DOCSLIB.ORG

Phase Behavior of a Family of Truncated Hard Cubes Anjan P

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Phase Behavior of a Family of Truncated Hard Cubes Anjan P THE JOURNAL OF CHEMICAL PHYSICS 142, 054904 (2015) Phase behavior of a family of truncated hard cubes Anjan P. Gantapara,1,a) Joost de Graaf,2 René van Roij,3 and Marjolein Dijkstra1,b) 1Soft Condensed Matter, Debye Institute for Nanomaterials Science, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands 2Institute for Computational Physics, Universität Stuttgart, Allmandring 3, 70569 Stuttgart, Germany 3Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands (Received 8 December 2014; accepted 5 January 2015; published online 5 February 2015; corrected 9 February 2015) In continuation of our work in Gantapara et al., [Phys. Rev. Lett. 111, 015501 (2013)], we inves- tigate here the thermodynamic phase behavior of a family of truncated hard cubes, for which the shape evolves smoothly from a cube via a cuboctahedron to an octahedron. We used Monte Carlo simulations and free-energy calculations to establish the full phase diagram. This phase diagram exhibits a remarkable richness in crystal and mesophase structures, depending sensitively on the precise particle shape. In addition, we examined in detail the nature of the plastic crystal (rotator) phases that appear for intermediate densities and levels of truncation. Our results allow us to probe the relation between phase behavior and building-block shape and to further the understanding of rotator phases. Furthermore, the phase diagram presented here should prove instrumental for guiding future experimental studies on similarly shaped nanoparticles and the creation of new materials. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4906753] I. INTRODUCTION pressures, i.e., at densities below close packing. Some of the first examples of the importance of studying larger (non- Material design based on nanoparticle assemblies have crystalline) assemblies appeared, when it was discovered been at the focus of materials science over the past that there exists a tetrahedron packing into a quasicrystal decade. In particular, the self-assembly of polyhedral colloidal arrangement with close-packed density much higher than nanoparticles into functional materials with targeted properties that of spheres.21,24–26 Simultaneously, the phase diagrams has attracted huge interest. Recent advances in experimental for hard superballs,27,28 a family of truncated tetrahedra,29 techniques led to the synthesis of a wide variety of polyhedron- and a family of truncated cubes30 were established. The shaped particles, such as cubes,1–4 truncated cubes,1,5–7 importance of mesophase structures was further underpinned truncated octahedra,5,8 octahedra,5 tetrahedra,9 superballs,10 by investigations of space filling polyhedra,31 truncated and rhombic dodecahedra.11,12 In addition to controlled cubes32 and bifrustums33 at an interface, and a large number synthesis, the ability to perform self-assembly experiments of polyhedral particles.34 with these polyhedral particles5,10,13–18 has made significant To date, only one experimental investigation of a family strides forward. of truncated particles has been undertaken. Using a polyol This motivated many physicists, mathematicians, and synthesis technique, Henzie et al.5 reported the shape- computer scientists to investigate and try to classify the close- controlled synthesis of monodisperse silver (Ag) nanocrystals packed structures exhibited by these particles. Initially, the including cubes, truncated cubes, cuboctahedra, truncated focus lay on the prediction of the maximum crystalline packing octahedra, and octahedra. They used these polyhedral particles for faceted particles, as these structures are likely to form to study the close-packed crystal structures via sedimentation upon deposition and evaporation, and also have interesting experiments and simulations. Henzie et al. created exotic geometric properties.19–21 Recent extensive investigations of superlattices with potential applications in nanophotonics, many particle shapes demonstrated the importance of shape photocatalysis, and plasmonics. Their results tested several for the high-density (close-packed) structures. In particular, de conjectures on the densest packings of hard polyhedra.19–21,35 Graaf et al.22 investigated the closed packed structures of 142 In addition to the close-packed structure studies in the bulk, convex polyhedra, as well as 17 nonconvex faceted shapes. they also investigated the influence of walls on the sedimented More recently, Chen et al.23 considered over 55 000 convex structures. shapes, using theoretical, numerical, and computational However, Henzie et al. did not examine the finite-pressure methods. behavior of the system. At finite pressures, the structures Advances in computer power and performance have made that form by self-assembly may differ substantially from it possible to perform simulations of these systems with large the packings achieved at high (sedimentation and solvent- numbers of particles and opened up the way for a thorough evaporation) pressures. For instance, simulations of superballs examination of the phase behavior of faceted colloids at finite and of truncated cubes exhibited plastic-crystal phases,28,30,31 while cubes, cuboids, and truncated cubes exhibit vacancy- 30,36,37 a)Electronic mail: [email protected] rich simple cubic and tetrahedra exhibit quasicrystalline b)Electronic mail: [email protected] mesophases.29 In fact, for almost all truncated particle shapes 0021-9606/2015/142(5)/054904/16/$30.00 142, 054904-1 © 2015 AIP Publishing LLC 054904-2 Gantapara et al. J. Chem. Phys. 142, 054904 (2015) studied thus far, the finite-pressure phases and those formed IV B, followed by a discussion of the full phase diagram in under close-packed conditions differ substantially.34 Sec.IVE. In Sec.IVF, we analyze the orientation distributions In this manuscript, we investigate the finite-pressure of particles in the various plastic-crystal phases observed in behavior of particles similar to those considered by Henzie the phase diagram. Finally, we discuss the results and draw et al. Here, we present a thorough investigation of the conclusions in Sec.V. phase behavior of a family of truncated hard cubes, which interpolates smoothly between cubes and octahedra (the mathematical dual of the cube) via cuboctahedra. We describe II. SIMULATION MODEL in detail the different phases, as well as the nature of the phase The particles that we investigated are completely specified transitions between these phases. This work is an extension by the level of truncation of a perfect cube, which we denote by of our previous investigation of these systems, see Ref. 30, s 2 0;1 , and the volume of the particle. We define our family and makes several minor updates on our previous results. In of truncated[ ] cubes using a simple mathematical expression the present paper, we put additional emphasis on the analysis for the location of the vertices. The line segments that connect of the plastic crystal or rotator phases and the computational these vertices can only be assigned in one (unique) way to details. obtain a truncated cube. The vertices of a truncated cube may We used Monte Carlo (MC) simulation studies and free- be written as a function of the shape parameter s 2 0; 1 : energy calculations to establish the phase diagram for this [ ] ! −1=3 ! !T system. This diagram exhibits a remarkably rich diversity 4 ( ) 1 1 1 1− s3 P ± − s ;± ;± in crystal structures that show a sensitive dependence on 3 D 2 2 2 the particle shape. Changes in phase behavior and crystal >8 1 > s 2 0; structures occur even for small variations in the level of > " 2 # v s = > ; (1) truncation. This is an unexpected result, since the particle > ! −1=3 { ( )} > 4 ( ) shape varies smoothly from that of a cube to that of an > −4λ3 P ± 1− λ ; ±λ; 0 T < 3 D octahedron by truncation. We also observed that for specific > ( ( ) ) > 1 levels of truncation, the particles possess an equation of state > λ ≡ 1− s 2 0; > " 2 # (EOS) that exhibits three distinct crystal phases as well as an > > isotropic fluid phase. where PD is: a permutation operation that generates all In addition, we found that for close-to-cubic particles that permutations of each element in the sets of 8 and 4 vertices form a vacancy-rich simple cubic (SC) phase, the equilibrium spanned by the ±-operations, respectively. The duplicate concentration of vacancies increases at a fixed packing fraction vertices that are a consequence of this definition are removed 30 φ upon increasing the level of truncation, see Gantapara et al. after letting PD act. The “T” indicates transposition. The The vacancy concentrations for truncated cubes for small prefactors ensure that the truncated cubes are normalized truncations are in agreement with the vacancy concentration to unit volume. Several Platonic and Archimedeanp solids of perfect cubes.36 However, our results differ from those are members of this family: s = 0 a cube, s = 2 − 2 =2 obtained by Monte Carlo simulations for parallel cuboids, ≈ 0:292 893 a truncated cube, s = 1=2 a cuboctahedron,( s = 2)=3 where the vacancy concentration remains constant, when the a truncated octahedron, and s = 1 an octahedron; these are shape is varied from a perfect cube to a sphere via rounded depicted in Fig. 1(a). cubes (so-called cuboids).37 Furthermore, we analyzed the orientation distribution properties of particles in the plastic-crystal phases observed III. SIMULATION METHODS in the truncated cubes phase diagram. The orientation A. Order parameters and correlations functions distribution function of plastic crystals of hard anisotropic particles is shown to be highly anisotropic and strongly peaked In this subsection, we describe different order parameters for specific orientations. Based on our results, we present that we used to quantify the positional and orientation order of a grouping of particles with different asphericity A values particles in our isothermal-isobaric Monte Carlo simulations according to their cubatic order S4 near (plastic-)crystal-fluid (also called NPT simulations; fixed pressure P, temperature transition regions. We find that particles with asphericity T, and number of particles N) of the truncated cubes.
Load more

Recommended publications
  • Smooth & Fractal Polyhedra Ergun Akleman, Paul Edmundson And
    Smooth & Fractal Polyhedra Ergun Akleman, Paul Edmundson and Ozan Ozener Visualization Sciences Program, Department of Architecture, 216 Langford Center, College Station, Texas 77843-3137, USA. email: [email protected]. Abstract In this paper, we present a new class of semi-regular polyhedra. All the faces of these polyhedra are bounded by smooth (quadratic B-spline) curves and the face boundaries are C1 discontinues every- where. These semi-regular polyhedral shapes are limit surfaces of a simple vertex truncation subdivision scheme. We obtain an approximation of these smooth fractal polyhedra by iteratively applying a new vertex truncation scheme to an initial manifold mesh. Our vertex truncation scheme is based on Chaikin’s construction. If the initial manifold mesh is a polyhedra only with planar faces and 3-valence vertices, in each iteration we construct polyhedral meshes in which all faces are planar and every vertex is 3-valence, 1 Introduction One of the most exciting aspects of shape modeling and sculpting is the development of new algorithms and methods to create unusual, interesting and aesthetically pleasing shapes. Recent advances in computer graphics, shape modeling and mathematics help the imagination of contemporary mathematicians, artists and architects to design new and unusual 3D forms [11]. In this paper, we present such a unusual class of semi-regular polyhedra that are contradictorily interesting, i.e. they are smooth but yet C1 discontinuous. To create these shapes we apply a polyhedral truncation algorithm based on Chaikin’s scheme to any initial manifold mesh. Figure 1 shows two shapes that are created by using our approach.
    [Show full text]
  • Archimedean Solids
    University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-2008 Archimedean Solids Anna Anderson University of Nebraska-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathmidexppap Part of the Science and Mathematics Education Commons Anderson, Anna, "Archimedean Solids" (2008). MAT Exam Expository Papers. 4. https://digitalcommons.unl.edu/mathmidexppap/4 This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in MAT Exam Expository Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Archimedean Solids Anna Anderson In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. Jim Lewis, Advisor July 2008 2 Archimedean Solids A polygon is a simple, closed, planar figure with sides formed by joining line segments, where each line segment intersects exactly two others. If all of the sides have the same length and all of the angles are congruent, the polygon is called regular. The sum of the angles of a regular polygon with n sides, where n is 3 or more, is 180° x (n – 2) degrees. If a regular polygon were connected with other regular polygons in three dimensional space, a polyhedron could be created. In geometry, a polyhedron is a three- dimensional solid which consists of a collection of polygons joined at their edges. The word polyhedron is derived from the Greek word poly (many) and the Indo-European term hedron (seat).
    [Show full text]
  • Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces
    Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces Pablo F. Damasceno1*, Michael Engel2*, Sharon C. Glotzer1,2,3† 1Applied Physics Program, 2Department of Chemical Engineering, and 3Department of Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA. * These authors contributed equally. † Corresponding author: [email protected] Dec 1, 2011 arXiv: 1109.1323v2 ACS Nano DOI: 10.1021/nn204012y ABSTRACT Polyhedra and their arrangements have intrigued humankind since the ancient Greeks and are today important motifs in condensed matter, with application to many classes of liquids and solids. Yet, little is known about the thermodynamically stable phases of polyhedrally-shaped building blocks, such as faceted nanoparticles and colloids. Although hard particles are known to organize due to entropy alone, and some unusual phases are reported in the literature, the role of entropic forces in connection with polyhedral shape is not well understood. Here, we study thermodynamic self-assembly of a family of truncated tetrahedra and report several atomic crystal isostructures, including diamond, β-tin, and high- pressure lithium, as the polyhedron shape varies from tetrahedral to octahedral. We compare our findings with the densest packings of the truncated tetrahedron family obtained by numerical compression and report a new space filling polyhedron, which has been overlooked in previous searches. Interestingly, the self-assembled structures differ from the densest packings. We show that the self-assembled crystal structures can be understood as a tendency for polyhedra to maximize face-to-face alignment, which can be generalized as directional entropic forces.
    [Show full text]
  • The Crystal Forms of Diamond and Their Significance
    THE CRYSTAL FORMS OF DIAMOND AND THEIR SIGNIFICANCE BY SIR C. V. RAMAN AND S. RAMASESHAN (From the Department of Physics, Indian Institute of Science, Bangalore) Received for publication, June 4, 1946 CONTENTS 1. Introductory Statement. 2. General Descriptive Characters. 3~ Some Theoretical Considerations. 4. Geometric Preliminaries. 5. The Configuration of the Edges. 6. The Crystal Symmetry of Diamond. 7. Classification of the Crystal Forros. 8. The Haidinger Diamond. 9. The Triangular Twins. 10. Some Descriptive Notes. 11. The Allo- tropic Modifications of Diamond. 12. Summary. References. Plates. 1. ~NTRODUCTORY STATEMENT THE" crystallography of diamond presents problems of peculiar interest and difficulty. The material as found is usually in the form of complete crystals bounded on all sides by their natural faces, but strangely enough, these faces generally exhibit a marked curvature. The diamonds found in the State of Panna in Central India, for example, are invariably of this kind. Other diamondsJas for example a group of specimens recently acquired for our studies ffom Hyderabad (Deccan)--show both plane and curved faces in combination. Even those diamonds which at first sight seem to resemble the standard forms of geometric crystallography, such as the rhombic dodeca- hedron or the octahedron, are found on scrutiny to exhibit features which preclude such an identification. This is the case, for example, witb. the South African diamonds presented to us for the purpose of these studŸ by the De Beers Mining Corporation of Kimberley. From these facts it is evident that the crystallography of diamond stands in a class by itself apart from that of other substances and needs to be approached from a distinctive stand- point.
    [Show full text]
  • Are Your Polyhedra the Same As My Polyhedra?
    Are Your Polyhedra the Same as My Polyhedra? Branko Gr¨unbaum 1 Introduction “Polyhedron” means different things to different people. There is very little in common between the meaning of the word in topology and in geometry. But even if we confine attention to geometry of the 3-dimensional Euclidean space – as we shall do from now on – “polyhedron” can mean either a solid (as in “Platonic solids”, convex polyhedron, and other contexts), or a surface (such as the polyhedral models constructed from cardboard using “nets”, which were introduced by Albrecht D¨urer [17] in 1525, or, in a more mod- ern version, by Aleksandrov [1]), or the 1-dimensional complex consisting of points (“vertices”) and line-segments (“edges”) organized in a suitable way into polygons (“faces”) subject to certain restrictions (“skeletal polyhedra”, diagrams of which have been presented first by Luca Pacioli [44] in 1498 and attributed to Leonardo da Vinci). The last alternative is the least usual one – but it is close to what seems to be the most useful approach to the theory of general polyhedra. Indeed, it does not restrict faces to be planar, and it makes possible to retrieve the other characterizations in circumstances in which they reasonably apply: If the faces of a “surface” polyhedron are sim- ple polygons, in most cases the polyhedron is unambiguously determined by the boundary circuits of the faces. And if the polyhedron itself is without selfintersections, then the “solid” can be found from the faces. These reasons, as well as some others, seem to warrant the choice of our approach.
    [Show full text]
  • Representing the Sporadic Archimedean Polyhedra As Abstract Polytopes$
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Discrete Mathematics 310 (2010) 1835–1844 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc Representing the sporadic Archimedean polyhedra as abstract polytopesI Michael I. Hartley a, Gordon I. Williams b,∗ a DownUnder GeoSolutions, 80 Churchill Ave, Subiaco, 6008, Western Australia, Australia b Department of Mathematics and Statistics, University of Alaska Fairbanks, PO Box 756660, Fairbanks, AK 99775-6660, United States article info a b s t r a c t Article history: We present the results of an investigation into the representations of Archimedean Received 29 August 2007 polyhedra (those polyhedra containing only one type of vertex figure) as quotients of Accepted 26 January 2010 regular abstract polytopes. Two methods of generating these presentations are discussed, Available online 13 February 2010 one of which may be applied in a general setting, and another which makes use of a regular polytope with the same automorphism group as the desired quotient. Representations Keywords: of the 14 sporadic Archimedean polyhedra (including the pseudorhombicuboctahedron) Abstract polytope as quotients of regular abstract polyhedra are obtained, and summarised in a table. The Archimedean polyhedron Uniform polyhedron information is used to characterize which of these polyhedra have acoptic Petrie schemes Quotient polytope (that is, have well-defined Petrie duals). Regular cover ' 2010 Elsevier B.V. All rights reserved. Flag action Exchange map 1. Introduction Much of the focus in the study of abstract polytopes has been on the regular abstract polytopes. A publication of the first author [6] introduced a method for representing any abstract polytope as a quotient of regular polytopes.
    [Show full text]
  • Chapter 2 Figures and Shapes 2.1 Polyhedron in N-Dimension in Linear
    Chapter 2 Figures and Shapes 2.1 Polyhedron in n-dimension In linear programming we know about the simplex method which is so named because the feasible region can be decomposed into simplexes. A zero-dimensional simplex is a point, an 1D simplex is a straight line segment, a 2D simplex is a triangle, a 3D simplex is a tetrahedron. In general, a n-dimensional simplex has n+1 vertices not all contained in a (n-1)- dimensional hyperplane. Therefore simplex is the simplest building block in the space it belongs. An n-dimensional polyhedron can be constructed from simplexes with only possible common face as their intersections. Such a definition is a bit vague and such a figure need not be connected. Connected polyhedron is the prototype of a closed manifold. We may use vertices, edges and faces (hyperplanes) to define a polyhedron. A polyhedron is convex if all convex linear combinations of the vertices Vi are inside itself, i.e. i Vi is contained inside for all i 0 and all _ i i 1. i If a polyhedron is not convex, then the smallest convex set which contains it is called the convex hull of the polyhedron. Separating hyperplane Theorem For any given point outside a convex set, there exists a hyperplane with this given point on one side of it and the entire convex set on the other. Proof: Because the given point will be outside one of the supporting hyperplanes of the convex set. 2.2 Platonic Solids Known to Plato (about 500 B.C.) and explained in the Elements (Book XIII) of Euclid (about 300 B.C.), these solids are governed by the rules that the faces are the regular polygons of a single species and the corners (vertices) are all alike.
    [Show full text]
  • Visualizing the Polychora with Hyperbolic Patchwork
    Bridges 2017 Conference Proceedings Crooked Houses: Visualizing the Polychora with Hyperbolic Patchwork Taneli Luotoniemi Dept. of Art, School of Arts, Design and Architecture, Aalto University Hämeentie 135 C, 00560 Helsinki, FINLAND [email protected] Abstract This paper presents kinetic models based on the 4-dimensional regular polytopes. The sequential ‘flattening’ is realized through the use of hyperbolic patchwork surfaces, which portray the bitruncated versions of the polychora. As pedagogical tools, these models offer a hands-on experience of 4D geometry. Introduction Four-dimensional space (hyperspace, 4-space) is the result of adding an extra spatial dimension perpendicular to our three dimensions of length, height and width. The research on its properties is made possible by generalizing the geometric principles acquired by studying more familiar spaces of lower dimensions. Originated in philosophy and mathematically formulated in geometry, the concept has roused interpretations in mysticism, in theoretical physics, in fiction and in visual arts. Lately the availability of digital visualization technologies has given artists a chance to study higher space with a greater fidelity to the precise geometry of the concept. Because the fourth dimension of space cannot be directly portrayed in our physical world, the focus of many inquiries into the subject has been on the challenge of developing a visual understanding of 4-dimensional reality. Luckily, just as 3-dimensional structures can be drawn, unfolded, sliced, photographed or otherwise projected onto a 2-dimensional medium like paper or a computer screen, these graphical techniques can be generalized to produce 3D models of 4-dimensional structures described by mathematicians. The precise subject matter of these visualizations is usually the family of regular polychora – 4-dimensional counterparts of the Platonic solids.
    [Show full text]
  • A Generalized Vertex Truncation Scheme to Construct Intriguing Polyhedral Shapes
    BRIDGES Mathematical Connections in Art, Music, and Science A Generalized Vertex Truncation Scheme to Construct Intriguing Polyhedral Shapes Ergun Aldeman, Paul Edmundson and Ozan Ozener Visualization Sciences Program, 216 Langford Center, Texas A&M University College Station, Texas 77843-3137, USA. email: [email protected] Abstract In this paper, we present a method to create a new class of polyhedra. All the faces of these polyhedra are bounded by smooth (quadratic B-spline) curves and the face boundaries are Cl discontinuous everywhere. These polyhedral shapes are limit surfaces of a generalized vertex truncation subdivision scheme. We obtain an approximation of these smooth and fractal polyhedra by iteratively applying a new vertex truncation scheme to an initial manifold mesh. Our vertex truncation scheme is based on Chaikin's construction. If the initial manifold mesh is a polyhedra only with planar faces and 3-valent vertices, in each iteration we construct a polyhedral mesh in which all faces are planar and every vertex is 3- valent, 1. Introduction One of the most exciting as~ects of shape modeling and sculpting is the development of new algorithms and methods to create unusual, interesting and aesthetically pleasing shapes. Recent advances in computer graphics, shape modeling and mathematics help the imagination of contemporary mathematicians, artists and architects to design new and unusual 3D forms [11]. In this paper, we present such an unusual class of polyhedra that are contradictorily interesting, i.e. they are smooth but yet Cl discontinuous. To create these shapes we apply a polyhedral truncation algorithm based on Chaikin's scheme to any initial manifold mesh.
    [Show full text]
  • Operation-Based Notation for Archimedean Graph
    Operation-Based Notation for Archimedean Graph Hidetoshi Nonaka Research Group of Mathematical Information Science Division of Computer Science, Hokkaido University N14W9, Sapporo, 060 0814, Japan ABSTRACT Table 1. The list of Archimedean solids, where p, q, r are the number of vertices, edges, and faces, respectively. We introduce three graph operations corresponding to Symbol Name of polyhedron p q r polyhedral operations. By applying these operations, thirteen Archimedean graphs can be generated from Platonic graphs that A(3⋅4)2 Cuboctahedron 12 24 14 are used as seed graphs. A4610⋅⋅ Great Rhombicosidodecahedron 120 180 62 A468⋅⋅ Great Rhombicuboctahedron 48 72 26 Keyword: Archimedean graph, Polyhedral graph, Polyhedron A 2 Icosidodecahedron 30 60 32 notation, Graph operation. (3⋅ 5) A3454⋅⋅⋅ Small Rhombicosidodecahedron 60 120 62 1. INTRODUCTION A3⋅43 Small Rhombicuboctahedron 24 48 26 A34 ⋅4 Snub Cube 24 60 38 Archimedean graph is a simple planar graph isomorphic to the A354 ⋅ Snub Dodecahedron 60 150 92 skeleton or wire-frame of the Archimedean solid. There are A38⋅ 2 Truncated Cube 24 36 14 thirteen Archimedean solids, which are semi-regular polyhedra A3⋅102 Truncated Dodecahedron 60 90 32 and the subset of uniform polyhedra. Seven of them can be A56⋅ 2 Truncated Icosahedron 60 90 32 formed by truncation of Platonic solids, and all of them can be A 2 Truncated Octahedron 24 36 14 formed by polyhedral operations defined as Conway polyhedron 4⋅6 A 2 Truncated Tetrahedron 12 18 8 notation [1-2]. 36⋅ The author has recently developed an interactive modeling system of uniform polyhedra including Platonic solids, Archimedean solids and Kepler-Poinsot solids, based on graph drawing and simulated elasticity, mainly for educational purpose [3-5].
    [Show full text]
  • 7 Dee's Decad of Shapes and Plato's Number.Pdf
    Dee’s Decad of Shapes and Plato’s Number i © 2010 by Jim Egan. All Rights reserved. ISBN_10: ISBN-13: LCCN: Published by Cosmopolite Press 153 Mill Street Newport, Rhode Island 02840 Visit johndeetower.com for more information. Printed in the United States of America ii Dee’s Decad of Shapes and Plato’s Number by Jim Egan Cosmopolite Press Newport, Rhode Island C S O S S E M R O P POLITE “Citizen of the World” (Cosmopolite, is a word coined by John Dee, from the Greek words cosmos meaning “world” and politês meaning ”citizen”) iii Dedication To Plato for his pursuit of “Truth, Goodness, and Beauty” and for writing a mathematical riddle for Dee and me to figure out. iv Table of Contents page 1 “Intertransformability” of the 5 Platonic Solids 15 The hidden geometric solids on the Title page of the Monas Hieroglyphica 65 Renewed enthusiasm for the Platonic and Archimedean solids in the Renaissance 87 Brief Biography of Plato 91 Plato’s Number(s) in Republic 8:546 101 An even closer look at Plato’s Number(s) in Republic 8:546 129 Plato shows his love of 360, 2520, and 12-ness in the Ideal City of “The Laws” 153 Dee plants more clues about Plato’s Number v vi “Intertransformability” of the 5 Platonic Solids Of all the polyhedra, only 5 have the stuff required to be considered “regular polyhedra” or Platonic solids: Rule 1. The faces must be all the same shape and be “regular” polygons (all the polygon’s angles must be identical).
    [Show full text]
  • Unfolding Polyhedra Via Cut-Tree Truncation
    CCCG 2007, Ottawa, Ontario, August 20–22, 2007 Unfolding Polyhedra via Cut-Tree Truncation Alex Benton∗ Joseph O’Rourke† Abstract y1 y a1 y We prove that an infinite class of convex polyhedra, pro- a 2 a a duced by restricted vertex truncations, always unfold with- u 2 u x b x b out overlap. The class includes the “domes,” providing a simpler proof that these unfold without overlap. c c v (a)v (b) 1 Introduction Figure 1: Sector nesting for triangle truncation. (a) Ver- 0 It is a long unresolved question whether or not every tex x of P is truncated to 4abc in P ; (b) The unfolding 0 convex polyhedron may be cut along edges and unfolded of P and of P in the vicinity of x. to a single, non-overlapping polygon [DO05, DO07]. (Henceforth, we use “unfolding” to mean this type of Let x be a leaf of the cut-tree T .A leaf trunca- “edge unfolding.”) Although an extensive exploration tion is a truncation of the vertex x of the polyhe- by Schlickenrieder [Sch97] failed to find either an algo- dron P to P 0, and a corresponding alteration of the rithm or a counterexample, only a few narrow classes cut-tree to T 0, so that, if y is the parent of x in T , of polyhedra are known to be unfoldable: prisms, pris- and (a, p1, p2 . , pk) is the polygon resulting from the moids, and “domes” [DO07, Chap. 22]. Here we slightly truncation, with a ∈ xy, then, T 0 follows the “claw” extend the latter class, via a new proof.
    [Show full text]