The Isometry Group of Tetrakis Hexahedron and Disdyakis Dodecahedron Spaces

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.

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.

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.

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.

The theory of convex sets is a vibrant and classical field of modern mathematics with rich applications. The more geometric aspects of convex sets are developed introducing some notions, but primarily polyhedra. A polyhedra, when it is convex, is an extremely important special solid in . Some examples of convex subsets of Euclidean 3-dimensional space are Platonic Solids, Archimedean Solids and Archimedean Duals or Catalan Solids. There are some relations between metrics and polyhedra. For example, it has been shown that cube, octahedron, deltoidal icositetrahedron are maximum, taxicab, Chinese Checker’s unit sphere, respectively. In this study, we give two new metrics to be their spheres truncated truncated dodecahedron and truncated truncated icosahedron.

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.”

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.

Unpacking this question requires defining its four technical terms. A polygon P is a planar shape whose boundary is composed of straight segments. It is a single-piece shape that could be cut out from a piece of paper by straight scissors cuts. A polyhedron Q is the 3D analog of a 2D polygon. It is a solid in space whose boundary is composed of polygonal faces. As we are concerned mainly with this surface boundary, we will use Q to refer to the surface rather than the solid. A convex polyhedron is one without dents or indentations. Examples include the five “regular” Platonic solids (tetrahedron, octahedron, cube, dodecahedron, icosahedron), the thirteen “semi-regular” Archimedean solids (truncated icosahedron (i.e., a soccer ball), etc.), or any of an infinite variety of irregular convex polyhedra. As we will only discuss convex polyhedra in this article, the “convex” qualification will be often left implicit. Finally, to fold a polygon to a polyhedron means to crease the polygon and fold it into 3D so that it forms precisely the surface of the polyhedron, without any wrapping overlap, and without leaving any gaps. Another way to view this is in reverse: a polygon P can fold to a polyhedron Q if Q could be cut open and unfolded flat to P . Two examples are shown in Figure 1. Note from (a) that creases of P , which become edges of Q, do not necessarily begin or end at vertices (corners) of P . Note from (b) that a nonconvex polygon might fold to a convex polyhedron. In the alternative cut-open-and-unfold view, the cuts in both these examples are along polyhedron edges. We will see that in general the cuts are arbitrary surface segments.

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, snubcuboctahedron and snubicosidodecahedron.

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.

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.

Simple geometry in complex organisms