

A Multicenter Metal–Organic Framework for Quantitative Detection of Multicomponent Organic Mixtures

Packing core-corona particles on a spherical surface.

We explore the non-trivial structures that can be obtained by the assembly of repulsive core-corona particles confined on a spherical surface. Using Monte Carlo simulations, we study the low-temperature equilibrium configurations as a function of the size of the confining (spherical) surface for a small number of particles (N ≤ 12) and obtain a large variety of minimal-energy arrangements including anisotropic and chiral structures. For a small cluster (N = 4), we construct a phase diagram in the confining surface radius vs corona range plane that showed regions where configurations with a certain energy are not accessible. Also, a phase diagram in the temperature and confining surface radius plane showed the presence of reentrant phases. The assembly of Platonic and Archimedean solids and the emergence of helical structures are also discussed. When the number of particles is large (N ≥ 100), apart from the appearance of defects, the overall configurations correspond closely to the ones formed in an unconfined two-dimensional case. Interestingly, the present model reproduces the symmetry of experimentally obtained small clusters of colloidal spheres confined at the surface of evaporating liquid droplets which cannot be explained in terms of packing of hard spheres. Thus, our simulations provide insight on the role that the softness of the particles may have in the assembly of clusters of nanoparticles.