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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.

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.

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.

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.