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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 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.

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.

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.

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.

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.

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.

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...

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]

We describe various types of maximum noninteger vertices. We identify types of polyhedron vertices by the number of fractional components contained in three-sections of fourindex matrices representing the polyhedron vertices.

SummaryThe Isoperimetric Quotient, or IQ, introduced by G. Polya, characterizes the degree of sphericity of a convex solid. This paper obtains closed form expressions for the surface area and volume of any Archimedean polyhedron in terms of the integers specifying the type and number of regular polygons occurring around each vertex. Similar results are obtained for the Catalan solids, which are the duals of the Archimedeans. These results are used to compute the IQs of the Archimedean and Catalan solids and it is found that nine of them have greater sphericity than the truncated icosahedron, the solid which serves as the geometric framework for a molecule of C-60, or “Buckyball.”

Dense packings have served as useful models of the structure of liquid, glassy and crystal states of matter, granular media, heterogeneous materials, and biological systems. Probing the symmetries and other mathematical properties of the densest packings is a problem of long-standing interest in discrete geometry and number theory. The preponderance of previous work has focused on spherical particles, and very little is known about dense polyhedral packings. We formulate the problem of generating dense packings of polyhedra within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem, which we call the Adaptive Shrinking Cell (ASC) scheme. This novel optimization problem is solved here (using a variety of multi-particle initial configurations) to find dense packings of each of the Platonic solids in three-dimensional Euclidean space. We find the densest known packings of tetrahedra, octahedra, dodecahedra and icosahedra with densities $0.782...$, $0.947...$, $0.904...$, and $0.836...$, respectively. Unlike the densest tetrahedral packing, which must be a non-Bravais lattice packing, the densest packings of the other non-tiling Platonic solids that we obtain are their previously known optimal (Bravais) lattice packings. Our simulations results, rigorous upper bounds that we derive, and theoretical arguments lead us to the strong conjecture that the densest packings of the Platonic and Archimedean solids with central symmetry are given by their corresponding densest lattice packings. This is the analog of Kepler's sphere conjecture for these solids., 18 pages, 4 figures. In the published version of this paper in Nature (described below), "P6" should be "A1" in Figure 1

We introduce the rainbowness of a polyhedron as the minimum number k such that any colouring of vertices of the polyhedron using at least k colours involves a face all vertices of which have different colours. We determine the rainbowness of Platonic solids, prisms, antiprisms and ten Archimedean solids. For the remaining three Archimedean solids this parameter is estimated.

It is customarily asserted that the Archimedean solids were rediscovered by Renaissance artists such as Piero della Francesca (ca. 1412-1492), Albrecht Durer (1471-1528), and Daniele Barbaro (1513-1570) step by step without knowledge of the ancient prehistory, which is only reported by Pappos in book V of his collectio (about 320 A.D.) and afterwards led to the modern name Archimedean solids. This story is well told in a fundamental paper (Field 1997). Let us summarize what has been said and generally accepted until now: 1 . Renaissance people were not, or at least not seriously, interested in the "combinatorial regularity"1 of these solids but merely in their ball-like shape, i.e. in the existence of a circumscribed sphere. In particular Durer in his Underweysung['525] pointed repeatedly to the fact that the solids treated by him riiren in einer holen kugel mit all iren ecken cm(touch a hollow sphere with all their vertices). Consequently alongside some Archimedean solids they also designed and studied some nonArchimedean solids. For example, Pacioli and Durer designed a "polyhedronal approximation" of the sphere with piecewise linear meridians and parallel circles, and the famous polyhedron in Durer's MELENCOLIA I has a circumscribed sphere (Schreiber 1999).

A new methodology for understanding the construction of polyhedral links has been developed on the basis of the Platonic and Archimedean solids by using our method of the ‘three-cross-curve and double-twist-line covering’. There are five classes of polyhedral links that can be explored: the tetrahedral and truncated tetrahedral links; the hexahedral and truncated hexahedral links; the dodecahedral and truncated dodecahedral links; the truncated octahedral and icosahedral links. Our results show that the tetrahedral and truncated tetrahedral links have T symmetry; the hexahedral and truncated hexahedral links, as well as the truncated octahedral links, O symmetry; the dodecahedral and truncated dodecahedral links, as well as the truncated icosahedral links, I symmetry, respectively. This study provides further insight into the molecular design, as well as theoretical characterization, of the DNA and protein catenanes.

et al., Keplerates are molecules that contain metal polyhedra that describe both Platonic and Archimedean solids; new copper keplerates are reported, with physical studies indicating that even where very high molecular symmetry is found, the low-temperature physics does not necessarily reflect this symmetry. New copper keplerates are reported, with physical studies indicating that even where very high molecular symmetry is found, the low-temperature physics does not necessarily reflect this symmetry., This work was supported by the EPSRC (UK), including the National EPR Facility, the National Crystallographic Service and for funding an X-ray diffractometer (grant number EP/K039547/1). EMP thanks the Panamanian agency SENACYT-IFARHU for funding. ME acknowledges financial support from MINECO through grant MAT2012-38318-C03-01. REPW thanks the Royal Society for a Wolfson Merit Award.JS thanks the Deutsche Forschungsgemeinschaft (SCHN/615-15)for continuous support. CH and JS thank for support through an exchange program between Germany and Bulgaria (DAAD PPP Bulgarien 57085392 &DNTS/Germany/01/2). Supercomputing time at the LRZ Garching is gratefully acknowledged. HN acknowledges financial support through a ICC-IMR project.

It is not too surprising that 5-fold symmetry is the symmetry of quasicrystals since icosahedral coordination is the most frequent atomic environment type (AET) in intermetallic phases. What about AETs with 7-fold symmetry? Whereas no regular or semi-regular polyhedra (Platonic or Archimedean solids) exist with rotational symmetry larger than 5-fold, polyhedra with only axial n-fold symmetry (e.g. pyramids) are possible for arbitrary n. Indeed, (distorted) seven-membered rings are very common in ternary borides and borocarbides such as the YCrB4, ThB4, ThMoB4, Y2ReB6 structure types. The possibility to find heptagonal quasicrystals related to these phases will be discussed.

We develop an analytical model to predict equilibrium shapes of two-component heterogeneous vesicles or capsules. Using a free energy functional including the bending energies of the two components and line tension contributions, the model describes shape transitions between spherical and polyhedral (faceted) states, complementing and extending results of previous numerical simulations. In the parameter space of relative area fraction, bending modulus ratio, and line tension, a region of polyhedral shapes occurs for weak line tension and large bending modulus ratio and is very robust towards changes in the modeling assumptions. At large enough line tension, the spherical shape fragments into two components. Within the polyhedral region, we compare the energies of all regular and semiregular polyhedra, as well as those of arbitrary prismatic shapes. We find that the largest bending modulus contrasts together with larger line tension favor polyhedra with small face number as optimal shapes. In this region, we also demonstrate the counter-intuitive result that the most symmetric polyhedra are not energetically optimal, with specific Archimedean solids and specific prismatic shapes beating more isotropic (e.g. Platonic) polyhedra. Furthermore, all polyhedra of lowest energy are found to be three-fold coordinated. The shape transition boundary for polyhedra can be computed analytically. The model can be utilized to predict heterogeneous vesicle shapes and to estimate physical properties of the components constituting observed vesicles.

We report star polyhedral gold nanocrystals synthesized by colloidal reduction with ascorbic acid in water at ambient conditions. We identify two distinct classes of star nanocrystals: multiple-twinned crystals with fivefold symmetry, and monocrystals. These respective classes correspond to icosahedra and cuboctahedra, two Archimedian solids, with preferential growth of their {1 1 1} surfaces. Due to this preferential growth, the {1 1 1} faces of the original Archimedean solids grow to become tetrahedral pyramids, the base of each pyramid being the original polyhedral face. By assuming a star morphology, gold nanocrystals increase their proportion of exposed {1 1 1} surfaces, which possess the lowest surface energy among low-index crystallographic planes for FCC crystals. Thus, we propose that the driving force for star nanocrystal formation could be the reduction in surface energy that the crystals experience. Interestingly, icosahedrally derived star nanocrystals possess a geometric morphology closely resembling the great stellated dodecahedron, a Kepler-Poinsot solid. r 2005 Elsevier B.V. All rights reserved. PACS: 61.46.+w; 81.07.Bc; 68.37.Lp; 68.37.Hk

Electron l shells of free atoms have been modeled on the basis of regular and semiregular polyhedra by determining stable systems of identical Coulomb particles with an increase in their number. It is shown that such stable systems correspond to the location of the maxima of electron density at the vertices of inversion-center polyhedra forming a sequence in which the first polyhedron has two vertices and the number of vertices in each subsequent polyhedron exceeds the number of vertices in the previous one by four. The electron s, p, d, and f shells are modeled by a dumbbell and trigonal, pentagonal, and heptagonal antiprisms, respectively. Thus, in addition to the quantum-mechanical properties, l shells should have the symmetry properties of these antiprisms. It is believed that consideration of the noncrystallographic (for the three-dimensional Euclidean space) fivefold and seven-fold symmetries of d and f shells of free atoms will make it possible to obtain a unified explanation for a number of phenomena in crystal structures and other ordered media.

We study a subclass of alternating links for which the complete hyperbolic metric can be realised directly by pairwise identification of faces of two ideal hyperbolic polyhedra. Our main result is a characterization of these links: essentially, the corresponding polyhedra are exactly the Archimedean solids with trivalent vertices. Furthermore, we show that the only knots which arise are the two dodecahedral knots, and the figure eight knot.

Our previous studies indicated that convex polyhedron PETs, such as dodecahedral PET, are able to achieve high performances similar to those of S-PETs. A convex polyhedron PET consisting of a couple of flat detectors is much easier to be constructed compared to an S-PET that needs detectors with curved surfaces. Typical convex polyhedrons suitable for PET include the dodecahedron (a regular convex polyhedron with 12 regular pentagonal faces) and the truncated icosahedron (an Archimedean solid with 12 regular pentagonal faces and 20 regular hexagonal faces). In both situations, pentagon-shaped and/or hexagon-shaped detector modules, instead of conventional square-shaped block detector module, are required. We presented two conceptual pentagonal detectors and two conceptual hexagonal detectors based on conventional lost-cost PMT-light-sharing scheme. Monte Carlo simulations were performed to investigate the crystal decoding performances of all four designs. The results show that high quality flood map can be archived by (1) increasing the areas of the PMT layer and (2) applying a light guide with optimized thickness in the detector modules. We conclude that that it is feasible to design and implement low-cost pentagonal or hexagonal detector modules for the convex polyhedron PETs.

Given a polyhedron, the number of its unfolding is obtained by the Matrix-Tree Theorem. For example, a cube has 384 ways of unfolding (i.e., cutting edges). By omitting mutually isomorphic unfoldings, we have 11 essentially different (i.e., nonisomorphic) unfoldings. In this paper, we address how to count the number of nonisomorphic unfoldings for any (i.e., including nonconvex) polyhedron. By applying this technique, we also give the numbers of nonisomorphic unfoldings of all regular-faced convex polyhedra (i.e., Platonic solids, Archimedean solids, Johnson-Zalgaller solids, Archimedean prisms, and antiprisms), Catalan solids, bipyramids and trapezohedra. For example, while a truncated icosahedron (a Buckminsterfullerene, or a soccer ball fullerene) has 375,291,866,372,898,816, 000 (approximately 3.75 ×1020) ways of unfolding, it has 3,127,432,220, 939,473,920 (approximately 3.13 ×1018) nonisomorphic unfoldings. A truncated icosidodecahedron has 21,789,262,703,685,125,511,464,767,107, 171,876,864,000 (approximately 2.18 ×1040) ways of unfolding, and has 181,577,189,197,376, 045,928,994,520,239,942,164,480 (approximately 1.82 ×1038) nonisomorphic unfoldings.

What is a polyhedron? The question is short, the answer is long. Although you may never have heard of the Polyhedron Kingdom before, it is nearly as vast and as varied as the animal, mineral, and vegetable kingdoms (and it overlaps all three of them). There are aristocrats and workers, families and individuals, old polyhedra with long and interesting histories and young polyhedra born yesterday or the day before. In this kingdom you can take a walking tour of polyhedral architecture, visit a nature preserve and an art gallery and an artisans’ polyhedra fair. As you stroll along you may even glimpse polyhedral ghosts from four-dimensional space.

We have recently devised organizing principles to obtain maximally dense packings of the Platonic and Archimedean solids, and certain smoothly-shaped convex nonspherical particles [Torquato and Jiao, Phys. Rev. E 81, 041310 (2010)]. Here we generalize them in order to guide one to ascertain the densest packings of other convex nonspherical particles as well as concave shapes. Our generalized organizing principles are explicitly stated as four distinct propositions. We apply and test all of these organizing principles to the most comprehensive set of both convex and concave particle shapes to date, including Catalan solids, prisms, antiprisms, cylinders, dimers of spheres and various concave polyhedra. We demonstrate that all of the densest known packings associated with this wide spectrum of nonspherical particles are consistent with our propositions. Among other applications, our general organizing principles enable us to construct analytically the densest known packings of certain convex nonspherical particles, including spherocylinders, "lens-shaped" particles, square pyramids and rhombic pyramids. Moreover, we show how to apply these principles to infer the high-density equilibrium crystalline phases of hard convex and concave particles. We also discuss the unique packing attributes of maximally random jammed packings of nonspherical particles., Comment: 37 pages, 14 figures

Albrecht Durer's Nets In 1525, the German painter and thinker Albrecht Durer published his masterwork on geometry, whose title translates as “On Teaching Measurement with a Compass and Straightedge.” The fourth part of this work concentrates on polyhedra: the Platonic solids, the Archimedean solids, and several polyhedra “discovered” by Durer himself. Figure 7.1 shows his famous engraving, “Melencolia I,” in which he used a polyhedron of his own invention a decade earlier. His book presented each polyhedron by drawing a net for it: an unfolding of the surface to a planar layout. The net makes the geometry of the faces and the number of each type of face immediately clear to the eye in a way that a 3D drawing, which necessarily hides part of the polyhedron, does not. Moreover, a net almost demands to be cut out and folded to form the 3D polyhedron. Figures 7.2 and 7.3 show two examples of Durer's nets. The first is a net of the snub cube , which consists of six squares and 32 equilateral triangles. The second is a net of the truncated icosahedron, consisting of 12 regular pentagons and 20 regular hexagons. We know the spherical version of this polyhedron as a soccer ball. Durer's nets, an apparently original representational invention, have since become a standard presentation method for describing polyhedra. For example, Figure 7.4 shows a modern display of nets for the so-called Archimedean (or semiregular ) solids.

Section 2.1 discusses the vectors and their use in analytic geometry. In Section 2.2 we study rigid motions of \({\mathbb{R}}^{n}\) (n≥2), including the complex and quaternionic approaches. Section 2.3 is devoted to the geometry on a sphere (induced by Euclidean geometry of the space), it and also introduces the stereographic projection. The study of polyhedra is organized in Section 2.4 in into the a sequence of themes: the notion of a polyhedron, Platonic solids, symmetries of geometrical figures, star-shaped polyhedra, and Archimedean solids. Section 2.5 (Appendix) surveys matrices and groups.

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

A packing is an arrangement of non-interpenetrable objects touching each other. In crystallography, dense packings play a special role for crystal structure modeling. Depending on the structure, it may be useful to describe it as packing of atoms or of larger structural units such as chains, columns, bands, layers or polyhedra. In this chapter, we will focus on polyhedra and their ways of space-filling packing. The emphasis is on periodic packings of Platonic and Archimedean solids. Quasiperiodic packings will be discussed on the example of polyhedra with icosahedral symmetry, in particular of triacontahedra and its related zonohedra.

The paper presents an interactive visualization tool, Archimedean Kaleidoscope (AK), aimed at supporting a learner's exploration of polyhedra. AK uses metamorphosis as a technique to help support the learner's mental construction of relationships among different polyhedra. AK uses the symmetric nature of the platonic solids as the foundation for exploring the way in which polyhedra are related. The high level of interactivity helps support the exploration of these relationships.

The radius of gyration (Rg) is one of the most common parameters to be extracted from small-angle X-ray/neutron scattering (SAXS, SANS) measurements of nanoparticles and combines information about size, shape, symmetry and homogeneity in one single value. The analytical expressions for Rg are well known for simple geometric shapes (spheres, ellipsoids, cylinders, cubes). In this work, the analytical equations for Rg for other homogeneous (constant electron or scattering length density) shapes like cones, pyramids, paraboloids, hemispheres or tori are derived and are compiled in this poster. In this approach, the Rg of different 3-dimensional objects can be composed of a 2-dimensional cross-sectional (Rc) and of a perpendicular (h) contribution. Thus, Rg2is the linear sum of both: Rg2= f1*Rc2+ f2*h2, with h being the height or diameter of the object in the perpendicular direction to the cross-section and f1 and f2 being multiplicative factors with values depending on the geometric shape. The cross-sectional area can be (semi-)circular, (semi-)elliptic, n-polygonal or rhombic, resulting in a conical, pyramidal, ellipsoidal or paraboloidal 3D-shape, depending on the perpendicular component. A mirror-symmetry in the cross-sectional plane may be present (e.g. ellipsoids, bi-cones or bi-pyramids) or absent (e.g. hemispheres or single cones or pyramids). General equations of Rc for regular (equilateral) n-polygons will be given, but also for non-equilateral polygonal (rectangular, triangular) and rhombic cross-sections. Furthermore, the analytical equations of Rg of nanoscaled particles of high symmetry, in particular of convex polyhedra like the 5 Platonic solids (tetra-, hexa-, octa-, dodeca- and icosa-hedron) or the 13 Archimedean solids and their duals (Catalan solids) are presented, for the solid, for the hollow (faces only) and as well as for the skeletal (edges only) and dot (vertices only) shape.

How can identical particles be crammed together as densely as possible? A combination of theory and computer simulations shows how the answer to this intricate problem depends on the shape of the particles. Models based on knowledge of the geometry of dense particle packing help explain the structure of many systems, including liquids, glasses, crystals, granular media and biological systems. Most previous work in this area has focused on spherical particles, but even for this idealized shape the problem is notoriously difficult — Kepler's conjecture on the densest packing of spheres was proved only in 2005. Little is known about the densest arrangements of the 18 classic geometric shapes, the Platonic and Archimedean solids, though they have been known since the time of the Ancient Greeks. Salvatore Torquato and Yang Jiao now report the densest known packings of the 5 Platonic solids (tetrahedron, cube, octahedron, dodecahedron and icosahedron) and 13 Archimedean polyhedra. The symmetries of the solids are crucial in determining their fundamental packing arrangements, and the densest packings of Platonic and Archimedean solids with central symmetry are conjectured to be given by their corresponding densest (Bravais) lattice packings.

Structures of polyoxometalates frequently are discovered to be based upon regular convex polyhedra, including the Platonic and Archimedean solids. A topological approach involving barycentric subdivision of the faces of such polyhedra, leads to their description as combinations of triangular building blocks assembled according to systematic rules. An analysis of the Keggin structure of [Mo12O36(PO4)]3-, is presented. As it turns out, it is the only spherical polyhedral structure of T symmetry built up from six 8-gons and eight 6-gons. Similarly, there is only one spherical polyhedral structure of T symmetry built up from eight 6-gons and twenty-four 4-gons satisfying also some obvious chemical combinatorial constraints. Such a structure is observed for [H9V18O42(VO4)]6-. Analysis of possible structures of lower symmetry (D3, D4), e.g. as observed for [V15O36(Hal)]6- and [H4V18O42(Hal)]9-, reveals the onset of combinatorial explosion. For example, there are 67 D3-structures satisfying the chemical condition.

We note that all regular and semiregular polytopes in arbitrary dimensions can be obtained from the Coxeter-Dynkin diagrams. The vertices of a regular or semiregular polytope are the weights obtained as the orbit of the Coxeter-Weyl group acting on the highest weight representing a selected irreducible representation of the Lie group. This paper, in particular, deals with the determination of the vertices of the Platonic and Archimedean solids from the Coxeter diagrams A3, B3, and H3 in the context of the quaternionic representations of the root systems and the Coxeter-Weyl groups. We use Lie algebraic techniques in the derivation of vertices of the polyhedra and show that the polyhedra possessing the tetrahedral, octahedral, and icosahedral symmetries are related to the Coxeter-Weyl groups representing the symmetries of the diagrams of A3, B3, and H3, respectively. This technique leads to the determination of the vertices of all Platonic and Archimedean solids except two chiral polyhedra, snubcuboctahedr...

Closed geometrical shapes may be obtained from a hexagonal tiling by substituting 12 pentagons for hexagons and fusing the appropriate edges. Such geometrical shapes describe the fullerenes, demonstrating that this mathematical rule has utility in structural chemistry. Closure from a hexagonal tiling may also be obtained by substituting six squares; or four triangles; or combinations of pentagons (one point each), squares (two points each), and triangles (three points each) so that the total is 12 points. Similar recipes exist for obtaining closed shapes from tetragonal or trigonal tilings. Structures thus obtained may evolve into additional structures having the same symmetry in a manner similar to the evolution of the Archimedean solids from the Platonic solids. Important examples include the deltahedra, a group of eight convex polyhedra consisting of all triangles and belonging to high-symmetry point groups. These polyhedra evolve into additional high symmetry shapes, many of which have already found u...

We construct a polyhedron with ten vertices of genus three which has three axes of symmetry. It is as symmetric as possible. Ten is the minimal number of vertices which a polyhedron of genus three can have. A modification of our polyhedron yields a symmetric polyhedral realization of Dyck's regular map.

The uniform polyhedra form a fascinating set of objects for display on graphics output devices. When animated on graphics screens they help in visualization of three-dimensional structures and in understanding the geometrical properties and relationships between the various polyhedra. However, the coordinates of the vertices of these objects—the Platonic solids, the Archimedean solids, the Johnson solids and the starred regular polyhedra—are mostly very difficult and tedious to determine algebraically. When the objective is simply to display these solids on a graphics screen, rather than explicit coordinate calculation, another approach can be adopted: the use of jet graphics software. Turtle graphics provides a compact method of describing and displaying the two-dimensional analogies of these solids, and this research investigated to what extent three-dimensional turtle graphics, also known as jet graphics, could describe and also display the regular three-dimensional polyhedra. A number of methods applicable to many of the solids were invented. In particular, methods were found that apply to the infinite classes of prisms and anti-prisms, pyramids and dipyramids, cupolas, bicupolas and gyrobicupolas as well as some starred versions of these polyhedra. A new class of polyhedra, the star antiprisms, are shown to be facially equilateral only for the case when the parallel faces correspond to pentagrams. A single iterative algorithm for the finite class of convex regular polyhedra was found. However, this method applies to all Platonic solids except the icosahedron, and generalizes the Petrie polygon rings to form new infinite classes of polyhedra such as Petrie-prisms. A recursive algorithm is described which correctly displays all regular polyhedra and is generalizable to semiregular and other classes of polyhedra.