DOCSLIB.ORG

Nucleation and Growth Model for {110}- and {111}-Truncated Nanoparticles

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Nucleation and Growth Model for {110}- and {111}-Truncated Nanoparticles Nucleation and growth model for {110}- and {111}-truncated nanoparticles Nicholas J. Jonesa),b) Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA Raja Swaminathan Intel Corporation, Chandler, Arizona 85226, USA Michael E. McHenryc) and David E. Laughlin Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA (Received 17 April 2015; accepted 18 August 2015) Nanoparticle-sized powders have seen more and more use in many of today’s applications. As particle size decreases, many properties change including the ability to embed the small particles in liquids and other media. With decreasing size, however, surface energy becomes more important and can dictate the final shape of the particle. In applications based on polar molecules attaching to the nanoparticle surface, the surface morphology can become a key design parameter. A nucleation and growth model has been constructed for truncated body-centered cubic derivative materials, along with an update to previously published work on face-centered cubic materials. The model shows that for (110)- and (111)-truncations of a cube with a specified surface energy for each surface, the critical nuclei and equilibrium growth shapes are the same, supporting the theory of self-similar growth that had only been mentioned previously, but never proven. In this analysis, saddle points play an important role in determining the critical nuclei for comparison with the equilibrium growth shapes. I. INTRODUCTION the resulting properties, which are different from bulk. In Nanomaterials research is a growing area that is this paper we describe the theory of nucleation and growth important for many diverse applications.1 Solid-state,2 of (100)- and (110)-faceted surfaces observed in body- liquid-phase,3 and gas-phase4 nucleation and growth the- centered cubic (BCC) MNPs. ories are important in understanding nanostructures and An understanding of the role of surface and interface controlling their properties. There is a large literature on structure and crystallography on properties of MNPs is an synthesis ! structure ! properties ! performance important, evolving area. In face-centered cubic (FCC) relationships in magnetic nanoparticles (MNPs), and the derivative nanocrystalline ferrite MNPs, for example, understanding of MNPs relies on studying the increasingly particle morphologies are observed to be faceted with fi 5–8 important influence of atoms at surfaces and interfaces, identi able crystallographic surface orientations. It has which may contribute differently to properties when been demonstrated that these crystallographic orientations compared with atoms in the bulk material. The theory of lead to distinct magnetic properties for the two different 6 fl MNP nucleation and growth can make important contri- surfaces. The surface morphology is also in uential in butions to the understanding of MNP morphologies and microstructural development in grain coarsening during sintering.9 In BCC FeCo/ferrite core–shell nanoparticles, the core is faceted and orientation relationships between the metal Contributing Editor: Suk-Joong Kang a) core and oxide shell have been determined (the shell can Address all correspondence to this author. 10,11 e-mail: [email protected] also be faceted, but only at low thicknesses). The b)Present Address: 9500 MacArthur Blvd., Naval Surface Warfare surface structure is important in understanding the Center, Carderock Division, Metallurgy and Fasteners Branch, oxidation process, and the core–shell morphology and West Bethesda, MD 20817 terminating surfaces control the functionalization of these c) This author was an editor of this focus issue the review and materials in a variety of applications.10,12–14 decision stage. For the JMR policy on review and publication of manuscripts authored by editors, please refer to http://www.mrs. Explanations for the evolution of nanoparticle morphol- org/jmr-editor-manuscripts/ ogies in terms of nucleation and growth models are DOI: 10.1557/jmr.2015.270 important for understanding the role of particle morphology Downloaded from http:/www.cambridge.org/coreJ. Mater.. Carnegie Res., Mellon Vol. 30, University No. 20,, on Oct 30 Dec 28, 2016 2015 at 19:34:10Ó Materials, subject to the Research Cambridge Society Core terms 2015 of use, available at http:/www.cambridge.org/core/terms 3011 . http://dx.doi.org/10.1557/jmr.2015.270 N.J. Jones et al.: Nucleation and growth model for {110}- and {111}-truncated nanoparticles on physical properties at different sizes.1 In this paper, we a 2nd power surface area term, so we get a traditional extend the nucleation and growth models of Ref. 4 to nucleation curve, for which the critical nuclei can be found address the observations of (100) and (110) surfaces in by setting the first derivative equal to zero (finding the FeCo MNPs. The theory of self-similar growth for homo- critical points), and solving for when the curvature (or geneous nucleation is also proven, graphically. While this second derivative) is less than zero. has been investigated for heterogeneous nucleation of While we use the maximum in the nucleation equation precipitates, the homogeneous nucleation of nanoparticles to give us the critical nuclei and nucleation energy barrier, is a simple concept (yet unaddressed in the literature) that we instead find the minimum in energy for the equilibrium has been mentioned, but not proven.15–17 An analogous 2D growth shape, minimizing the total surface energy. Since model can analytically show that the critical nuclei and we are maximizing in one case and minimizing in another, equilibrium growth shapes are self-similar, although such both with two different equations, it seems reasonable to a model has not been presented here. assume that the shape of the critical nuclei and equilibrium We have calculated both the critical nuclei and growth shapes will be different, which is contrary to what equilibrium growth shapes of BCC FeCo MNPs with has been previously stated by Christian and Aaronson.15,18 (100) and/or (110) terminating surfaces by considering This assumption is based upon the traditional arguments the Helmholtz free energies of perfect and truncated using a dimension of the shape, whether that be the radius cubes and rhombic dodecahedra. This model considers of a sphere or the edge length of a polyhedra, in a 2D/3D volume and surface energy terms in predicting the critical system. It is inadequate, however, to compare edge lengths nucleus shapes. Equilibrium growth shapes are obtained when dealing with the possibility of different shapes. This by minimizing the total surface energy associated with will be expanded upon below, with a three-dimensional (100) and (110) surfaces while constraining the volume to analysis for both {110} and {111} faceting of cubes, but a constant size. Finally, there is a reanalysis of nucleation Fig. 1 shows this graphically for FCC based structures. and growth for FCC-type materials4 in light of the BCC When plotting the Helmholtz free energy versus the analysis. equivalent edge length of multiple polyhedra [Fig. 1(a)], it seems that there is one shape with a minimum nucle- II. REQUIREMENTS FOR NUCLEATION AND ation energy barrier (the cube), and another which will EQUILIBRIUM GROWTH SHAPES provide the minimum energy growth shape at a certain edge length (the cuboctahedron). While this is surely In order for a material to nucleate, there is a surface easier to plot, this is misleading and incorrect. energy barrier that must be overcome. The creation of To compare different shapes, we need to compare surfaces, which always costs energy, can be considered equivalent volumes, otherwise the energies we are com- an introduction of two-dimensional defects. It is for this paring are due to different amounts of atoms attaching to reason that heterogeneous nucleation is often preferred the nucleus, which will alter the energies being compared. over homogeneous nucleation, because in heterogeneous By comparing volumes, we remove the volume free nucleation there is already a surface present which can energy as a variable and we also change the traditional assist in reducing the energy barrier to nucleate new shape of the nucleation curves [Fig. 1(b)], but we can surfaces. Irrespective of the mode of nucleation, the now do both nucleation and growth analyses at the same shape of the critical nuclei can be determined to be the time. The minimum energy barrier for nucleation will not nuclei shape that has the lowest free energy of formation. change, since we are only transforming the x-axis While this has often been thought to be the same as the dimensions; we don’t change the maxima in energy, but equilibrium growth shape (often termed “self-similar only their lateral position with respect to each other. The growth”), this has not been proven analytically for size of the critical nucleus doesn’t change, but we are three-dimensional objects.15 using the volume to compare the polyhedra to each other. Beginning very simply, we can look at the general By comparing volumes, however, we can set a constant equations for traditional nucleation and growth shown volume and compare the energies for the different shapes, below [Eqs. (1) and (2)]. which is a Wulff Construction, in essence (rather than removing the volume term from the equation, we are just Nucleation : À VDfv þ +rA ; ð1Þ making it a constant).19,20 We can now see that the critical nucleus is not different from the equilibrium Growth : +rA : ð2Þ growth shape but will be the same [Fig. 1(b)], since the curve with the minimum nucleation energy barrier will The nucleation equation has both a volume energy always be less than any other curve. This is the origin of term and a surface energy term, while the equilibrium the difference between this model and the old model,4 and growth shape only has a surface energy term. The thus constitutes an improved formalism for self-similar nucleation equation has a 3rd power volume term, and growth.
Load more

Recommended publications
  • Avenues for Polyhedral Research
    Avenues for Polyhedral Research by Magnus J. Wenninger In the epilogue of my book Polyhedron Models I alluded Topologie Structurale #5, 1980 Structural Topology #5,1980 to an avenue for polyhedral research along which I myself have continued to journey, involving Ar- chimedean stellations and duals (Wenninger 1971). By way of introduction to what follows here it may be noted that the five regular convex polyhedra belong to a larger set known as uniform’ polyhedra. Their stellated forms are derived from the process of extending their R&urn6 facial planes while preserving symmetry. In particular the tetrahedron and the hexahedron have no stellated forms. The octahedron has one, the dodecahedron Cet article est le sommaire d’une etude et de three and the icosahedron a total of fifty eight (Coxeter recherches faites par I’auteur depuis une dizaine 1951). The octahedral stellation is a compound of two d’annees, impliquant les polyedres archimediens Abstract tetrahedra, classified as a uniform compound. The &oil& et les duals. La presentation en est faite a three dodecahedral stellations are non-convex regular partir d’un passe historique, et les travaux This article is a summary of investigations and solids and belong to the set of uniform polyhedra. Only d’auteurs recents sont brievement rappel& pour study done by the author over the past ten years, one icosahedral stellation is a non-convex regular la place importante qu’ils occupent dans I’etude involving Archimedean stellations and duals. The polyhedron and hence also a uniform polyhedron, but des formes polyedriques. L’auteur enonce une presentation is set into its historical background, several other icosahedral stellations are uniform com- regle &n&ale par laquelle on peut t,rouver un and the works of recent authors are briefly sket- pounds, namely the compound of five octahedra, five dual a partir de n’importe quel polyedre uniforme ched for their place in the continuing study of tetrahedra and ten tetrahedra.
    [Show full text]
  • Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity Eppur Si Muove: Biomimetic Embedding of N-Tuple Helices in Spherical Polyhedra - /
    Alternative view of segmented documents via Kairos 23 October 2017 | Draft Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity Eppur si muove: Biomimetic embedding of N-tuple helices in spherical polyhedra - / - Introduction Symbolic stars vs Strategic pillars; Polyhedra vs Helices; Logic vs Comprehension? Dynamic bonding patterns in n-tuple helices engendering n-fold rotating symbols Embedding the triple helix in a spherical octahedron Embedding the quadruple helix in a spherical cube Embedding the quintuple helix in a spherical dodecahedron and a Pentagramma Mirificum Embedding six-fold, eight-fold and ten-fold helices in appropriately encircled polyhedra Embedding twelve-fold, eleven-fold, nine-fold and seven-fold helices in appropriately encircled polyhedra Neglected recognition of logical patterns -- especially of opposition Dynamic relationship between polyhedra engendered by circles -- variously implying forms of unity Symbol rotation as dynamic essential to engaging with value-inversion References Introduction The contrast to the geocentric model of the solar system was framed by the Italian mathematician, physicist and philosopher Galileo Galilei (1564-1642). His much-cited phrase, " And yet it moves" (E pur si muove or Eppur si muove) was allegedly pronounced in 1633 when he was forced to recant his claims that the Earth moves around the immovable Sun rather than the converse -- known as the Galileo affair. Such a shift in perspective might usefully inspire the recognition that the stasis attributed so widely to logos and other much-valued cultural and heraldic symbols obscures the manner in which they imply a fundamental cognitive dynamic. Cultural symbols fundamental to the identity of a group might then be understood as variously moving and transforming in ways which currently elude comprehension.
    [Show full text]
  • Temperature-Tuned Faceting and Shape-Changes in Liquid Alkane Droplets
    Published in Langmuir 33, 1305 (2017) Temperature-tuned faceting and shape-changes in liquid alkane droplets Shani Guttman,† Zvi Sapir,†,¶ Benjamin M. Ocko,‡ Moshe Deutsch,† and Eli Sloutskin∗,† Physics Dept. and Institute of Nanotechnology, Bar-Ilan University, Ramat-Gan 5290002, Israel, and NSLS-II, Brookhaven National Laboratory, Upton NY 11973, USA E-mail: [email protected] Abstract Recent extensive studies reveal that surfactant-stabilized spherical alkane emulsion droplets spontaneously adopt polyhedral shapes upon cooling below a temperature Td, while still remaining liquid. Further cooling induces growth of tails and spontaneous droplet splitting. Two mechanisms were offered to account for these intriguing effects. One assigns the effects to the formation of an intra-droplet frame of tubules, consist- ing of crystalline rotator phases with cylindrically curved lattice planes. The second assigns the sphere-to-polyhedron transition to the buckling of defects in a crystalline interfacial monolayer, known to form in these systems at some Ts > Td. The buck- ling reduces the extensional energy of the crystalline monolayer’s defects, unavoidably formed when wrapping a spherical droplet by a hexagonally-packed interfacial mono- layer. The tail growth, shape changes, and droplet splitting were assigned to the ∗To whom correspondence should be addressed †Physics Dept. and Institute of Nanotechnology, Bar-Ilan University, Ramat-Gan 5290002, Israel ‡NSLS-II, Brookhaven National Laboratory, Upton NY 11973, USA ¶Present address: Intel (Israel) Ltd., Kiryat Gat, Israel 1 vanishing of the surface tension, γ. Here we present temperature-dependent γ(T ), optical microscopy measurements, and interfacial entropy determinations, for several alkane/surfactant combinations. We demonstrate the advantages and accuracy of the in-situ γ(T ) measurements, done simultaneously with the microscopy measurements on the same droplet.
    [Show full text]
  • A Tourist Guide to the RCSR
    A tourist guide to the RCSR Some of the sights, curiosities, and little-visited by-ways Michael O'Keeffe, Arizona State University RCSR is a Reticular Chemistry Structure Resource available at http://rcsr.net. It is open every day of the year, 24 hours a day, and admission is free. It consists of data for polyhedra and 2-periodic and 3-periodic structures (nets). Visitors unfamiliar with the resource are urged to read the "about" link first. This guide assumes you have. The guide is designed to draw attention to some of the attractions therein. If they sound particularly attractive please visit them. It can be a nice way to spend a rainy Sunday afternoon. OKH refers to M. O'Keeffe & B. G. Hyde. Crystal Structures I: Patterns and Symmetry. Mineral. Soc. Am. 1966. This is out of print but due as a Dover reprint 2019. POLYHEDRA Read the "about" for hints on how to use the polyhedron data to make accurate drawings of polyhedra using crystal drawing programs such as CrystalMaker (see "links" for that program). Note that they are Cartesian coordinates for (roughly) equal edge. To make the drawing with unit edge set the unit cell edges to all 10 and divide the coordinates given by 10. There seems to be no generally-agreed best embedding for complex polyhedra. It is generally not possible to have equal edge, vertices on a sphere and planar faces. Keywords used in the search include: Simple. Each vertex is trivalent (three edges meet at each vertex) Simplicial. Each face is a triangle.
    [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]
  • Icosahedral Polyhedra from 6 Lattice and Danzer's ABCK Tiling
    Icosahedral Polyhedra from 퐷6 lattice and Danzer’s ABCK tiling Abeer Al-Siyabi, a Nazife Ozdes Koca, a* and Mehmet Kocab aDepartment of Physics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khoud, 123 Muscat, Sultanate of Oman, bDepartment of Physics, Cukurova University, Adana, Turkey, retired, *Correspondence e-mail: [email protected] ABSTRACT It is well known that the point group of the root lattice 퐷6 admits the icosahedral group as a maximal subgroup. The generators of the icosahedral group 퐻3 , its roots and weights are determined in terms of those of 퐷6 . Platonic and Archimedean solids possessing icosahedral symmetry have been obtained by projections of the sets of lattice vectors of 퐷6 determined by a pair of integers (푚1, 푚2 ) in most cases, either both even or both odd. Vertices of the Danzer’s ABCK tetrahedra are determined as the fundamental weights of 퐻3 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 (푚1, 푚2 ) with coefficients from Fibonacci sequence. Tiling procedure both for the ABCK tetrahedral and the < 퐴퐵퐶퐾 > octahedral tilings in 3D space with icosahedral symmetry 퐻3 and those related transformations in 6D space with 퐷6 symmetry are specified by determining the rotations and translations in 3D and the corresponding group elements in 퐷6 .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.
    [Show full text]
  • DETECTION of FACES in WIRE-FRAME POLYHEDRA Interactive Modelling of Uniform Polyhedra
    DETECTION OF FACES IN WIRE-FRAME POLYHEDRA Interactive Modelling of Uniform Polyhedra Hidetoshi Nonaka Hokkaido University, N14W9, Sapporo, 060 0814, Japan Keywords: Uniform polyhedron, polyhedral graph, simulated elasticity, interactive computing, recreational mathematics. Abstract: This paper presents an interactive modelling system of uniform polyhedra including regular polyhedra, semi-regular polyhedra, and intersected concave polyhedra. In our system, user can virtually “make” and “handle” them interactively. The coordinate of vertices are computed without the knowledge of faces, solids, or metric information, but only with the isomorphic graph structure. After forming a wire-frame polyhedron, the faces are detected semi-automatically through user-computer interaction. This system can be applied to recreational mathematics, computer assisted education of the graph theory, and so on. 1 INTRODUCTION 2 UNIFORM POLYHEDRA This paper presents an interactive modelling system 2.1 Platonic Solids of uniform polyhedra using simulated elasticity. Uniform polyhedra include five regular polyhedra Five Platonic solids are listed in Table 1. The (Platonic solids), thirteen semi-regular polyhedra symbol Pmn indicates that the number of faces (Archimedean solids), and four regular concave gathering around a vertex is n, and each face is m- polyhedra (Kepler-Poinsot solids). Alan Holden is describing in his writing, “The best way to learn sided regular polygon. about these objects is to make them, next best to handle them (Holden, 1971).” Traditionally, these Table 1: The list of Platonic solids. objects are made based on the shapes of faces or Symbol Polyhedron Vertices Edges Faces solids. Development figures and a set of regular polygons cut from card boards can be used to P33 Tetrahedron 4 6 4 assemble them.
    [Show full text]
  • Rhombic Dodecahedron
    Rhombic Dodecahedron Figure 1 Rhombic Dodecahedron. Vertex labels as used for the corresponding vertices of the 120 Polyhedron. COPYRIGHT 2007, Robert W. Gray Page 1 of 14 Encyclopedia Polyhedra: Last Revision: September 5, 2007 RHOMBIC DODECAHEDRON Figure 2 Cube (blue) and Octahedron (green) define rhombic Dodecahedron. Figure 3 “Long” (green) and “short” (blue) face diagonals. COPYRIGHT 2007, Robert W. Gray Page 2 of 14 Encyclopedia Polyhedra: Last Revision: September 5, 2007 RHOMBIC DODECAHEDRON Topology: Vertices = 14 Edges = 24 Faces = 12 diamonds Lengths: EL ≡ Edge length of rhombic Dodecahedron. 22 FDL ≡ Long face diagonal = EL ≅ 1.632 993 162 EL 3 1 FDS ≡ Short face diagonal = FDL ≅ 0.707 106 781 FDL 2 2 FDS = EL ≅ 1.154 700 538 EL 3 DFVL ≡ Vertex at the end of a long face diagonal 1 2 = FDL = EL ≅ 0.816 496 581 EL 2 3 DFVS ≡ Vertex at the end of a short face diagonal 1 = FDL ≅ 0.353 553 391 FDL 22 1 = EL ≅ 0.577 350 269 EL 3 COPYRIGHT 2007, Robert W. Gray Page 3 of 14 Encyclopedia Polyhedra: Last Revision: September 5, 2007 RHOMBIC DODECAHEDRON 3 DFE = FDL ≅ 0.306 186 218 FDL 42 1 = EL 2 1 DVVL = FDL ≅ 0.707 106 781 FDL 2 2 = EL ≅ 1.154 700 538 EL 3 3 DVVS = FDL ≅ 0.612 372 4 FDL 22 = EL 11 DVE = FDL ≅ 0.586 301 970 FDL 42 11 = EL ≅ 0.957 427 108 EL 23 1 DVF = FDL 2 2 = EL ≅ 0.816 496 581 EL 3 COPYRIGHT 2007, Robert W. Gray Page 4 of 14 Encyclopedia Polyhedra: Last Revision: September 5, 2007 RHOMBIC DODECAHEDRON Areas: 2 2 1 Area of one diamond face = FDL ≅ 0.353 553 391 FDL 22 22 2 2 = EL ≅ 0.942 809 042 EL 3 2 2 Total face area = 32 FDL ≅ 4.242 640 687 FDL 2 2 = 82 EL ≅ 4.242 640 687 EL Volume: 16 3 3 Cubic measure volume equation = EL ≅ 3.079 201 436 EL 33 3 Synergetics' Tetra-volume equation = 6 FDL Angles: Face Angles: ⎛⎞2 2arcsin θS ≡ Face angle at short vertex = ⎜⎟ ≅ 109.471 220 634° ⎝⎠3 ⎛⎞22 θ ≡ Face angle at long vertex = arcsin ≅ 70.528 779 366° L ⎜⎟ ⎝⎠3 Sum of face angles = 4320° Central Angles: ⎛⎞1 All central angles are = arccos ≅ 54.735 610 317° ⎜⎟ ⎝⎠3 Dihedral Angles: All dihedral angles are = 120° COPYRIGHT 2007, Robert W.
    [Show full text]
  • Dissection of Rhombic Solids with Octahedral Symmetry to Archimedean Solids, Part 2
    Dissection of rhombic solids with octahedral symmetry to Archimedean solids, Part 2 Izidor Hafner In [5] we found "best approximate" by "rhombic solids" for certain Platonic and Archimedean solids with octahedral symmetry. In this paper we treat the remaining solids: truncated cube, rhombicubocahedron and truncated cuboctahedron. For this purpose we extend the notion of rhombic solid so, that it is also possible to add to them 1/4 and 1/8 of the rhombic dodecahedron and 1/6 of the cube. These last solids can be dissected to cubes. The authors of [1, pg. 331] produced the table of the Dehn invariants for the non-snub unit edge Archimedean polyhedra. We have considered the icosahedral part in [7]. Results from both papers show, that each best aproximate has a surplus or a deficit measured in tetrahedrons (T): -2, -1, 0, 1, 2. Actualy 1/4 of the tetrahedron appears as a piece. The table of Dehn invariant for the non-snub unit edge Archimedean polyhedra (octahedral part) [1]: Tetrahedron -12(3)2 Truncated tetrahedron 12(3)2 Cube 0 Truncated cube -24(3)2 Octahedron 24(3)2 Truncated octahedron 0 Rhombicuboctahedron -24(3)2 ??? Cuboctahedron -24(3)2 Truncated cuboctahedron 0 Along with the above table, we produced the following one: Tetrahedron 1 Truncated tetrahedron -1 Cube 0 Truncated cube 2 Octahedron -2 Truncated octahedron 0 Rhombicuboctahedron -2 Cuboctahedron 2 Truncated cuboctahedron 0 For instance: the tetrahedron has a surplus of 1 tetrahedron relative to empty polyhedron, the octahedron has deficit of 2 tetrahedra relative to rhombic dodecahedron. Our truncated cube, rhombicuboctahedron and truncated cuboctahedron are not Archimedean, but can be enlarged to Archimedean by addition of some prisms, so the results in the table are valid for Archimedean solids as well.
    [Show full text]
  • Local Symmetry Preserving Operations on Polyhedra
    Local Symmetry Preserving Operations on Polyhedra Pieter Goetschalckx Submitted to the Faculty of Sciences of Ghent University in fulfilment of the requirements for the degree of Doctor of Science: Mathematics. Supervisors prof. dr. dr. Kris Coolsaet dr. Nico Van Cleemput Chair prof. dr. Marnix Van Daele Examination Board prof. dr. Tomaž Pisanski prof. dr. Jan De Beule prof. dr. Tom De Medts dr. Carol T. Zamfirescu dr. Jan Goedgebeur © 2020 Pieter Goetschalckx Department of Applied Mathematics, Computer Science and Statistics Faculty of Sciences, Ghent University This work is licensed under a “CC BY 4.0” licence. https://creativecommons.org/licenses/by/4.0/deed.en In memory of John Horton Conway (1937–2020) Contents Acknowledgements 9 Dutch summary 13 Summary 17 List of publications 21 1 A brief history of operations on polyhedra 23 1 Platonic, Archimedean and Catalan solids . 23 2 Conway polyhedron notation . 31 3 The Goldberg-Coxeter construction . 32 3.1 Goldberg ....................... 32 3.2 Buckminster Fuller . 37 3.3 Caspar and Klug ................... 40 3.4 Coxeter ........................ 44 4 Other approaches ....................... 45 References ............................... 46 2 Embedded graphs, tilings and polyhedra 49 1 Combinatorial graphs .................... 49 2 Embedded graphs ....................... 51 3 Symmetry and isomorphisms . 55 4 Tilings .............................. 57 5 Polyhedra ............................ 59 6 Chamber systems ....................... 60 7 Connectivity .......................... 62 References
    [Show full text]
  • Logical Geometry of the Rhombic Dodecahedron of Oppositions
    Logical Geometry of the Rhombic Dodecahedron of Oppositions Hans Smessaert Introduction: Aristotelian subdiagrams 2 3 squares embedded in (strong) Jacoby-Sesmat-Blanché hexagon (JSB) 3 squares embedded in Sherwood-Czezowski hexagon (SC) Logical Geometry of RDH H. Smessaert Introduction: Aristotelian subdiagrams 3 4 hexagons embedded in Buridan octagon Logical Geometry of RDH H. Smessaert Introduction: Aristotelian subdiagrams in RDH 4 Internal structure of bigger/3D Aristotelian diagrams ? Some initial results: 4 weak JSB-hexagons in logical cube (Moretti-Pellissier) 6 strong JSB hexagons in bigger 3D structure with 14 formulas/vertices tetra-hexahedron (Sauriol) tetra-icosahedron (Moretti-Pellissier) nested tetrahedron (Lewis, Dubois-Prade) rhombic dodecahedron = RDH (Smessaert-Demey) joint work Greater complexity of RDH exhaustive analysis of internal structure ?? Main aim of this talk tools and techniques for such an analysis examine larger substructures (octagon, decagon, dodecagon, ...) distinguish families of substructures (strong JSB, weak JSB, ...) establish the exhaustiveness of the typology Logical Geometry of RDH H. Smessaert Structure of the talk 5 1 Introduction 2 The Rhombic Dodecahedron of Oppositions RDH 3 Sigma-structures 4 Families of Sigma-structures: the CO-perspective 5 Complementarities between families of Sigma-structures 6 Conclusion Logical Geometry of RDH H. Smessaert Structure of the talk 6 1 Introduction 2 The Rhombic Dodecahedron of Oppositions RDH 3 Sigma-structures 4 Families of Sigma-structures: the CO-perspective 5 Complementarities between families of Sigma-structures 6 Conclusion Logical Geometry of RDH H. Smessaert Rhombic Dodecahedron (RDH) 7 cube + octahedron = cuboctahedron =dual) rhombic dodecahedron Platonic Platonic Archimedean Catalan 6 faces 8 faces 14 faces 12 faces 8 vertices 6 vertices 12 vertices 14 vertices Logical Geometry of RDH H.
    [Show full text]
  • Paper Models of Polyhedra
    Paper Models of Polyhedra Gijs Korthals Altes Polyhedra are beautiful 3-D geometrical figures that have fascinated philosophers, mathematicians and artists for millennia Copyrights © 1998-2001 Gijs.Korthals Altes All rights reserved . It's permitted to make copies for non-commercial purposes only email: [email protected] Paper Models of Polyhedra Platonic Solids Dodecahedron Cube and Tetrahedron Octahedron Icosahedron Archimedean Solids Cuboctahedron Icosidodecahedron Truncated Tetrahedron Truncated Octahedron Truncated Cube Truncated Icosahedron (soccer ball) Truncated dodecahedron Rhombicuboctahedron Truncated Cuboctahedron Rhombicosidodecahedron Truncated Icosidodecahedron Snub Cube Snub Dodecahedron Kepler-Poinsot Polyhedra Great Stellated Dodecahedron Small Stellated Dodecahedron Great Icosahedron Great Dodecahedron Other Uniform Polyhedra Tetrahemihexahedron Octahemioctahedron Cubohemioctahedron Small Rhombihexahedron Small Rhombidodecahedron S mall Dodecahemiododecahedron Small Ditrigonal Icosidodecahedron Great Dodecahedron Compounds Stella Octangula Compound of Cube and Octahedron Compound of Dodecahedron and Icosahedron Compound of Two Cubes Compound of Three Cubes Compound of Five Cubes Compound of Five Octahedra Compound of Five Tetrahedra Compound of Truncated Icosahedron and Pentakisdodecahedron Other Polyhedra Pentagonal Hexecontahedron Pentagonalconsitetrahedron Pyramid Pentagonal Pyramid Decahedron Rhombic Dodecahedron Great Rhombihexacron Pentagonal Dipyramid Pentakisdodecahedron Small Triakisoctahedron Small Triambic
    [Show full text]