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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. .

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

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.

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.

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]

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.

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

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

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

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

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

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

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

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

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

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Unlike the situation in the classical theory of convex polytopes, there is a wealth of semi-regular abstract polytopes, including interesting examples exhibiting some unexpected phenomena. We prove that even an equifacetted semi-regular abstract polytope can have an arbitrary large number of flag orbits or face orbits under its combinatorial automorphism group., Comment: Glasnik Matematicki, to appear, 10 pp

Semi-Equivelar maps are generalizations of maps on the surface of Archimedean Solids to surfaces other than \(2\)-Sphere. We classify some semi-equivelar maps on surface of Euler characteristic \(-1\) and show that none of these are vertex transitive. We establish existence of \(12\)-covered triangulations for this surface. We further construct double cover of these maps to show existence of semi-equivelar maps on the surface of double torus. We also construct several semi-equivelar maps on the surfaces of Euler characteristics \(-8\) and \(-10\) and on non-orientable surface of Euler characteristics \(-2\).

The archimedean solids Cubus simus (snub cube) and Dodecaedron simum (snub dodecahedron) cannot be constructed by ruler and compass. We explain that for general reasons their vertices can be constructed via paper folding on the faces of a cube, respectively dodecahedron, and we present explicit folding constructions. The construction of the Cubus simus is particularly elegant. We also review and prove the construction rules of the other Archimedean solids., Comment: 13 pages, 12 figures, v2: as published in Geometriae Dedicata

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.

Supramolecular coordination cages provide unique restricted inner cavities that can be exploited for molecular recognition purposes and catalysis. Their syntheses often involve complex self-organization processes and rely on the identification of preorganized, kinetically stable building units that provide ligand-accessible coordination sites. Here we report a highly effective protocol for the successive buildup of symmetrical nanoscopic polyoxometalate (POM) cages. Our methodology takes advantage of a supramolecular templating effect and utilizes the structure-directing influence of octahedral {X(x)(H(2)O)(6-x)} (X = Br(-), Cl(-); x = 2, 4, 6) assemblies that reside inside the hollow cluster shells and determine the arrangement of di- and tetranuclear vanadate units. The approach allows the preparation of a series of high-nuclearity POM cages that are characterized by {V(16)As(8)}, {V(16)As(10)}, {V(20)As(8)}, and {V(24)As(8)} core structures. In the latter cluster cage, the vanadium centers adopt a truncated octahedral topology. The formation of this Archimedean body is the direct result of the assembly of six square {V(4)O(8)} units that cap the vertices of the encapsulated Platonic {Cl(6)} octahedron. To the best of our knowledge, this {V(24)As(8)} cage is the largest hybrid vanadate cluster reported to date.

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

The spherical capsule skeleton of the host–guest system [{Mo6O19}2−⊂{MoVI72FeIII30O252(CH3COO)20(H2O)92}]4−1a—built up by 12 {(MoVI)MoVI5} type pentagonal units linked by 30 FeIII centers which span the unique icosahedral Archimedean solid, the icosidodecahedron—can now be constructed deliberately and with a simpler composition than before from an acidified aqueous molybdate solution containing the mentioned (virtual) pentagonal units; the encapsulated hexamolybdate—normally not formed in water—is built up in an unprecedented way concomitant with capsule growth, while being directed by the corresponding internal electrophilic surface functionalities.

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.

The problem of classifying orientable vertex-transitive maps on a surface with genus two is considered. We construct and classify all simple orientable vertex-transitive maps, with face width at least 3 which can be viewed as generalisations of classical Archimedean solids. The proof is computer-aided. The developed method applies to higher genera as well.

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.

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.

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

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.

Colloidal self-assembly has enormous potential as a bottom-up means of structure fabrication. Here we demonstrate hierarchical self-assembly of rationally designed charge-stabilised colloidal magnetic particles into ground state structures that are topologically equivalent to a snub cube and a snub dodecahedron, the only two chiral Archimedean solids, for size-selected clusters. These spherical structures open up in response to an external magnetic field and demonstrate controllable porosity. Such features are critical to their applications as functional materials.

Archimedes is generally regarded as the greatest mathematician of antiquity and alongside Isaac Newton and C F Gauss as the top three of all times. He was also an excellent theoreticiancum-engineer who identified mathematical prob lems in his work on mechanics, got hints on their solution through engineering techniques and then solved those mathematical problems, many a time discovering fundamental results in mathematics, for instance, the concepts oflimits andintegration. In his own words,“… which I first dis covered by means of mechanics and then exhibited by means of geometry”. In this article we briefly describe some of his main contributions to mathematics.

Modern nanoparticle research in the field of small metallic systems has confirmed that many nanoparticles take on some Platonic and Archimedean solids related shapes. A Platonic solid looks the same from any vertex, and intuitively they appear as good candidates for atomic equilibrium shapes. A very clear example is the icosahedral ( Ih) particle that only shows {111} faces that contribute to produce a more rounded structure. Indeed, many studies report the Ihas the most stable particle at the size range r≤20 Å for noble gases and for some metals. In this review, we report on the structure and shape of mono- and bimetallic nanoparticles in the wide size range from 1–300 nm. First, we present AuPd nanoparticles in the 1–2 nm size range that show dodecahedral atomic growth packing, one of the Platonic solid shapes that have not been identified before in this small size range for metallic particles. Next, with particles in the size range of 2–5 nm, we present an energetic surface reconstruction phenomenon observed also on bimetallic nanoparticle systems of AuPd and AuCu , similar to a re-solidification effect observed during cooling process in lead clusters. These binary alloy nanoparticles show the fivefold edges truncated, resulting in {100} faces on decahedral structures, an effect largely envisioned and reported theoretically, with no experimental evidence in the literature before. Next nanostructure we review is a monometallic system in the size range of ≈5 nm that we termed the decmon. We present here some detailed geometrical analysis and experimental evidence that supports our models. Finally, in the size range of 100–300 nm, we present icosahedrally derived star gold nanocrystals which resembles the great stellated dodechaedron, which is a Kepler–Poisont solid. We conclude then that the shape or morphology of some mono- and bimetallic particles evolves with size following the sequence from atoms to the Platonic solids, and with a slightly greater particle's size, they tend to adopt Archimedean related shapes. If the particle's size is still greater, they tend to adopt shapes beyond the Archimedean (Kepler–Poisont) solids, reaching at the very end the bulk structure of solids. We demonstrate both experimentally and by means of computational simulations for each case that this structural atomic growth sequence is followed in such mono- and bimetallic nanoparticles.

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

A covalently closed molecular complex whose double-helical edges have the connectivity of a truncated octahedron has been assembled from DNA on a solid support. This three-connected Archimedean solid contains six squares and eight hexagons, formed from 36 edges arranged about 24 vertices. The vertices are the branch points of four-arm DNA junctions, so each vertex has an extra exocyclic arm associated with it. The construct contains six single-stranded cyclic DNA molecules that form the squares and the extra arms; in addition, there are eight cyclic strands that correspond to the eight hexagons. The molecule contains 1440 nucleotides in the edges and 1110 in the extra arms; the estimated molecular weight for the 2550 nucleotides in the construct is 790 kDa