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Polynuclear metal clusters, particularly those with aesthetically appealing structures, are of increasing interest. In this study, by employing C4 symmetrical macrocyclic silsesquioxane ligand, a novel tridecanuclear Gd(III) cluster (Gd13) with a {Gd13O8} core featuring unprecedent body-centered Archimedean cuboctahedron conformation is realized using an efficient solvothermal method instead of conventional Schlenk techniques. To the best of our knowledge, Gd13 cluster is thus far the highest nuclearity of lanthanide-silsesquioxane. Moreover, a magnetic study reveals a weak antiferromagnetic interaction between adjacent Gd ions in the cluster. In addition, Gd13 exhibits cryogenic magnetocaloric effects with a maximum −ΔSM value of 20 J kg−1 K−1 at 2 K and ΔH = 7.0 T.

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.

Summary The design principle of supramolecular architectures through a bottom-up strategy has always been guided by polyhedra such as Platonic solids, Archimedean solids, and prisms (antiprisms) with high symmetry. Non-uniform convex polyhedra with regular faces, i.e., Johnson solids, are rarely constructed experimentally although their topologies have been predicted through mathematical theory. Herein, self-assembly of a giant triangular orthobicupola [Zn24612], one of 92 Johnson solids J27, has been achieved from the coordination between 24 Zn2+ and 12 tetrakis-terpyridine building blocks 6 in DMF (N,N-Dimethylformamide). Different from classical Oh or Td symmetric cuboctahedron, the structure of this triangular orthobicupola revealed D3 symmetry which has been unambiguously determined by synchrotron X-ray analysis, NMR, and mass spectrometry. Surprisingly, the self-assembly process has been found to be solvent dependent. The same Zn2+ and tetrakis-terpyridine ligand 6 assembled into another pentagonal prism with the composition of [Zn20610] when using methanol as the reaction solvent.

The synthesis of molecular cages consisting of fully fused, π-conjugated rings is rare due to synthetic challenges including preorganization, large strain, and poor solubility. Herein, we report such an example in which a tris-2-aminobenzophenone precursor undergoes acid-mediated self-condensation to form a truncated tetrahedron, one of the 13 Archimedean solids. Formation of eight-membered [1,5]diazocine rings provides preorganization and releases the strain while still maintains weak π-conjugation of the backbone. Thorough characterizations were performed by X-ray, NMR, and UV-vis analysis, assisted by theoretical calculations. The cage exhibits a rigid backbone structure with a well-defined cavity that confines a magnetically shielded environment. The solvent molecule, o-dichlorobenzene, is precisely encapsulated in the cavity at a 1:1 ratio with multiple π···π, C-H···π, and halogen···π interactions with the cage skeleton, implying its template effect for the cage closing reaction. Our synthetic strategy opens the opportunity to access more complex, fully fused, three-dimensional π-conjugated cages.

Silver clusters with more than 2 concentric silver shells are scarce. Here, we enable self-assembly and crystallize SD/Ag192a, a highly symmetric silver chalcogenide cluster (SCC) with 192 silver cations in 6 shells and 136 anionic groups in 14 shells. All but 1 of these 20 concentric shells are Platonic or Archimedean solids. All have octahedral or tetrahedral symmetry and align the maximum number of their 2-, 3-, and 4-fold axes of rotational symmetry, thus identifying the cluster as a Keplerate. A rhombic dodecahedron supershell, formed from the first 3 anionic shells, is the keystone for the entire structure. But, nearly all of the edges in these polyhedral shells are too long to represent bonds. What mechanism of coordination chemistry holds the shells together? Like Na+ ions held electrostatically inside adjacent cube-shaped anionic compartments in a crystal of NaCl, individual Ag+ ions sit inside adjacent octahedron-shaped anionic compartments that fill space. Similarly, like Cl- ions in NaCl, individual anionic groups sit inside adjacent cationic (Ag+) compartments, mostly uniform polyhedra, that also fill space.

The paper is devoted to the study of the metric properties of regular and semiregular polyhedra in Euclidean spaces. In the first part, we prove that every regular polytope of dimension greater or equal than 4, and different from 120-cell in $$\mathbb {E}^4 $$ is such that the set of its vertices is a Clifford–Wolf homogeneous finite metric space. The second part of the paper is devoted to the study of special properties of Archimedean solids. In particular, for each Archimedean solid, its description is given as the convex hull of the orbit of a suitable point of a regular tetrahedron, cube or dodecahedron under the action of the corresponding isometry group.

Semi Equivelar maps are generalizations of Archimedean solids to surfaces other than 2-sphere. Semi Equivelar Maps were introduced by Upadhyay et. al. in 2014. They also studied semi equivelar maps on the surface of Euler characteristics $$\chi = -1$$ . In this article we classify all the semi equivelar maps on this surface with upto 12 vertices. We show that there are exactly four such maps. We also prove that there are at least 10 semi equivelar maps on this surface. We compute their Automorphism groups and show that none of these maps are vertex transitive.

The aim of this work is to investigate some properties of the truncated octahedron metric introduced in the space in further studies on metric geometry. With this metric, the 3-dimensional analytical space is a Minkowski geometry which is a non-Euclidean geometry in a finite number of dimensions. In a Minkowski geometry, the unit ball is a certain symmetric closed convex set instead of the usual sphere in Euclidean space. The unit ball of the truncated octahedron geometry is a truncated octahedron which is an Archimedean solid. In this study, first, metric properties of truncated octahedron distance, d_TO, in R^2 has been examined by metric approach. Then, by using synthetic approach some distance formulae in R_TO^3, 3-dimensional analytical space furnished with the truncated octahedron metric has been found.

Semi-equivelar maps are generalizations of Archimedean solids. We classify all the semi-equivelar maps on the surface of Euler Characteristics-2 with vertices up to 12. We calculate their automorphism groups and study their vertex-transitivity.

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.

Innovative spatial forms appear and develop at the joint of science, art, and architecture. Geometry is the most important, fundamental components of architectural forming. Now, having passed the stages of passion for the large-span shells, the sky-scrapers, typical inexpensive buildings, architectural bionics and ergonomics; pneumatic, membrane, wire rope and shrouds erections, the architects and designers payed attention at analytically non-given forms of erections and at the polyhedron. It is noticeably especially at the last 10-15 years. In a paper, the problems of application of the polyhedron and their modifications in architecture, building, and technics are analyzed. They consider prisms, pyramids, prismatoids, Platonic and several Archimedean solids, quasi-polyhedrons, and some figures constituted on their base. Polyhedral domes, umbrella shells, and hipped plate constructions are presented too. Large quantity of the illustrations devoted to the architecture of buildings and erections, to the landscape architecture and to the sculptural compositions is presented for the confirmation of increasing interest to these structures. 31 titles of the used original sources are given.

Geometric algebras known as a generalization of Grassmann algebras complex numbers and quaternions are presented by Clifford (1878). Geometric algebra describing the geometric symmetries of both physical space and spacetime is a strong language for physics. Groups generated from `Clifford numbers` are firstly defined by Lipschitz (1886). They are used for defining rotations in a Euclidean space. In this work, Clifford algebra is identified. The energy of classic particles with Clifford algebra are defined. This calculations are applied to some Archimedean solids. Also, the vertices of Archimedean solids presented in the Cartesian coordinates are calculated.

A brief history of the development of the regular polyhedrons theory is given. The work introduces the reader to modelling of the two most complex regular polyhedrons – Platonic solids: icosahedron and dodecahedron, in AutoCAD package. It is suggested to apply the method of the icosahedron and dodecahedron building using rectangles with their sides’ ratio like in the golden section, having taken the icosahedron’s golden rectangles as a basis. This method is well-known-of and is used for icosahedron, but is extremely rarely applied to dodecahedron, as in the available literature it is suggested to build the latter one as a figure dual to icosahedron. The work provides information on the first mentioning of this building method by an Italian mathematician L. Pacioli in his Divine Proportion book. In 1937, a Soviet mathematician D.I. Perepelkin published a paper On One Building Case of the Regular Icosahedron and Regular Dodecahedron, where he noted that this “method is not very well known of” and provided a building based “solely on dividing an intercept in the golden section ratio”. Taking into account the simplicity and good visualization of the building based on golden rectangles, a computer modeling of icosahedron and dodecahedron inscribed in a regular hexahedron is performed in the article. Given that, if we think in terms of the golden section concepts, the bigger side of the rectangle equals a whole intercept – side of the regular hexahedron, and the smaller sides of the icosahedron and dodecahedron rectangles are calculated as parts of the golden section ratio (of the bigger part and the smaller one, respectively). It is demonstrated how, using the scheme of a wireframe image of the dual connection of these polyhedrons as a basis, to calculate the sides of the rectangles in the golden section ratio in order to build an “infinite” cascade of these dual figures, as well as to build the icosahedron and dodecahedron circumscribed about the regular hexahedron. The method based on using the golden-section rectangles is also applied to building semiregular polyhedrons – Archimedean solids: a truncated icosahedron, truncated dodecahedron, icosidodecahedron, rhombicosidodecahedron, and rhombitruncated icosidodecahedron, which are based on icosahedron and dodecahedron.

Reticular chemistry has been an important guiding principle for the design of metal-organic frameworks (MOFs). This approach utilizes discrete building units (molecules and clusters) that are connected through strong bonds into extended networks assisted by topological considerations. Although the simple design principle of connecting points and lines has proved successful, new design strategies are still needed to further explore the structures and functions of MOFs. Herein, we report the design and synthesis of two mixed-ligand MOFs, [(CH

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.

Parastichy, the spiral arrangement of plant organs, is an example of the long-range apparent order seen in biological systems. These ordered arrangements provide scientists with both an aesthetic challenge and a mathematical inspiration. Synthetic efforts to replicate the regularity of parastichy may allow for molecular-scale control over particle arrangement processes. Here we report the packing of a supramolecular truncated cuboctahedron (TCO) into double-helical (DH) nanowires on a graphite surface with a non-natural parastichy pattern ascribed to the symmetry of the TCOs and interactions between TCOs. Such a study is expected to advance our understanding of the design inputs needed to create complex, but precisely controlled, hierarchical materials. It is also one of the few reported helical packing structures based on Platonic or Archimedean solids since the discovery of the Boerdijk-Coxeter helix. As such, it may provide experimental support for studies of packing theory at the molecular level.

The contribution deals with tessellations of convex uniform honeycombs that consist of only Platonic and Archimedean solids. The Archimedean truncated cuboctahedron is the hull of a 3D model of the 9D cube. Its edges have nine different orientations and are parallel to those of solids applied in the considered space-filling mosaics. The segments connecting the centroids of same cells of a tessellation, have also nine different orientations. These belong to two sets and can be the initial edges of two different 3D models of the 9D cube. Sequences of elements along the segments can be considered a compound edge of a compound model. The elements of the models can be marked by different colours according to their shapes, orientation and roles. The above described correlations make it possible to mark all compound models of all considered convex uniform honeycombs and their tessellations in all original ones by colouring. Using the compound models of the solids, new compound solids and tessellations can be built and this process can be repeated. This way, fractals or fractal like structures can be created. Differently oriented planes cut out periodical plane-tiling patterns from the gained spatial tessellations. An intersecting plane moved parallel to itself creates a series of patterns transforming into each other. These can be queued up as frames of an animation or give the level line depictions of the tessellations.

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.

Algebras of distributions of binary isolating and semi-isolating formulas are derived objects for given theory and reflect binary formula relations between realizations of 1-types. These algebras are associated with the following natural classification questions: 1) for a given class of theories, determine which algebras correspond to the theories from this class and classify these algebras; 2) to classify theories from a given class depending on the algebras defined by these theories of isolating and semi-isolating formulas. Here the description of a finite algebra of binary isolating formulas unambiguously entails a description of the algebra of binary semi-isolating formulas, which makes it possible to track the behavior of all binary formula relations of a given theory. In the article we describe algebras of binary formulas for the theories of Archimedean solids. For the obtained algebras, Cayley tables are given. It is shown that these algebras are exhausted by described algebras for a truncated cube, truncated octahedron, rhombocuboctahedron, icosododecahedron, truncated tetrahedron, cubooctahedron, flatnosed cube, flat-nosed dodecahedron, truncated cubooctahedron, rhomboicosododecahedron, truncated icosahedron, truncated dodecahedron, rhombo-truncated icosododecahedron.

In solid tessellations or three-dimensional honeycombs, polyhedra fit together to fill space, so that every face of each polyhedron belongs to another polyhedron. Solid and plane tessellations are intrinsically connected, since any section cut through a solid tessellation always produces some kind of plane tessellation. To clarify this relation, we will mention a short list of convex polyhedra that fill space monohedrally and illustrate the convex uniform honeycombs, focusing on those with structural potential to outline spaceframes. With regular plane tessellations as starting point, we hint at the geometrical possibilities in which the Platonic and two Archimedean solids are explorable in topological interlocking, aiming to expand the repertoire of blocks for monohedral topological interlocking assemblies. This has possible applications in architecture, in relation, for example, to space frames.

We say that a polytope has the Rupert property if we can make a hole large enough in to permit another copy of to pass through. In this article, we show that among the 13 Archimedean solids, 8 have...

Semi-equivelar maps are generalizations of maps on the surfaces of Archimedean solids to surfaces other than the 2-sphere. The well known 11 types of normal tilings of the plane suggest the possible types of semi-equivelar maps on the torus and the Klein bottle. In this article we classify (up to isomorphism) semi-equivelar maps on the torus and the Klein bottle with few vertices.

In this paper are presented two domed minimal housing units: Ghibli and D-Home. Both prototypes were outlined from the discretization of a domed space, using the geometry of the truncated icosahedron, an Archimedean solid that allows the discretization and tessellation of a concentric sphere with only two types of flat surfaces: pentagonal and hexagonal. In this way, both manufacturing and assembly operations will be facilitated and hastened by the presence of only two structural modules. The prototypes are verged in two different constructive solutions—D-Home uses a lightweight timber system, while Ghibli is an experimentation that employs a massive system. The prototypes show the possibilities offered by digital fabrication combined with subtractive and additive manufacturing techniques. The prototypes also show, specifically, how important is for a designer to master geometry and thus be able to create visually complex and unitary solutions, at the same time, reducing costs and facilitating the assembly.

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. .

Rational control of the self-assembly of large structures is one of the key challenges in chemistry, and is believed to become increasingly difficult and ultimately impossible as the number of components involved increases. So far, it has not been possible to design a self-assembled discrete molecule made up of more than 100 components. Such molecules-for example, spherical virus capsids-are prevalent in nature, which suggests that the difficulty in designing these very large self-assembled molecules is due to a lack of understanding of the underlying design principles. For example, the targeted assembly of a series of large spherical structures containing up to 30 palladium ions coordinated by up to 60 bent organic ligands was achieved by considering their topologies. Here we report the self-assembly of a spherical structure that also contains 30 palladium ions and 60 bent ligands, but belongs to a shape family that has not previously been observed experimentally. The new structure consists of a combination of 8 triangles and 24 squares, and has the symmetry of a tetravalent Goldberg polyhedron. Platonic and Archimedean solids have previously been prepared through self-assembly, as have trivalent Goldberg polyhedra, which occur naturally in the form of virus capsids and fullerenes. But tetravalent Goldberg polyhedra have not previously been reported at the molecular level, although their topologies have been predicted using graph theory. We use graph theory to predict the self-assembly of even larger tetravalent Goldberg polyhedra, which should be more stable, enabling another member of this polyhedron family to be assembled from 144 components: 48 palladium ions and 96 bent ligands.

It is well known that the point group of the root lattice D6 admits the icosahedral group as a maximal subgroup. The generators of the icosahedral group H3 , its roots, and weights are determined in terms of those of D6 . Platonic and Archimedean solids possessing icosahedral symmetry have been obtained by projections of the sets of lattice vectors of D6 determined by a pair of integers (m1,m2) in most cases, either both even or both odd. Vertices of the Danzer’s ABCK tetrahedra are determined as the fundamental weights of H3 , and it is shown that the inflation of the tiles can be obtained as projections of the lattice vectors characterized by the pair of integers, which are linear combinations of the integers (m1,m2) with coefficients from the Fibonacci sequence. Tiling procedure both for the ABCK tetrahedral and the &lt, ABCK&gt, octahedral tilings in 3D space with icosahedral symmetry H3, and those related transformations in 6D space with D6 symmetry are specified by determining the rotations and translations in 3D and the corresponding group elements in D6. The tetrahedron K constitutes the fundamental region of the icosahedral group and generates the rhombic triacontahedron upon the group action. Properties of “K-polyhedron”, “B-polyhedron”, and “C-polyhedron” generated by the icosahedral group have been discussed.

Starting with a cube and its symmetry group one can get a set of related polyhedra via adding congruent pyramids to its faces. The height and the rotation angle of the added pyramids give rise to a two-parameter set of such polyhedra. Thereby occur Archimedian solids and their duals, as e.g. an “icosi-tetra deltahedron”, but also starshaped solids. This approach can also be applied when taking a regular tetrahedron or a regular pentagon-dodecahedron as start figure. A hypercube in \( {\mathbb{R}}^{n} \) (an “n-cube”), too, suits as start object and gives rise to interesting polytopes (c.f. [1, 2, 3]). The cube’s geodesics and (inner) billiards, especially the closed ones, are already well-known (see [4, 5]). Hereby, a ray’s incoming angle equals its outcoming angle. There are many practical applications of reflections in a cube’s corner, as e.g. the cat’s eye and retroreflectors or reflectors guiding ships through bridges. Geodesics on a cube can be interpreted as billiards in the circumscribed rhombi-dodecahedron. This gives a hint, how to treat geodesics on arbitrary poly-hedra. Generalising reflections to refractions means that one has to apply Snellius’ refraction law saying that the sine-ratio of incoming and outcoming angles is constant. Application of this law (or a convenient modification of it) to geodesics on a polyhedron will result in trace polygons, which might be called “quasi-geodesics”. The concept “pseudo-geodesic”, coined for curves c on smooth surfaces \( {\Phi } \), is defined by the property of c that its osculating planes enclose a constant angle with the normals n of \( {\Phi } \). Again, this concept can be modified for polyhedrons, too. We look for these three types of traces of rays in and on a 3-cube and a 4-cube.

We study the property of $\ensuremath{\alpha}$ aggregates distributed on Archimedean solids within a microscopic framework, which takes full account of the Pauli principle. We pay special attention to the Coulomb energy for such exotic shapes of nuclei, and we discuss the advantage of $\ensuremath{\alpha}$ clusters with geometric configurations compared with the uniform density distributions in reducing the repulsive effect. We consider four kinds of configurations of $\ensuremath{\alpha}$ clusters distributed on Archimedean solids: dual polyhedra composed of a dodecahedron and an icosahedron, an octacontahedron, and two types of truncated icosahedrons, that is, two kinds of Archimedean solids. The latter two are an icosidodecahedron and a fullerene shape. Putting an $\ensuremath{\alpha}$ cluster on each vertex of the polyhedra, four $\ensuremath{\alpha}$-cluster aggregates correspond to the following four nuclei: Gd (64 protons), Po (84 protons), Nd (60 protons), and a nucleus with 120 protons, respectively.

The close relationship between art and science is a distinctive feature of the Renaissance, and, as we have observed again and again, Leonardo represents one of the highest points of this synthesis. The scientific knowledge that he obtained via his studies became a fundamental instrument also for his work as an artist and vice versa. As discussed in the previous chapter, it was thanks to his dear friend Luca Pacioli that Leonardo was able to fully appreciate the beauty of geometry and arithmetic. Luca Pacioli’s support was also critical, more generally, in enabling him to achieve a satisfactory knowledge of mathematics.

Starting with a cube and its symmetry group one can get related polyhedrons via corner and edge trimming and dualizing processes or via adding congruent polyhedrons to its faces. These processes deliver a subset of Archimedian solids and their duals, but also starshaped solids. Thereby, polyhedrons with congruent faces are of special interest, as these faces could be used as tiles in mosaics, either in a Euclidean or at least in a non-Euclidean plane. Obviously this approach can be applied when taking a regular tetrahedron or a regular pentagon-dodecahedron as start figure. But a hypercube in R^n (an “n-cube”), too, suits as start object and gives rise to interesting polytopes. The cube’s geodesics and (inner) billiards, especially the closed ones, are already well-known (see references). Hereby, a ray’s incoming angle equals its outcoming angle. There are many practical applications of reflections in a cube’s corner, as e.g. the cat’s eye and retroreflectors or reflectors guiding ships through bridges. Geodesics on a cube can be interpreted as billiards in the circumscribed rhombi-dodecahedron. This gives a hint, how to treat geodesics on arbitrary polyhedrons. Generalising reflections to refractions means that one has to apply Snellius’ refraction law saying that the sine-ratio of incoming and outcoming angles is constant. Application of this law (or a convenient modification of it) to geodesics on a polyhedron will result in trace polygons, which might be called “quasi-geodesics”. The concept “pseudo-geodesic”, coined for curves c on smooth surfaces Φ, is defined by the property of c that its osculating planes enclose a constant angle with the normals n of Φ. Again, this concept can be modified for polyhedrons, too. We look for these three types of traces of rays in and on a 3-cube and a 4-cube.

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'

Structural DNA nanotechnology and the DNA origami technique, in particular, have provided a range of spatially addressable two- and three-dimensional nanostructures. These structures are, however, typically formed of tightly packed parallel helices. The development of wireframe structures should allow the creation of novel designs with unique functionalities, but engineering complex wireframe architectures with arbitrarily designed connections between selected vertices in three-dimensional space remains a challenge. Here, we report a design strategy for fabricating finite-size wireframe DNA nanostructures with high complexity and programmability. In our approach, the vertices are represented by n × 4 multi-arm junctions (n = 2-10) with controlled angles, and the lines are represented by antiparallel DNA crossover tiles of variable lengths. Scaffold strands are used to integrate the vertices and lines into fully assembled structures displaying intricate architectures. To demonstrate the versatility of the technique, a series of two-dimensional designs including quasi-crystalline patterns and curvilinear arrays or variable curvatures, and three-dimensional designs including a complex snub cube and a reconfigurable Archimedean solid were constructed.

Polyhedral transformation enables large volumetric change amongst Platonic and Archimedean solids. It has great potential in applications where transportability and protection of payload are critical design features, e.g., small or micro satellites, space habitats or planetary rovers. However, existing designs introduce many degrees of freedoms, making the control of transformation process extremely difficult and cumbersome, and thus the practical applicability of the mechanisms is restricted. This paper develops a kinematic method to enable polyhedral transformation with a single degree of freedom. The approach has been implemented for the transformation between truncated octahedron and cube. Motion analysis indicates that transformation path is unique without singularity, which is further demonstrated with physical validation models. We envisage that our method could be suitable for extension to other paired polyhedron sets.

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.

The spontaneous formation of discrete spherical nanosized molecules is prevalent in nature, but the authentic structural mimicry of such highly symmetric polyhedra from edge sharing of regular polygons has remained elusive. Here we present a novel ball-shaped {(HNEt3)[Ag37S4(SC6H4(t)Bu)24(CF3COO)6(H2O)12]} cluster () that is assembled via a one-pot process from polymeric {(HNEt3)2[Ag10(SC6H4(t)Bu)12]}n and CF3COOAg. Single crystal X-ray analysis confirmed that is a Td symmetric spherical molecule with a [Ag36(SC6H4(t)Bu)24] anion shell enwrapping a AgS4 tetrahedron. The shell topology of belongs to one of 13 Archimedean solids, a truncated tetrahedron with four edge-shared hexagons and trigons, which are supported by a AgS4 Platonic solid in the core. Interestingly, the cluster emits green luminescence centered at 515 nm at room temperature. Our investigations have provided a promising synthetic protocol for a high-nuclearity silver cluster based on underlying geometrical principles.

Buckminsterfullerene (C60) represents a perfect combination of geometry and molecular structural chemistry. It has inspired many creative ideas for building fullerene-like nanopolyhedra. These include other fullerenes, virus capsids, polyhedra based on DNA, and synthetic polynuclear metal clusters and cages. Indeed, the regular organization of large numbers of metal atoms into one highly complex structure remains one of the foremost challenges in supramolecular chemistry. Here we describe the design, synthesis, and characterization of a Ag180 nanocage with 180 Ag atoms as 4-valent vertices (V), 360 edges (E), and 182 faces (F)––sixty 3-gons, ninety 4-gons, twelve 5-gons, and twenty 6-gons––in agreement with Euler’s rule V − E + F = 2. If each 3-gon (or silver Trigon) were replaced with a carbon atom linked by edges along the 4-gons, the result would be like C60, topologically a truncated icosahedron, an Archimedean solid with icosahedral (Ih) point-group symmetry. If C60 can be described mathematically as a curling up of a 6.6.6 Platonic tiling, the Ag180 cage can be described as a curling up of a 3.4.6.4 Archimedean tiling. High-resolution electrospray ionization mass spectrometry reveals that {Ag3}n subunits coexist with the Ag180 species in the assembly system before the final crystallization of Ag180, suggesting that the silver Trigon is the smallest building block in assembly of the final cage. Thus, we assign the underlying growth mechanism of Ag180 to the Silver-Trigon Assembly Road (STAR), an assembly path that might be further employed to fabricate larger, elegant silver cages.

Sulfonates were incorporated into six novel high-nuclearity silver(I) thiolate clusters under the guidance of anion templates varied from S2- , SO42- , α-[Mo5 O18 ]6- , β-[Mo5 O18 ]6- , [Mo2 O8 ]4- , to [Mo4 O14 (SO4 )]6- . Single crystal X-ray analysis revealed that SD/Ag1, SD/Ag3, SD/Ag5, and SD/Ag6 are discrete [S@Ag60 ], [α-Mo5 O18 @Ag36 ], [Mo2 O8 @Ag30 ]2 , and [Mo4 O14 (SO4 )@Ag73 ] clusters, respectively, whereas SD/Ag2 and SD/Ag4 are one-dimensional (1D) chains based on the [SO4 @Ag20 ] and [β-Mo5 O18 @Ag36 ] cluster subunits, respectively. Their silver skeletons are protected exteriorly by thiolates and sulfonates and interiorly supported by diverse anions as templates. Structurally, cluster SD/Ag1 is a typical core-shell structure comprised of an inner Ag12 cuboctahedron and an outer Ag48 shell. The sulfate-templated drum-like Ag20 cluster subunits are bridged by PhSO3- to give a 1D chain of SD/Ag2. Complex SD/Ag3 and SD/Ag4 are spindle-like Ag36 clusters with isomeric [Mo5 O18 ]6- inside, and the latter is further extended to a 1D chain through PhSO3- bridges. A pair of [Mo2 O8 ]4- templated gourd-like Ag30 clusters are dimerized in a head-to-head fashion to form SD/Ag5. Complex SD/Ag6 is the largest cluster in this family and doubly templated by unprecedented [Mo4 O14 (SO4 )]6- anions. Geometrically, the silver shells of SD/Ag1-SD/Ag5 show the polyhedral features of Johnson solids, instead of the usual Platonic or Archimedean solids. Solution behaviors and luminescent properties were also investigated in detail.

A polyhedron is a three-dimensional figure bounded by a finite number of polygonal faces. Its literal meaning is a many-faced figure because poly means many and hedron means face. Thus a tetrahedron is a four-faced figure, which can only be the triangular pyramid. The plural form of polyhedron is polyhedra. Some human beings are bihedra.

Biochemists have coaxed a protein to assemble into a snub-cube-shaped cage. Snub cubes, which have 38 faces—6 squares and 32 equilateral triangles—are one of the 13 Archimedean solids, first descri...

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.

Understanding the self-assembly of small structural units into large supramolecular assemblies remains one of the great challenges in structural chemistry. We have discovered that tetrahedral supramolecular cages, exhibiting the shapes of Archimedean solids, can be self-assembled by hydrogen bonding interactions using tricationic N-donors (1 or 2) in cooperation with water (W). Single crystal X-ray analysis shows that cage (2)4(W)6, assembled in an aqueous solution of cation 2 and KPF6, consists of four tripodal trications linked by six water monomers and resembles the shape of a truncated tetrahedron. Similarly, cage (1)4(W6)4 spontaneously self-assembles in an aqueous solution of cation 1 and NH4PF6 and consists of four tripodal cations and four water hexamers. Here, each of the four (H2O)6 units act as tritopic nodes between three distinct tripodal cations forming a polyhedron similar to the cantellated tetrahedron. These two well-defined cages are assembled via total of 12 and 36 hydrogen bonds, respe...

Breaking edges off regular truncated cuboctahedra leads to a metal-organic framework (MOF) with polybenzene topology. This fully characterized MOF with lowest connectivity has the largest BET surface area among interpenetrated MOFs.

In this paper we determine new upper bounds for the maximal density of translative packings of superballs in three dimensions (unit balls for the $l^p_3$-norm) and of Platonic and Archimedean solids having tetrahedral symmetry. Thereby, we improve Zong's recent upper bound for the maximal density of translative packings of regular tetrahedra from $0.3840\ldots$ to $0.3745\ldots$, getting closer to the best known lower bound of $0.3673\ldots$ We apply the linear programming bound of Cohn and Elkies which originally was designed for the classical problem of densest packings of round spheres. The proofs of our new upper bounds are computational and rigorous. Our main technical contribution is the use of invariant theory of pseudo-reflection groups in polynomial optimization., Comment: 30 pages, 6 tables, 3 figures, (v3) comments of referees incorporated

We present explicit geometric decompositions of the hyperbolic complements of alternating $k$-uniform tiling links, which are alternating links whose projection graphs are $k$-uniform tilings of $S^2$, $\mathbb{E}^2$, or $\mathbb{H}^2$. A consequence of this decomposition is that the volumes of spherical alternating $k$-uniform tiling links are precisely twice the maximal volumes of the ideal Archimedean solids of the same combinatorial description, and the hyperbolic structures for the hyperbolic alternating tiling links come from the equilateral realization of the $k$-uniform tiling on $\mathbb{H}^2$. In the case of hyperbolic tiling links, we are led to consider links embedded in thickened surfaces $S_g \times I$ with genus $g \ge 2$ and totally geodesic boundaries. We generalize the bipyramid construction of Adams to truncated bipyramids and use them to prove that the set of possible volume densities for all hyperbolic links in $S_g \times I$, ranging over all $g \ge 2$, is a dense subset of the interval $[0, 2v_\text{oct}]$, where $v_\text{oct} \approx 3.66386$ is the volume of the ideal regular octahedron., Comment: This version contains some clarification, and corrects some errata. arXiv admin note: text overlap with arXiv:1603.03715

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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