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The article is devoted to modeling and visualization of the formation of flat-nosed (snub-nosed) dodecahedron (snub dodecahedron). The purpose of the research is to model the snub dodecahedron (flat-nosed dodecahedron) and visualize the process of its formation. The formation of the faces of the flat-nosed dodecahedron consists in the truncation of the edges and vertices of the Platonic dodecahedron with the subsequent rotation of the new faces around their centers. The values of the truncation of the dodecahedron edges, the angle of rotation of the faces and the length of the edge of the flat-nosed dodecahedron are the parameters of three equations composed as the distances between the vertices of triangles located between the faces of the snub dodecahedron. The solution of these equations was carried out by the method of successive approximations. The results of the calculations were used to create an electronic model of the flat-nosed dodecahedron and visualize its formation. The task was generally achieved in the AutoCAD system using programs in the AutoLISP language. Software has been created for calculating the parameters of modeling a snub dodecahedron and visualizing its formation.

Polyhedra are ubiquitous in chemistry, biology, mathematics and other disciplines. Coordination-driven self-assembly has created molecules mimicking Platonic, Archimedean and even Goldberg polyhedra, however, nesting multiple polyhedra in one cluster is challenging, not only for synthesis but also for determining the alignment of the polyhedra. Here, we synthesize a nested Ag90 nanocluster under solvothermal condition. This pseudo-Th symmetric Ag90 ball contains three concentric Ag polyhedra with apparently incompatible symmetry. Specifically, the inner (Ag6) and middle (Ag24) shells are octahedral (Oh), an octahedron (a Platonic solid with six 3.3.3.3 vertices) and a truncated octahedron (an Archimedean solid with twenty-four 4.6.6 vertices), whereas the outer (Ag60) shell is icosahedral (Ih), a rhombicosidodecahedron (an Archimedean solid with sixty 3.4.5.4 vertices). The Ag90 nanocluster solves the apparent incompatibility with the most symmetric arrangement of 2- and 3-fold rotational axes, similar to the arrangement in the model called Kepler’s Kosmos, devised by the mathematician John Conway.

The theory of convex sets is a vibrant and classicalfield of modern mathematics with rich applications. The more geometric aspectsof convex sets are developed introducing some notions, but primarily polyhedra.A polyhedra, when it is convex, is an extremely important special solid in . Some examples of convex subsets of Euclidean3-dimensional space are Platonic Solids, Archimedean Solids and ArchimedeanDuals or Catalan Solids. There are some relations between metrics andpolyhedra. For example, it has been shown that cube, octahedron, deltoidalicositetrahedron are maximum, taxicab, Chinese Checker’s unit sphere,respectively. In this study, we give two new metrics to be their spherestruncated truncated dodecahedron and truncated truncated icosahedron., Konveks kümeler teorisi, zengin uygulamalara sahipmodern matematiğin canlı ve klasik bir alanıdır. Konveks kümelerin geometrikyönleri, bazı kavramlar, fakat öncelikle çokyüzlülerin tanıtılmasıylageliştirilmiştir. Konveks olduğunda bir çokyüzlü, de çok önemlibir özel cisimdir. Öklid 3 boyutlu uzayın konveks alt kümelerinin bazıörnekleri Platonik cisimler, Arşimet cisimleri ve Arşimet dualleri veya Katalancisimleridir. Metriklerle çokyüzlüler arasında bazı ilişkiler vardır. Örneğin,küp, sekizyüzlü, deltoidal icositetrahedron'un sırasıyla, maksimum, Taksi, Çindama uzaylarının birim küresi olduğu görülmektedir. Bu çalışmada, kürelerinintruncated truncated dodecahedron ve truncated truncated icosahedron olan ikiyeni metrik tanıtıldı.

The construction of Bell inequalities based on Platonic and Archimedean solids (Quantum 4 (2020), 293) is generalized to the case of orbits generated by the action of some finite groups. A number of examples with considerable violation of Bell inequalities is presented., 15 pages, 6 figures, modified version as suggested by the reviewers, accepted in Quantum 2021-11-18, Licence statement "arXiv.org perpetual, non-exclusive license" changed to CC-BY 4.0 as demanded by Quantum

Inspired by the structures of virus capsids, chemists have long pursued the synthesis of their artificial molecular counterparts through self–assembly. Building nanoscale hierarchical structures to simulate double-shell virus capsids is believed to be a daunting challenge in supramolecular chemistry. Here, we report a double-shell cage wherein two independent metal–organic polyhedra featuring Platonic and Archimedean solids are nested together. The inner (3.2 nm) and outer (3.3 nm) shells do not follow the traditional “small vs. large” pattern, but are basically of the same size. Furthermore, the assembly of the inner and outer shells is based on supramolecular recognition, a behavior analogous to the assembly principle found in double-shell viruses. These two unique nested characteristics provide a new model for Matryoshka–type assemblies. The inner cage can be isolated individually and proves to be a potential molecular receptor to selectively trap guest molecules., Supramolecular constructs that mimic complex biological assemblies are synthetically challenging. Here, the authors present a double-shell cage wherein two independent metal-organic polyhedra are nested together in a manner analogous to that found in double-shell virus capsids.

There are two main motivations in this article. First,we give the new metrics and the metric spaces whose unit spheres are Rectified Archimedean Solids. Then, using the general technique which is quite simple, we show that isometry group of $\mathbb{R}^{3}$ endowed with these new metrics are the semi direct product of the translation group $T(3)$ of $\mathbb{R}^{3}\ $ with the Euclidean symmetry groups of Rectified Archimedean Solids. .

A vertex-transitive map $X$ is a map on a surface on which the automorphism group of $X$ acts transitively on the set of vertices of $X$. If the face-cycles at all the vertices in a map are of same type then the map is called a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse of this is not true in general. In particular, there are semi-equivelar maps on the torus, on the Klein bottle and on the surfaces of Euler characteristics $-1$ $\&$ $-2$ which are not vertex-transitive. It is known that the boundaries of Platonic solids, Archimedean solids, regular prisms and antiprisms are vertex-transitive maps on $\mathbb{S}^2$. Here we show that there is exactly one semi-equivelar map on $\mathbb{S}^2$ which is not vertex-transitive. More precisely, we show that a semi-equivelar map on $\mathbb{S}^2$ is the boundary of a Platonic solid, an Archimedean solid, a regular prism, an antiprism or the pseudorhombicuboctahedron. As a consequence, we show that all the semi-equivelar maps on $\mathbb{RP}^2$ are vertex-transitive. Moreover, every semi-equivelar map on $\mathbb{S}^2$ can be geometrized, i.e., every semi-equivelar map on $\mathbb{S}^2$ is isomorphic to a semi-regular tiling of $\mathbb{S}^2$. In the course of the proof of our main result, we present a combinatorial characterization in terms of an inequality of all the types of semi-equivelar maps on $\mathbb{S}^2$. Here, we present self-contained combinatorial proofs of all our results., Final version. To appear in `Discrete mathematics'

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&mdash, thus &ldquo, EP&rdquo, &mdash, and display icosahedral symmetry. However, the latter two (star polyhedra) are non-convex and &ldquo, pathological&rdquo, 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&mdash, E edge length equations (enforcing equilaterality) and 2 E &minus, 3 F face (torsion) equations (enforcing planarity)&mdash, and of variables ( 3 V &minus, 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&mdash, two, four and one, respectively, for the (graphs of the) dodecahedron, the icosidodecahedron and the rhombicosidodecahedron.

We study the dimension of graphs of the Archimedean solids. For most of these graphs we find the exact value of their dimension by finding unit-distance embeddings in the euclidean plane or by proving that such an embedding is not possible.

The paper analyzes the possibility of obtaining polyhedral shapes formed by combining polyhedral surfaces based on the segment surface of elongated concave pyramids of the second sort (CeP II-10, type A and type B). In previous research, CP II type A and CP II type B were elaborated in detail. In this paper we discuss further potential of these polyhedral surfaces, on the example of combining them with Archimedean solid - Truncated dodecahedron (U26). The faces of this solid consist of 12 decagons and 20 triangles. On the decagonal faces, decagonal polygons of the CeP II segments (CP II-10) can be added, which provides the new polyhedral composite forms that are, furthermore, concave deltahedra. There are considered possibilities of obtaining polyhedral shapes by combining sheet segments CP II-10-A, as well as of CP II-10-B with U26. Finally, a couple of new shape suggestions are given: compound polyhedra, obtained by intersection of paired composite concave polyhedra originated in the described manner. [https://www.mas.bg.ac.rs/_media/istrazivanje/fme/vol45/2/contents_fme_vol_45_no_2.pdf] [https://www.mas.bg.ac.rs/istrazivanje/fme/start]

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Grothendieck's dessins d'enfants arise with ever-increasing frequency in many areas of 21st century mathematical physics. In this paper, we review the connections between dessins and the theory of Hecke groups. Focussing on the restricted class of highly symmetric dessins corresponding to the so-called Archimedean solids, we apply this theory in order to provide a means of computing representatives of the associated conjugacy classes of Hecke subgroups in each case. The aim of this paper is to demonstrate that dessins arising in mathematical physics can point to new and hitherto unexpected directions for further research. In addition, given the particular ubiquity of many of the dessins corresponding to the Archimedean solids, the hope is that the computational results of this paper will prove useful in the further study of these objects in mathematical physics contexts., Comment: 28 pages, 1 figure. v2: Substantial streamlining. v3: Minor changes; matches published version, Frontiers in Physics 3:91, 2015

The use of the novel pro-ligand H4L combining the complimentary phenolic oxime and diethanolamine moieties in one organic framework, results in the formation of the first example of a [MnIII 12] truncated tetrahedron and an extremely rare example of a Mn cage conforming to an Archimedean solid.

First we prove that the class $C_{I}$ of centrally symmetric convex polyhedra with regular polygonal faces consists of 4 of the 5 Platonic, 9 of the 13 Archimedean, 13 of the 92 Johnson solids and two infinite families of $2n$-prisms and $(2n+1)$-antiprisms. Then we show how the presented maps of their halves (obtained by identification of all pairs of antipodal points) in the projective plane can be used for obtaining their flag graphs and symmetry-type graphs. Finally, we study some linear dependence relations between polyhedra of the class $C_{I}$.

We formulate the problem of generating dense packings of nonoverlapping, non-tiling polyhedra within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem, which we call the Adaptive Shrinking Cell (ASC) scheme. This novel optimization problem is solved here (using a variety of multi-particle initial configurations) to find the dense packings of each of the Platonic solids in three-dimensional Euclidean space R3 , except for the cube, which is the only Platonic solid that tiles space. We find the densest known packings of tetrahedra, icosahedra, dodecahedra, and octahedra with densities 0.823, 0.836, 0.904, and 0.947, respectively. It is noteworthy that the densest tetrahedral packing possesses no long-range order. Unlike the densest tetrahedral packing, which must not be a Bravais lattice packing, the densest packings of the other non-tiling Platonic solids that we obtain are their previously known optimal (Bravais) lattice packings. We also derive a simple upper bound on the maximal density of packings of congruent nonspherical particles, and apply it to Platonic solids, Archimedean solids, superballs and ellipsoids. Our simulation results, rigorous upper bounds, and other theoretical arguments lead us to the conjecture that the densest packings of Platonic and Archimedean solids with central symmetry are given by their corresponding densest lattice packings. In addition, we conjecture that the optimal packing of any convex, congruent polyhedron without central symmetry generally is not a lattice packing., To appear in Phys. Rev. E. The paper is a sequel of a recent Nature paper, i.e., Nature 460, 876 (2009). In this paper, we reported a new packing of tetrahedra with higher density 0.823 than that reported in Nature (0.782). The paper is 52 pages, including 14 figures and 4 tables

A dissection of a convex d-polytope is a partition of the polytope into d-simplices whose vertices are among the vertices of the polytope. Triangulations are dissections that have the additional property that the set of all its simplices forms a simplicial complex. The size of a dissection is the number of d-simplices it contains. This paper compares triangulations of maximal size with dissections of maximal size. We also exhibit lower and upper bounds for the size of dissections of a 3-polytope and analyze extremal size triangulations for specific non-simplicial polytopes: prisms, antiprisms, Archimedean solids, and combinatorial d-cubes., 19 pages

A family of nanoscale-sized supramolecular cage compounds with a polyhedral framework is prepared by self-assembly from tritopic building blocks and rectangular corner units via noncovalent coordination interactions. These highly symmetrical cage compounds are described as face-directed, self-assembled truncated tetrahedra with Tdsymmetry.

The notion of a Belyi function is a main technical tool which relates the combinatorics of maps (i.e., graphs embedded into surfaces) to Galois theory, algebraic number theory, and the theory of Riemann surfaces. In this paper we compute Belyi functions for a class of semi-regular maps which correspond to the so-called Archimedean solids.

This account covers template effects that lead to the formation of polynuclear coordination compounds, with a focus on how the interplay of the structure-directing effects of multiple agents (including inorganic anions and organic ligands) combine to influence the geometry and nuclearity of metal-containing assemblies. Simple anions and ligands impart defined geometric shapes (triangles, squares, hexagons) onto oligonuclear assemblies, and these shapes can combine to produce a range of more sophisticated metallo-supramolecular architectures (including Platonic and Archimedean solids). An understanding of the structure-directing effects of simple templates is therefore necessary to predict or rationalise such polynuclear architectures, which may have wide-ranging applications in catalysis, in magnetism, as sensors, etc. This article primarily focuses on halide-templated assemblies of first row transition metal ions (particularly manganese coordination clusters and vanadates), but the principles discussed herein are applicable to a much wider range of coordination compounds.

We study and classify regular and semi-regular tessellations of Riemann surfaces of various genera and investigate their corresponding supersymmetric gauge theories. These tessellations are generalizations of brane tilings, or bipartite graphs on the torus as well as the Platonic and Archimedean solids on the sphere. On higher genus they give rise to intricate patterns. Special attention will be paid to the master space and the moduli space of vacua of the gauge theory and to how their geometry is determined by the tessellations., 89 pages, 43 figures; v2: references and comments added, typos fixed