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Metal–organic polyhedra are a member of metal–organic materials, and are together with metal–organic frameworks utilized as emerging porous platforms for numerous applications in energy- and bio-related sciences. However, metal–organic polyhedra have been significantly underexplored, unlike their metal–organic framework counterparts. In this review, we will cover the topologies and the classification of metal–organic polyhedra and share several suggestions, which might be useful to synthetic chemists regarding the future directions in this rapid-growing field. [ABSTRACT FROM AUTHOR]

When the principle of modular assembly of semi-regulated polyhedra is applied in architectural design, we obtain functional and structural autonomous architectural structures, based on a common matrix system. This system is subdivided into modules that allow independent changes or adjustments and offers great adaptability to the environment. The development of computer technology and 3D printing enables tessellation of more spectacular geometric modules. The studied surface is much easier to control both in terms of shape and structure. Therefore, research on the applications of modular assembly in the design of architectural geometry it is a great necessity for architectural studies. This paper aims to present several buildings that use the three-dimensional assembly of semi-regular polyhedra, explore the feasibility and significance of application of modular assembly in architectural geometry design and can be considered the beginning of a much more detailed ongoing research. [ABSTRACT FROM AUTHOR]

Previous discoveries of the first author (1984-88) on so-called hyperbolic football manifolds and our recent works (2016-17) on locally extremal ball packing and covering hyperbolic space H³ with congruent balls had led us to the idea that our "experience space in small size" could be of hyperbolic structure. In this paper we construct a new infinite series of oriented hyperbolic space forms so-called cobweb (or tube) manifolds Cw(2z; 2z; 2z) = Cw(2z), 3 ≤ z odd, which can describe nanotubes, very probably. So we get a structure of rotational order z = 5; 7 ..., as new phenomena. Although the theoretical basis of compact manifolds of constant curvature (i.e. space forms) are well-known (100 years old), we are far from an overview. So our new very natural hyperbolic infinite series Cw(2z) seems to be very timely also for crystallographic applications. Mathematical novelties are foreseen as well, for future investigations. [ABSTRACT FROM AUTHOR]

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The dispersion in the dot size, shape, and composition leads to a difficult comparison with experimental spectroscopy and transport data even if the growth conditions are similar. In this work, an extensive analysis of the influence of the dot size and shape on the electron and hole energy states and on transition energies is carried out using a unified model of the semiconductor band structure. In this study we obtain the electron energy spectra for three-dimensional small InAs/GaAs quantum dots of several different truncated shapes described in the literature: tetrahedral, pyramidal with base of different geometry, etc. Also, in order to give an idea of the flexibility of the method, the icosahedral geometry is analyzed. The combination of theoretical results using a unified model for all the geometries with structural techniques will allow a more precise analysis of experimental samples. [ABSTRACT FROM AUTHOR]

A correction to the article "Dense packings of the Platonic and Archimedean solids" that was published in a previous issue is presented.

Thiacalix[4]arenes as a family of promising ligands have been widely used to construct polynuclear metal clusters, but scarcely employed in silver nanoclusters. Herein, an anion-templated Ag88 nanocluster (SD/Ag88a) built from p-tert-butylthiacalix[4]arene (H4TC4A) is reported. Single-crystal X-ray diffraction reveals that C4-symmetric SD/Ag88a resembles a metal-organic super calix comprised of eight TC4A4− as walls and 88 silver atoms as base, which can be deconstructed to eight [CrO4@Ag11(TC4A)(EtS)4(OAc)] secondary building units arranged in an annulus encircling a CrO42− in the center. Local and global anion template effects from chromates are individually manifested in SD/Ag88a. The solution stability and hierarchical assembly mechanism of SD/Ag88a are studied by using electrospray mass spectrometry. The Ag88 nanocluster represents the highest nuclearity metal cluster capped by TC4A4−. This work not only exemplify the specific macrocyclic effects of TC4A4− in the construction of silver nanocluster but also realize the shape heredity of TC4A4− to overall silver super calix. The assembly of giant silver clusters by using macrocylic multidentate ligand remains a challenge. Here, the authors synthesize a chromate-templated 88-nuclei silver super calix and reveal the role of anion templating effects and a hierarchical assembly mechanism. [ABSTRACT FROM AUTHOR]

Polyhedrons have been studied by mathematicians and geometers during many years, because of their symmetries. The theory of convex sets is a vibrant and classical field of modern mathematics with rich applications. The more geometric aspects of convex sets are developed introducing some notions, but primarily polyhedra. A polyhedra, when it is convex, is an extremely important special solid in Rn. Some examples of convex subsets of Euclidean 3-dimensional space are Platonic Solids, Archimedean Solids and Archimedean Duals or Catalan Solids. There are some relations between metrics and polyhedra. For example, it has been shown that cube, octahedron, deltoidal icositetrahedron are maximum, taxicab, Chinese Checker’s unit sphere, respectively. In this study, I introduce two new metrics, and show that the spheres of the 3-dimensional analytical space furnished by these metrics are rhombicuboctahedron and rhombicosidodecahedron. Also some properties about these metrics are given. [ABSTRACT FROM AUTHOR]

Stable metal clusters that can resist both highly concentrated acid and alkali are unknown. Herein, we present a discrete neutral cluster, Hf13(μ4‐O)8(OCH3)36 (1), which features extraordinary chemical stability by preserving its crystalline state in concentrated aqueous solutions of both acid (10 m HNO3) and alkali (20 m boiling NaOH). Importantly, 1 can serve as a luminescent probe for detecting both concentrated alkali (20 m NaOH) and strong acid (1 m HNO3) with high selectivity and repeatability. DFT studies of the electronic structure and bonding revealed that 1 has an extremely large HOMO–LUMO gap due to strong d π–p π bonding that accounts for the ultrahigh stability. [ABSTRACT FROM AUTHOR]

This paper studies colorings of patterns with multiple orbits, particularly those colorings where the orbits share colors. The main problem is determining when such colorings become perfect. This problem is attacked by characterizing all perfect colorings of patterns through the construction of sufficient and necessary conditions for a coloring to be perfect. These results are then applied on symmetrical objects to construct both perfect and non‐perfect colorings. [ABSTRACT FROM AUTHOR]

Slake Durability Index test is being widely used to determine the disintegration characteristics of the weak rocks when they are subjected to drying and wetting cycles. In the test standards, samples are recommended to have rounded or close to spherical shape. In this study, a new method called as Pasha Method was proposed and applied to prepare equal-sized spherical samples. Slake Durability Index tests were performed on the spherical and rounded sample sets prepared from different rock types. According to Slake Durability Index test results in all rocks, the difference between the first cycle and second cycle results of the rounded samples and the spherical samples is found as 0.05–8.04% and 0.1–13.63%, respectively. When only the spherical test results were considered, the difference in the first cycle and second cycle values is found to be 0.28–2.19% and 0.44–7.57%, respectively. Applying this new method allows researchers to perform this test on equal-sized spherical samples, resulting in the minimization of the shape effect on the test results. [ABSTRACT FROM AUTHOR]

The authors study the class of finite homogeneous metric spaces and some of its important subclasses that have natural definitions in terms of the metrics and well-studied analogs in the class of Riemannian manifolds. The relationships between these classes are explored. The examples of the corresponding spaces are built, some of which are the vertex sets of the special convex polytopes in Euclidean space. We describe the classes on using the language of graph theory, which enables us to provide some examples of finite metric spaces with unusual properties. Several unsolved problems are posed. [ABSTRACT FROM AUTHOR]

We demonstrate that, when spanning two sources of attractants on polyhedrons, the slime mould Physarum polycephalum follows the geodesic lines. Truncated tetrahedron, truncated cubes and snub cubes of different sizes have been in 3D printed. Two sites on each polyhedron were chosen at random. The slime mould was inoculated in one site and an oat flake was placed in the target site. In 38 of 63 experiments the slime mould spanned to sites propagating along the geodesic lines with maximal deflection of c. 1 cm. [ABSTRACT FROM AUTHOR]

Symmetrical protein cages have evolved to fulfil diverse roles in nature, including compartmentalization and cargo delivery1, and have inspired synthetic biologists to create novel protein assemblies via the precise manipulation of protein–protein interfaces. Despite the impressive array of protein cages produced in the laboratory, the design of inducible assemblies remains challenging2,3. Here we demonstrate an ultra-stable artificial protein cage, the assembly and disassembly of which can be controlled by metal coordination at the protein–protein interfaces. The addition of a gold (i)-triphenylphosphine compound to a cysteine-substituted, 11-mer protein ring triggers supramolecular self-assembly, which generates monodisperse cage structures with masses greater than 2 MDa. The geometry of these structures is based on the Archimedean snub cube and is, to our knowledge, unprecedented. Cryo-electron microscopy confirms that the assemblies are held together by 120 S–Aui–S staples between the protein oligomers, and exist in two chiral forms. The cage shows extreme chemical and thermal stability, yet it readily disassembles upon exposure to reducing agents. As well as gold, mercury(ii) is also found to enable formation of the protein cage. This work establishes an approach for linking protein components into robust, higher-order structures, and expands the design space available for supramolecular assemblies to include previously unexplored geometries. An artificial protein cage is readily assembled by metal ion coordination and disassembled by reducing agents, and displays excellent chemical and thermal stability. [ABSTRACT FROM AUTHOR]

We study combinatorial properties of polytopes realizable in the Lobachevsky space L 3 as polytopes of finite volume with right dihedral angles. On the basis of E. M. An-dreev's theorem we prove that cutting off ideal vertices of right-angled polytopes defines a one-to-one correspondence with strongly cyclically four-edge-connected polytopes different from the cube and the pentagonal prism. We show that any polytope of the latter family can be obtained by cutting off a matching of a polytope from the same family or of the cube with at most two nonadjacent orthogonal edges cut, in such a way that each quadrangle results from cutting off an edge. We refine D. Barnette's construction of this family of polytopes and present its application to right-angled polytopes. We refine the known construction of ideal right-angled polytopes using edge twists and describe its connection with D. Barnette's construction via perfect matchings. We make a conjecture on the behavior of the volume under operations and give arguments to support it. [ABSTRACT FROM AUTHOR]

Attempting to solve the Four Color Problem in 1940, Henry Lebesgue gave an approximate description of the neighborhoods of 5-vertices in the class P5 of 3-polytopes with minimum degree 5. This description depends on 32 main parameters. Not many precise upper bounds on these parameters have been obtained as yet, even for restricted subclasses in P5. Given a 3-polytope P, by w(P) denote the minimum of the maximum degree-sum (weight) of the neighborhoods of 5-vertices (minor 5-stars) in P. In 1996, Jendrol' and Madaras showed that if a polytope P in P5 is allowed to have a 5-vertex adjacent to four 5-vertices (called a minor (5, 5, 5, 5, ∞)-star), then w(P) can be arbitrarily large. For each P* in P5 with neither vertices of degree 6 and 7 nor minor (5, 5, 5, 5, ∞)-star, it follows from Lebesgue's Theorem that w(P*) ≤ 51. We prove that every such polytope P* satisfies w(P*) ≤ 42, which bound is sharp. This result is also best possible in the sense that if 6-vertices are allowed but 7-vertices forbidden, or vice versa; then the weight of all minor 5-stars in P5 under the absence of minor (5, 5, 5, 5, ∞)-stars can reach 43 or 44, respectively. [ABSTRACT FROM AUTHOR]

The combination of profound chirality and high symmetry on the nm-scale is unusual and would open exciting avenues, both fundamental and applied. Here we show how the unique electronic structure and bonding of quasi-2D gold makes this possible. We report a chiral symmetry breaking, i.e., the spontaneous formation of a chiral-icosahedral shell (I-Au60) from achiral (Uh) precursor forms, accompanied by a contraction in the Au-Au bonding and hence the radius of this perfect golden sphere, in which all 60 sites are chemically equivalent. This structure, which resembles the most complex of semi-regular (Archimedean) polyhedra (34.5*), may be viewed as an optimal solution to the topological problem: how to close a 60-vertex 2D (triangular) net in 3D. The singular rigidity of the I-Au60 manifests in uniquely discrete structural, vibrational, electronic, and optical signatures, which we report herein as a guide to its experimental detection and ultimately its isolation in material forms. [ABSTRACT FROM AUTHOR]

The 3-dimensional model of any k-dimensional cube can be constructed by starting k edges whose Minkowski sum can be called zonotope. Combined 2

Numerous examples of cuboctahedra found in medieval-era buildings whose dates range from the early twelfth to the early fifteenth century across in Turkey indicate the significant use of such geometrical entities. Here we focus particularly on cuboctahedra with carved-out surfaces. The results show that although the unit cell, which is a combination of cubes and tetrahedra, sufficiently explains all examples, the octahemioctahedron and stella octangula strengthen the possibility of tetrahedral packing with its dual network and indicate a “vector matrix”, as suggested by R. Buckminster Fuller. Therefore, their prevalent use as a “geometric solid” in a hollow cube frame and their appearance as an envelope of either tetrahedral packing or highly complex surfaces reveal almost 800-hundred-year-old examples of cuboctahedra as a Vector Equilibrium (VE) producing Isotropic Vector Matrix (IVM). [ABSTRACT FROM AUTHOR]

A discrete 78-nucleus silver–sulfur nanocluster with a sulfate-centered multishell structure was isolated and characterized. Its crystal structure revealed 18 and 60 Ag atoms in the inner and outer shell, respectively. The inner shell of 18-nuclearity Ag atoms is a very rare convex polyhedron featuring an elongated triangular orthobicupola. The incorporation of a sulfate anion and multishell arrangement in the nanocluster led to a dramatic decrease in the band gap (Eg = 1.40 eV). Our study showed that simple anions can also induce the formation of high-nuclearity silver clusters with excellent optical properties. [ABSTRACT FROM AUTHOR]

A Pogorelov polytope is a combinatorial simple 3-polytope realizable in the Lobachevsky (hyperbolic) space as a bounded right-angled polytope. These polytopes are exactly simple 3-polytopes with cyclically 5-edge connected graphs. A Pogorelov polytope has no 3- and 4-gons and may have any prescribed numbers of k-gons, k ≥ 7. Any simple polytope with only 5-, 6- and at most one 7-gon is Pogorelov. For any other prescribed numbers of k-gons, k ≥ 7, we give an explicit construction of a Pogorelov and a non-Pogorelov polytope. Any Pogorelov polytope different from k-barrels (also known as Löbel polytopes, whose graphs are biladders on 2k vertices) can be constructed from the 5- or the 6-barrel by cutting off pairs of adjacent edges and connected sums with the 5-barrel along a 5-gon with the intermediate polytopes being Pogorelov. For fullerenes, there is a stronger result. Any fullerene different from the 5-barrel and the (5, 0)-nanotubes can be constructed by only cutting off adjacent edges from the 6-barrel with all the intermediate polytopes having 5-, 6- and at most one additional 7-gon adjacent to a 5-gon. This result cannot be literally extended to the latter class of polytopes. We prove that it becomes valid if we additionally allow connected sums with the 5-barrel and 3 new operations, which are compositions of cutting off adjacent edges. We generalize this result to the case when the 7-gon may be isolated from 5-gons. [ABSTRACT FROM AUTHOR]

The origins of the chiroptical activities of inorganic nanostructures have perplexed scientists, and deracemization of high-nuclearity metal nanoclusters (NCs) remains challenging. Here, we report a single-crystal structure of Rac-Ag 70 that contains enantiomeric pairs of 70-nuclearity silver clusters with 20 free valence electrons (Ag 70 ), and each of these clusters is a doubly truncated tetrahedron with pseudo-T symmetry. A deracemization method using a chiral metal precursor not only stabilizes Ag 70 in solution but also enables monitoring of the gradual enlargement of the electronic circular dichroism (CD) responses and anisotropy factor g abs . The chiral crystals of R/S-Ag 70 in space group P2 1 containing a pseudo-T-symmetric enantiomeric NC show significant kernel-based and shell-based CD responses. The small symmetry breaking of T d symmetry arising from local distortion of Ag-S motifs and rotation of the apical Ag 3 trigons results in large chiroptical responses. This work opens an avenue to construct chiral medium/large-sized NCs and nanoparticles, which are promising for asymmetric catalysis, nonlinear optics, chiral sensing, and biomedicine., (© 2022. The Author(s).)

Abstract: Sulfonates were incorporated into six novel high‐nuclearity silver(I) thiolate clusters under the guidance of anion templates varied from S2−, SO42−, α‐[Mo5O18]6−, β‐[Mo5O18]6−, [Mo2O8]4−, to [Mo4O14(SO4)]6−. Single crystal X‐ray analysis revealed that SD/Ag1, SD/Ag3, SD/Ag5, and SD/Ag6 are discrete [S@Ag60], [α‐Mo5O18@Ag36], [Mo2O8@Ag30]2, and [Mo4O14(SO4)@Ag73] clusters, respectively, whereas SD/Ag2 and SD/Ag4 are one‐dimensional (1D) chains based on the [SO4@Ag20] and [β‐Mo5O18@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 [Mo5O18]6− inside, and the latter is further extended to a 1D chain through PhSO3− bridges. A pair of [Mo2O8]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 [Mo4O14(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. [ABSTRACT FROM AUTHOR]

Soft bioelectronic devices provide new opportunities for next-generation implantable devices owing to their soft mechanical nature that leads to minimal tissue damages and immune responses. However, a soft form of the implantable optoelectronic device for optical sensing and retinal stimulation has not been developed yet because of the bulkiness and rigidity of conventional imaging modules and their composing materials. Here, we describe a highdensity and hemispherically curved image sensor array that leverages the atomically thin MoS2-graphene heterostructure and strain-releasing device designs. The hemispherically curved image sensor array exhibits infrared blindness and successfully acquires pixelated optical signals. We corroborate the validity of the proposed soft materials and ultrathin device designs through theoretical modeling and finite element analysis. Then, we propose the ultrathin hemispherically curved image sensor array as a promising imaging element in the soft retinal implant. The CurvIS array is applied as a human eye-inspired soft implantable optoelectronic device that can detect optical signals and apply programmed electrical stimulation to optic nerves with minimum mechanical side effects to the retina. [ABSTRACT FROM AUTHOR]

Aigner and Fromme initiated the systematic study of the cop number of a graph by proving the elegant and sharp result that in every connected planar graph, three cops are sufficient to win a natural pursuit game against a single robber. This game, introduced by Nowakowski and Winkler, is commonly known as Cops and Robbers in the combinatorial literature. We extend this study to directed planar graphs, and establish separation from the undirected setting. We exhibit a geometric construction that shows that a sophisticated robber strategy can indefinitely evade three cops on a particular strongly connected planar-directed graph. [ABSTRACT FROM AUTHOR]

Aristotelian diagrams visualize the logical relations among a finite set of objects. These diagrams originated in philosophy, but recently, they have also been used extensively in artificial intelligence, in order to study (connections between) various knowledge representation formalisms. In this paper, we develop the idea that Aristotelian diagrams can be fruitfully studied as geometrical entities. In particular, we focus on four polyhedral Aristotelian diagrams for the Boolean algebra B4, viz. the rhombic dodecahedron, the tetrakis hexahedron, the tetraicosahedron and the nested tetrahedron. After an in-depth investigation of the geometrical properties and interrelationships of these polyhedral diagrams, we analyze the correlation (or lack thereof) between logical (Hamming) and geometrical (Euclidean) distance in each of these diagrams. The outcome of this analysis is that the Aristotelian rhombic dodecahedron and tetrakis hexahedron exhibit the strongest degree of correlation between logical and geometrical distance; the tetraicosahedron performs worse; and the nested tetrahedron has the lowest degree of correlation. Finally, these results are used to shed new light on the relative strengths and weaknesses of these polyhedral Aristotelian diagrams, by appealing to the congruence principle from cognitive research on diagram design. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The article addresses the genesis and visualization of the capstone image to Kepler’s polyhedral hypothesis of the planetary intervals from his first major work, Mysterium Cosmographicum (1596). The contention is that the famous Tabula III was directed less by Kepler than it was an initiative spearheaded by Georg Gruppenbach, the printer of Mysterium, and Kepler’s mentor Michael Mäistlin, who sought to produce a marketable broadsheet that would appeal to the contemporary German fashion for illustrations of polyhedral geometry. More generally, the article seeks to redefine the key role played by the printing workshop and the decorative arts in the theory’s inception and ultimate graphic manifestation. [ABSTRACT FROM AUTHOR]

Vertices and symmetries of regular and irregular chiral polyhedra are represented by quaternions with the use of Coxeter graphs. A new technique is introduced to construct the chiral Archimedean solids, the snub cube and snub dodecahedron together with their dual Catalan solids, pentagonal icositetrahedron and pentagonal hexecontahedron. Starting with the proper subgroups of the Coxeter groups W(A1 ⊕A1 ⊕A1), W(A3), W(B3) and W(H3), we derive the orbits representing the respective solids, the regular and irregular forms of a tetrahedron, icosahedron, snub cube, and snub dodecahedron. Since the families of tetrahedra, icosahedra and their dual solids can be transformed to their mirror images by the proper rotational octahedral group, they are not considered as chiral solids. Regular structures are obtained from irregular solids depending on the choice of two parameters. We point out that the regular and irregular solids whose vertices are at the edge mid-points of the irregular icosahedron, irregular snub cube and irregular snub dodecahedron can be constructed. [ABSTRACT FROM AUTHOR]

In 1940, in attempts to solve the Four Color Problem, Henry Lebesgue gave an approximate description of the neighborhoods of 5-vertices in the class P of 3-polytopes with minimum degree 5. This description depends on 32 main parameters. Very few precise upper bounds on these parameters have been obtained as yet, even for restricted subclasses in P . Given a 3-polytope P, denote the minimum of the maximum degrees (height) of the neighborhoods of 5-vertices (minor 5-stars) in P by h( P). Jendrol' and Madaras in 1996 showed that if a polytope P in P is allowed to have a 5-vertex adjacent to four 5-vertices (called a minor (5, 5, 5, 5,∞)- star), then h( P) can be arbitrarily large. For each P in P with neither vertices of the degree from 6 to 8 nor minor (5, 5, 5, 5,∞)-star, it follows from Lebesgue's Theorem that h( P ) ≤ 17. We prove in particular that every such polytope P satisfies h( P ) ≤ 12, and this bound is sharp. This result is best possible in the sense that if vertices of one of degrees in {6, 7, 8} are allowed but those of the other two forbidden, then the height of minor 5-stars in P under the absence of minor (5, 5, 5, 5,∞)- stars can reach 15, 17, or 14, respectively. [ABSTRACT FROM AUTHOR]