The Building Game: From Enumerative Combinatorics to Conformational Diffusion

Understanding the nature of dense particle packings is a subject of intense research in the physical, mathematical, and biological sciences. 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 nonoverlapping, nontiling 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 optimization problem is solved here (using a variety of multiparticle 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 nontiling 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. Provided that what we term the "asphericity" (ratio of the circumradius to inradius) is sufficiently small, the upper bounds are relatively tight and thus close to the corresponding densities of the optimal lattice packings of the centrally symmetric Platonic and Archimedean solids. 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. This can be regarded to be the analog of Kepler's sphere conjecture for these solids.The truncated tetrahedron is the only nonchiral Archimedean solid that is not centrally symmetric [corrected], the densest known packing of which is a non-lattice packing with density at least as high as 23/24=0.958 333... . We discuss the validity of our conjecture to packings of superballs, prisms, and antiprisms as well as to high-dimensional analogs of the Platonic solids. In addition, we conjecture that the optimal packing of any convex, congruent polyhedron without central symmetry generally is not a lattice packing. Finally, we discuss the possible applications and generalizations of the ASC scheme in predicting the crystal structures of polyhedral nanoparticles and the study of random packings of hard polyhedra.

Two-point resistances in Archimedean resistor networks

Dense particle packings have served as useful models of the structures of liquid, glassy and crystalline 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 interest in discrete geometry and number theory. Previous work has focused mainly on spherical particles-very little is known about dense polyhedral packings. Here we formulate the generation of dense packings of polyhedra as an optimization problem, using an adaptive fundamental cell subject to periodic boundary conditions (we term this the 'adaptive shrinking cell' scheme). Using a variety of multi-particle initial configurations, we find the densest known packings of the four non-tiling Platonic solids (the tetrahedron, octahedron, dodecahedron and icosahedron) in three-dimensional Euclidean space. The densities are 0.782..., 0.947..., 0.904... and 0.836..., respectively. 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. Combining our simulation results with derived rigorous upper bounds and theoretical arguments leads us to the 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 analogue of Kepler's sphere conjecture for these solids.

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.

Catalan solids are the duals of the Archimedean solids, the vertices of which can be obtained from the Coxeter–Dynkin diagrams A3, B3, and H3 whose simple roots can be represented by quaternions. The respective Weyl groups W(A3), W(B3), and W(H3) acting on the highest weights generate the orbits corresponding to the solids possessing these symmetries. Vertices of the Platonic and Archimedean solids result from the orbits derived from fundamental weights. The Platonic solids are dual to each other; however, the duals of the Archimedean solids are the Catalan solids whose vertices can be written as the union of the orbits, up to some scale factors, obtained by applying the above Weyl groups on the fundamental highest weights (100), (010), and (011) for each diagram. The faces are represented by the orbits derived from the weights (010), (110), (101), (011), and (111), which correspond to the vertices of the Archimedean solids. Representations of the Weyl groups W(A3), W(B3), and W(H3) by the quaternions simplify the calculations with no reference to the computer calculations.

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.

What Exactly does a Random Walk Simulation Simulate?

A copolymer is a chain of repetitive units (monomers) that are almost identical, but they differ in their degree of affinity for certain solvents. This difference leads to striking phenomena when the polymer fluctuates in a nonhomogeneous medium, for example, made of two solvents separated by an interface. One may observe, for instance, the localization of the polymer at the interface between the two solvents. A discrete model of such system, based on the simple symmetric random walk on $\mathbb {Z}$, has been investigated in [Bolthausen and den Hollander, Ann. Probab. 25 (1997), 1334-1366], notably in the weak polymer-solvent coupling limit, where the convergence of the discrete model toward a continuum model, based on Brownian motion, has been established. This result is remarkable because it strongly suggests a universal feature of copolymer models. In this work, we prove that this is indeed the case. More precisely, we determine the weak coupling limit for a general class of discrete copolymer models, obtaining as limits a one-parameter [$\alpha \in(0,1)$] family of continuum models, based on $\alpha$-stable regenerative sets.

In 1525, the German painter and thinker Albrecht Dürer 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 Dürer 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 Dürer'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.

For a random walk defined for a doubly infinite sequence of times, we let the time parameter itself be an integer-valued process, and call the orginal process a random walk at random time. We find the scaling limit which generalizes the so-called iterated Brownian motion. Khoshnevisan and Lewis [Ann. Appl. Probab. 9 (1999) 629-667] suggested "the existence of a form of measure-theoretic duality" between iterated Brownian motion and a Brownian motion in random scenery. We show that a random walk at random time can be considered a random walk in "alternating" scenery, thus hinting at a mechanism behind this duality. Following Cohen and Samorodnitsky [Ann. Appl. Probab. 16 (2006) 1432-1461], we also consider alternating random reward schema associated to random walks at random times. Whereas random reward schema scale to local time fractional stable motions, we show that the alternating random reward schema scale to indicator fractional stable motions. Finally, we show that one may recursively "subordinate" random time processes to get new local time and indicator fractional stable motions and new stable processes in random scenery or at random times. When $\alpha=2$, the fractional stable motions given by the recursion are fractional Brownian motions with dyadic $H\in(0,1)$. Also, we see that "un-subordinating" via a time-change allows one to, in some sense, extract Brownian motion from fractional Brownian motions with $H<1/2$.

While physical models are classical, random walks are statistical models. Historically, Karl Pearson was the first to propose the random walk. This study will focus on the essential of the random walk, and additionally, this essay will offer some application guidelines and explain when the random walk is applicable. The random walk means that the performance of an event is based on the past, as people cannot predict the future development steps and directions. The random walk, which dates to the 16th century, is a significant area of probability with many applications. The random walk and Brownian motion are the main topics of this paper. The four applications of random walks that are highlighted in this paper are the following. They include the multi-particle random walk immune algorithm, the density peak clustering algorithm, a review of random walk-based community discovery techniques, and interactive image segmentation that combines random walk and random forest. This work paves the way to unravel the importance of random walk.

Modern research in the field of small metallic systems has confirmed that many nanoparticles take 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 good example is the icosahedral (Ih) particle that only shows {111} faces that produce a more rounded structure. Indeed, many studies report the Ih as the most stable particle at the size range r≤20 A for noble gases and for some metals. In this chapter, we discuss the structure and shape of mono- and bimetallic nanoparticles in the size range from 1–300 nm. First, AuPd nanoparticles (1–2 nm) that show dodecahedral atomic growth packing. Next, in the range of 2–5 nm, we discuss a surface reconstruction phenomenon observed also on AuPd and AuCu nanoparticles. 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, we review a monometallic system (≈5 nm) that we termed the decmon. Finally, we present icosahedrally derived star gold nanocrystals (100–300 nm) which resemble the great stellated dodechaedron, a Kepler-Poisont solid. We conclude that the shape or morphology of some mono- and bimetallic particles evolves with size following the sequence from atoms to the Platonic solids. As the size increases, they tend to adopt Archimedean related shapes and then beyond the Archimedean (Kepler-Poisont) solids, up to the bulk structure of solids.

The construction of Bell inequalities based on Platonic and Archimedean solids (Quantum 4 (2020), 293) is generalized to the case of orbits generated by the action of some finite groups. A number of examples with considerable violation of Bell inequalities is presented.

This chapter is about polyhedra. A ‘polyhedron’ is a three-dimensional object with polygons as its faces (such as a cube with its six square faces). Examples include the five regular polyhedra or ‘Platonic solids’ (the tetrahedron, cube, octahedron, dodecahedron, and icosahedron) whose faces are polygons of the same type, and the thirteen semi-regular polyhedra or ‘Archimedean solids’, where each face is a regular polygon but the faces are not all the same. This chapter also presents some other types of polyhedra, and comments on their appearance in nature as crystals or chemical molecules (such as ‘fullerenes or ‘buckyballs’). The numbers V of vertices, E of edges, and F of faces of a polyhedron are related by ‘Euler’s polyhedron formula’V−E+F=2. This chapter presents the history of this equation, outlines a proof of it, and uses it to prove that there are five regular polyhedra, and that polyhedra consisting only of pentagons and hexagons with three polygons at each vertex (such as a soccer ball) have exactly twelve pentagons. This chapter concludes with a surprising result on the number of regions formed by joining points on a circle.

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