Archimedean Solids Research Articles

Notes on the illumination parameters of convex bodies

Convex bodies with large illumination parameters are constructed in each dimension. The exact values of the illumination parameters of the centrally symmetric Platonic solids are calculated, and estimates on the illumination parameters of the centrally symmetric Archimedean solids are given.

Effects of temperature and ferromagnetism on the γ-Ni/γ′-Ni3Al interfacial free energy from first principles calculations

The temperature dependencies of the γ(f.c.c.)-Ni/γ′-Ni3Al(L12) interfacial free energy for the {100}, {110}, and {111} interfaces are calculated using first-principles calculations, including both coherency strain energy and phonon vibrational entropy. Calculations performed including ferromagnetic effects predict that the {100}-type interface has the smallest free energy at different elevated temperatures, while alternatively the {111}-type interface has the smallest free energy when ferromagnetism is absent; the latter result is inconsistent with experimental observations of γ′-Ni3Al-precipitates in Ni–Al alloys faceted strongly on {100}-type planes. The γ(f.c.c.)-Ni/γ′-Ni3Al interfacial free energies for the {100}, {110}, and {111} interfaces decrease with increasing temperature due to vibrational entropy. The predicted morphology of γ′-Ni3Al(L12) precipitates, based on a Wulff construction, is a Great Rhombicuboctahedron (or Truncated Cuboctahedron), which is one of the 13 Archimedean solids, with 6-{100}, 12-{110}, and 8-{111} facets. The first-principles calculated morphology of a γ′-Ni3Al(L12) precipitate is in agreement with experimental three-dimensional atom-probe tomographic observations of cuboidal L12 precipitates with large {100}-type facets in a Ni-13.0 at.% Al alloy aged at 823 K for 4096 h. At 823 K this alloy has a lattice parameter mismatch of 0.004 ± 0.001 between the γ(f.c.c.)-Ni-matrix and the γ′-Ni3Al-precipitates.

Nano-scaling law: geometric foundation of thiolated gold nanomolecules

Thiolated gold nanomolecules show a power correlation between the number of gold atoms and the thiolate ligands with a 2/3 scaling similar to Platonic and Archimedean solids. Nanomolecule stability is influenced by a universal geometric factor that is foundational to its stability through the Euclidean surface rule, in addition to the electronic shell closing factor and staple motif requirements.

Design Optimization for DNA Nanostructures

This paper is concerned with minimizing the cost of self-assembling DNA nanostructures by minimizing the number of different components used in the construction. We first describe the nanostructures, then give a combinatorial formalization of the assembly process and demonstrate that the octet truss provides an accurate geometric framework for current branched junction molecule assembly. We choose the octet truss because it is highly symmetric and has an appropriate number of edges for the application. We develop a method of differentiating among branched junction molecules, the basic building blocks of the nanostructures, within this structure. In the mathematical model, we represent the branched junction molecules graphically with „tiles‟. We use this approach to find the minimum number of tiles necessary to construct Platonic and Archimedean solids naturally occurring within the octet truss. This will be useful and cost efficient for the chemists and biologists who actually build these branched junction molecules because once a branched junction molecule is created, a lab can make many copies of it.

How Spherical Are the Archimedean Solids and Their Duals?

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.”

Konstrukcije u prostoru Platonskih i Arhimedskih čvrstih tela

This paper gives a classification of Platonic and Archimedean solids related to more-dimensional cubes. A general infinite net structure for space-filling mosaics created from the above solids has been derived and the node elements for grid structures based on our groups of solids have been described within the frame of constructive geometry. The topic is related to arts and design and raises further geometrical questions.

Publisher's Note: Dense packings of polyhedra: Platonic and Archimedean solids [Phys. Rev. E80, 041104 (2009)

Received 15 November 2010DOI:https://doi.org/10.1103/PhysRevE.82.059904©2010 American Physical Society

Erratum: Dense packings of polyhedra: Platonic and Archimedean solids [Phys. Rev. E80, 041104 (2009)

Erratum: Dense packings of polyhedra: Platonic and Archimedean solids [Phys. Rev. E<b>80</b>, 041104 (2009)

Catalan solids derived from three-dimensional-root systems and quaternions

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.

Evolution of gold nanoparticles through Catalan, Archimedean, and Platonic solids

This paper reports the transformation of rhombic dodecahedral gold nanoparticles through two different pathways depending on the water content. The gold nanoparticles were prepared by reducing HAuCl4 with N,N-dimethylforamide in the presence of poly(vinyl pyrrolidone) and water. Not only the concentration of PVP but also the water content has a critical role in determining the shape transformation of the gold nanocrystals. At low water content the rhombic dodecahedron is transformed into a rhombicuboctahedron to a truncated octahedron and then to an octahedron, while at higher water content the rhombic dodecahedron is transformed into a rhombicuboctahedron and then to a cube or a truncated cube and then to a cuboctahedron and to a truncated octahedron. In addition, we demonstrated cell imaging by using the truncated octahedral, rhombic dodecahedral and octahedral gold nanoparticles as a contrast agent.

Dense packings of polyhedra: Platonic and Archimedean solids

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.

Dense packings of the Platonic and Archimedean solids

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.

A tight squeeze

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.

On the 3D Visualisation of Logical Relations

The central aim of this paper is to present a Boolean algebraic approach to the classical Aristotelian Relations of Opposition, namely Contradiction and (Sub)contrariety, and to provide a 3D visualisation of those relations based on the geometrical properties of Platonic and Archimedean solids. In the first part we start from the standard Generalized Quantifier analysis of expressions for comparative quantification to build the Comparative Quantifier Algebra CQA. The underlying scalar structure allows us to define the Aristotelian relations in Boolean terms and to propose a 3D visualisation by transforming a cube into an octahedron. In part two, the architecture of the CQA is shown to carry over, both to the classical quantifiers of Predicate Calculus and to the modal operators—which are given a Generalized Quantifier style re-interpretation. In this way we provide an algebraic foundation for Blanche’s Aristotelian hexagon as well as a 3D alternative to his 2D star-like visualisation. In a final part, a richer scalar structure is argued to underly the realm of Modality, thus generalizing the 3D algebra with eight (23) operators to a 4D algebra with sixteen (24) operators. The visual representation of the latter structure involves a transformation of the hypercube to a rhombic dodecahedron. The resulting 3D visualisation allows a straightforward embedding, not only of the classical Blanche star of Aristotelian relations or the paracomplete and paraconsistent stars of Beziau (Log Investig 10, 218–232, 2003) but also of three additional isomorphic Aristotelian constellations.

On rainbowness of semiregular polyhedra

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.

New light on the rediscovery of the Archimedean solids during the Renaissance

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).

Alicia Boole Stott, a geometer in higher dimension

In this paper we present the life and work of Alicia Boole Stott, an Irish woman who made a significant contribution to the study of four-dimensional geometry. Although she never studied mathematics, she taught herself to “see” the fourth dimension and developed a new method of visualizing four-dimensional polytopes. In particular, she constructed three-dimensional sections of these four-dimensional objects, which resulted in a series of Archimedean solids. The presence in the University of Groningen of an extensive collection of these three-dimensional models, together with related drawings, reveals a collaboration between Boole Stott and the Groningen professor of geometry, P.H. Schoute. This collaboration lasted more than 20 years and combined Schoute's analytical methods with Boole Stott's unusual ability to visualize the fourth dimension. After Schoute's death in 1913 Boole Stott was isolated from the mathematical community until about 1930, when she was introduced to the geometer H.S.M. Coxeter, with whom she collaborated until her death in 1940.

92.24 Counting faces on Archimedean solids

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Architecture of Platonic and Archimedean polyhedral links

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.

Polyhedra obtained from Coxeter groups and quaternions

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.