On Contact Numbers of Finite Lattice Sphere Packings of Balls
|
|
|
- Aubrey McKinney
- 5 years ago
- Views:
Transcription
1 On Contact Numbers of Finite Lattice Sphere Packings of 2-27 Balls István Szalkai University of Pannonia, Veszprém, Hungary March 16, 216 Abstract Empirical constructions having maximal contact numbers of unit balls as putative best ones are presented for 2-27 balls. 1 Introduction Consider n balls of the same (unit) radius in the 3 -dimensional Euclidean space when n is any but fixed number. If we assume that each pair of balls have at most one common (tangential, "touching" or "kissing") point, the number of such points is called the contact number of this ball-configuration. For each fixed number nn we may ask for configurations which have maximal contact number, these maximal numbers are denoted by c(n). Many recent contributions deal with the maximal contact number of balls, using theoretical and empirical approach as well (see eg. all the references listed). The possible configurations have many applications e.g. in material science and other fields of applied physics and chemistry. This enlarged interest lead us to publish our 5 year old empirical investigations. The computer runs investigated all possible cases, so we may think our maximal ones are the real values of c(n). Our investigations are still in progress, so further results will be contributed soon. Harboth [7] proved that in the plane the hexagonal grid (lattice) is the optimal configuration for congruent circles. This means, that the centers of the circles fit on the grid 2 1 i j : i, 3 j Z (1) For balls in three dimension we suspect that optimal configurations can be found in the 3 -dimensional hexagonal grid as follows. First we place some balls in planes in the above planar manner, we call this set a layer. Second, put such layers onto each other: the layers are translated with the vector 1, 1/ 3, 8/ 3 T to get the neighbour layer (like the usual placement of melons). We use the following hexagonal coordinates for the centers of the balls: P[i, j, k] H T refers to the k -th layer, so k is the vertical coordinate, and i, j are the horizontal ones as in (1). So, P has the the following (usual) Cartesian coordinates ( in the subscript):
2 for k even : H j i k k j i (2) for k = 2l+1 even : H j i l k j i / 3 1/ 1 (3) In the tables in Section 2 we present the centers F 1 through F n of the balls of our configurations in hexagonal [i, j, k] H T coordinates, which can be transformed to Cartesian ones by (2) and (3) for visualization. Theorem 4.1 of [5] states for n : ) ( / 2 n n c n The similar (also triangular) barycentric coordinate system is also used in chemistry, see e.g. [13]. 2 Results First we summarize in a table the maximal contact numbers c(n) (found by computer), which can be considered putative maximal ones: n c(n) c(n)-3n / ) ( 6 n n c n In what follows we give one example for each of the above results. F 1 through F n are the centres of the balls, the rows below them are their hexagonal coordinates as defined in the Introduction. ouble vertical lines separate the layers, which can be distinguished by the third coordinates. The colors of balls and their connections of separate layers are changing from layer to layer. After the tables all the contacts (F i, F j ) of the configuration are listed.
3 2 balls (F1,F2), (F1,F3), (F1,F7), (F1,F8), (F2,F3), (F2,F4), (F2,F5), (F2,F7), (F2,F9), (F2,F1), (F3,F5), (F3,F6), (F3,F8), (F3,F1), (F3,F11), (F4,F5), (F4,F9), (F4,F12), (F5,F6), (F5,F1), (F5,F12), (F5,F13), (F6,F11), (F6,F13), (F6,F14), (F7,F8), (F7,F9), (F7,F1), (F7,F15), (F7,F16), (F8,F1), (F8,F11), (F8,F15), (F8,F17), (F9,F1), (F9,F12), (F9,F16), (F9,F18), (F1,F11), (F1,F12), (F1,F13), (F1,F16), (F1,F17), (F1,F19), (F11,F13), (F11,F14), (F11,F17), (F11,F2), (F12,F13), (F12,F18), (F12,F19), (F13,F14), (F13,F19), (F13,F2), (F14,F2), (F15,F16), (F15,F17), (F16,F17), (F16,F18), (F16,F19), (F17,F19), (F17,F2), (F18,F19), (F19,F2), Total: c(2) = 64.
4 21 balls (F1,F2), (F1,F3), (F1,F4), (F1,F9), (F1,F1), (F2,F4), (F2,F5), (F2,F1), (F2,F11), (F3,F4), (F3,F6), (F3,F9), (F3,F12), (F4,F5), (F4,F6), (F4,F7), (F4,F1), (F4,F12), (F4,F13), (F5,F7), (F5,F8), (F5,F11), (F5,F13), (F5,F14), (F6,F7), (F6,F12), (F6,F15), (F7,F8), (F7,F13), (F7,F15), (F7,F16), (F8,F14), (F8,F16), (F8,F17), (F9,F1), (F9,F12), (F1,F11), (F1,F12), (F1,F13), (F1,F18), (F11,F13), (F11,F14), (F11,F19), (F12,F13), (F12,F15), (F12,F18), (F13,F14), (F13,F15), (F13,F16), (F13,F18), (F13,F19), (F13,F2), (F14,F16), (F14,F17), (F14,F19), (F14,F21), (F15,F16), (F15,F2), (F16,F17), (F16,F2), (F16,F21), (F17,F21), (F18,F19), (F18,F2), (F19,F2), (F19,F21), (F2,F21), Total: c(21) = 67.
5 22 balls (F1,F2), (F1,F3), (F1,F4), (F1,F8), (F1,F9), (F2,F4), (F2,F5), (F2,F9), (F2,F1), (F3,F4), (F3,F6), (F3,F8), (F3,F11), (F4,F5), (F4,F6), (F4,F7), (F4,F9), (F4,F11), (F4,F12), (F5,F7), (F5,F1), (F5,F12), (F5,F13), (F6,F7), (F6,F11), (F6,F14), (F7,F12), (F7,F14), (F7,F15), (F8,F9), (F8,F11), (F8,F16), (F8,F18), (F9,F1), (F9,F11), (F9,F12), (F9,F16), (F9,F17), (F9,F19), (F1,F12), (F1,F13), (F1,F17), (F1,F2), (F11,F12), (F11,F14), (F11,F18), (F11,F19), (F11,F21), (F12,F13), (F12,F14), (F12,F15), (F12,F19), (F12,F2), (F12,F22), (F13,F15), (F13,F2), (F14,F15), (F14,F21), (F14,F22), (F15,F22), (F16,F17), (F16,F18), (F16,F19), (F17,F19), (F17,F2), (F18,F19), (F18,F21), (F19,F2), (F19,F21), (F19,F22), (F2,F22), (F21,F22), Total: c(22) = 72.
6 23 balls (F1,F2), (F1,F3), (F1,F4), (F1,F9), (F1,F1), (F2,F4), (F2,F5), (F2,F1), (F2,F11), (F3,F4), (F3,F6), (F3,F9), (F3,F12), (F4,F5), (F4,F6), (F4,F7), (F4,F1), (F4,F12), (F4,F13), (F5,F7), (F5,F8), (F5,F11), (F5,F13), (F5,F14), (F6,F7), (F6,F12), (F6,F15), (F7,F8), (F7,F13), (F7,F15), (F7,F16), (F8,F14), (F8,F16), (F9,F1), (F9,F12), (F9,F17), (F9,F19), (F1,F11), (F1,F12), (F1,F13), (F1,F17), (F1,F18), (F1,F2), (F11,F13), (F11,F14), (F11,F18), (F11,F21), (F12,F13), (F12,F15), (F12,F19), (F12,F2), (F12,F22), (F13,F14), (F13,F15), (F13,F16), (F13,F2), (F13,F21), (F13,F23), (F14,F16), (F14,F21), (F15,F16), (F15,F22), (F15,F23), (F16,F23), (F17,F18), (F17,F19), (F17,F2), (F18,F2), (F18,F21), (F19,F2), (F19,F22), (F2,F21), (F2,F22), (F2,F23), (F21,F23), (F22,F23), Total: c(23) = 76.
7 24 balls (F1,F2), (F1,F3), (F1,F4), (F1,F9), (F1,F1), (F2,F4), (F2,F5), (F2,F1), (F2,F11), (F3,F4), (F3,F6), (F3,F9), (F3,F12), (F4,F5), (F4,F6), (F4,F7), (F4,F1), (F4,F12), (F4,F13), (F5,F7), (F5,F8), (F5,F11), (F5,F13), (F5,F14), (F6,F7), (F6,F12), (F6,F15), (F7,F8), (F7,F13), (F7,F15), (F7,F16), (F8,F14), (F8,F16), (F9,F1), (F9,F12), (F9,F17), (F9,F19), (F1,F11), (F1,F12), (F1,F13), (F1,F17), (F1,F18), (F1,F2), (F11,F13), (F11,F14), (F11,F18), (F11,F21), (F12,F13), (F12,F15), (F12,F19), (F12,F2), (F12,F22), (F13,F14), (F13,F15), (F13,F16), (F13,F2), (F13,F21), (F13,F23), (F14,F16), (F14,F21), (F14,F24), (F15,F16), (F15,F22), (F15,F23), (F16,F23), (F16,F24), (F17,F18), (F17,F19), (F17,F2), (F18,F2), (F18,F21), (F19,F2), (F19,F22), (F2,F21), (F2,F22), (F2,F23), (F21,F23), (F21,F24), (F22,F23), (F23,F24), Total: c(24) = y.
8 25 balls (F1,F2), (F1,F3), (F1,F4), (F1,F9), (F1,F1), (F2,F4), (F2,F5), (F2,F1), (F2,F11), (F3,F4), (F3,F6), (F3,F9), (F3,F12), (F4,F5), (F4,F6), (F4,F7), (F4,F1), (F4,F12), (F4,F13), (F5,F7), (F5,F8), (F5,F11), (F5,F13), (F5,F14), (F6,F7), (F6,F12), (F6,F15), (F7,F8), (F7,F13), (F7,F15), (F7,F16), (F8,F14), (F8,F16), (F8,F17), (F9,F1), (F9,F12), (F9,F18), (F9,F2), (F1,F11), (F1,F12), (F1,F13), (F1,F18), (F1,F19), (F1,F21), (F11,F13), (F11,F14), (F11,F19), (F11,F22), (F12,F13), (F12,F15), (F12,F2), (F12,F21), (F12,F23), (F13,F14), (F13,F15), (F13,F16), (F13,F21), (F13,F22), (F13,F24), (F14,F16), (F14,F17), (F14,F22), (F14,F25), (F15,F16), (F15,F23), (F15,F24), (F16,F17), (F16,F24), (F16,F25), (F17,F25), (F18,F19), (F18,F2), (F18,F21), (F19,F21), (F19,F22), (F2,F21), (F2,F23), (F21,F22), (F21,F23), (F21,F24), (F22,F24), (F22,F25), (F23,F24), (F24,F25), Total: c(25) = 84.
9 26 balls (F1,F2), (F1,F4), (F1,F1), (F2,F3), (F2,F4), (F2,F5), (F2,F1), (F2,F11), (F3,F5), (F3,F6), (F3,F11), (F3,F12), (F4,F5), (F4,F7), (F4,F1), (F4,F13), (F5,F6), (F5,F7), (F5,F8), (F5,F11), (F5,F13), (F5,F14), (F6,F8), (F6,F9), (F6,F12), (F6,F14), (F6,F15), (F7,F8), (F7,F13), (F7,F16), (F8,F9), (F8,F14), (F8,F16), (F8,F17), (F9,F15), (F9,F17), (F9,F18), (F1,F11), (F1,F13), (F1,F19), (F1,F21), (F11,F12), (F11,F13), (F11,F14), (F11,F19), (F11,F2), (F11,F22), (F12,F14), (F12,F15), (F12,F2), (F12,F23), (F13,F14), (F13,F16), (F13,F21), (F13,F22), (F13,F24), (F14,F15), (F14,F16), (F14,F17), (F14,F22), (F14,F23), (F14,F25), (F15,F17), (F15,F18), (F15,F23), (F15,F26), (F16,F17), (F16,F24), (F16,F25), (F17,F18), (F17,F25), (F17,F26), (F18,F26), (F19,F2), (F19,F21), (F19,F22), (F2,F22), (F2,F23), (F21,F22), (F21,F24), (F22,F23), (F22,F24), (F22,F25), (F23,F25), (F23,F26), (F24,F25), (F25,F26), Total: c(26) = 87.
10 27 balls (F1,F2), (F1,F4), (F1,F1), (F2,F3), (F2,F4), (F2,F5), (F2,F1), (F2,F11), (F3,F5), (F3,F6), (F3,F11), (F3,F12), (F4,F5), (F4,F7), (F4,F1), (F4,F13), (F5,F6), (F5,F7), (F5,F8), (F5,F11), (F5,F13), (F5,F14), (F6,F8), (F6,F9), (F6,F12), (F6,F14), (F6,F15), (F7,F8), (F7,F13), (F7,F16), (F8,F9), (F8,F14), (F8,F16), (F8,F17), (F9,F15), (F9,F17), (F9,F18), (F1,F11), (F1,F13), (F1,F19), (F1,F2), (F1,F22), (F11,F12), (F11,F13), (F11,F14), (F11,F2), (F11,F21), (F11,F23), (F12,F14), (F12,F15), (F12,F21), (F12,F24), (F13,F14), (F13,F16), (F13,F22), (F13,F23), (F13,F25), (F14,F15), (F14,F16), (F14,F17), (F14,F23), (F14,F24), (F14,F26), (F15,F17), (F15,F18), (F15,F24), (F15,F27), (F16,F17), (F16,F25), (F16,F26), (F17,F18), (F17,F26), (F17,F27), (F18,F27), (F19,F2), (F19,F22), (F2,F21), (F2,F22), (F2,F23), (F21,F23), (F21,F24), (F22,F23), (F22,F25), (F23,F24), (F23,F25), (F23,F26), (F24,F26), (F24,F27), (F25,F26), (F26,F27), Total: c(27) = 9.
11 References [1] Arkus,N., Manoharan,V., Brenner,M.: eriving Finite Sphere Packings, SIAM J. iscrete Math. Vol. 25, No. 4, pp , [2] Bezdek,K.: Lectures on Sphere Arrangements - the iscrete Geometric Side, Springer, 213. [3] Bezdek,K., Reid,S.: Contact graphs of unit sphere packings revisited, J. Geom. 14 (213), no. 1, [4] Bezdek,K., Szalkai,B., Szalkai,I.: On contact numbers of totally separable unit sphere packings, iscrete Mathematics, Vol. 339, No. 2, (215), Pages [5] Bezdek,K., Khan,M.A.: Contact numbers for sphere packings, [6] Blair,., Santangelo,C.., Machta,J.: Packing Squares in a Torus, J. Statistical Mechanics: Th. and Experiment, P118 (212), [7] Harborth, H.: Lösung zu Problem 664A, Elem. Math. 29, 1415 (1974). [8] Holmes-Cerfon,M.: Enumerating nonlinearly rigid sphere packings, [9] Hoy,R.S., Harwayne-Gidansky,J., O'Hern,C.S.: Structure of finite sphere packings via exact enumeration: Implications for colloidal crystal nucleation, Physical Review E 85 (212), [1] Neves, E.J.: A discrete variational problem related to Ising droplets at low temperatu res, J. Statistical Physics, 8 (1995), [11] Reid, S.: Regular Totally Separable Sphere Packings, RESEARCHJune 215, OI: /RG , [12] Reid, S.: On Contact Numbers of Finite Lattice Sphere Packings and the Maximal Coor dination of Monatomic Crystals, [13] Szalkai,I.: Handling Multicomponent Systems in R n, J. Math.Chem. 25 (1999),
Sphere Packings, Coverings and Lattices
Sphere Packings, Coverings and Lattices Anja Stein Supervised by: Prof Christopher Smyth September, 06 Abstract This article is the result of six weeks of research for a Summer Project undertaken at the
NOTE. On the Maximum Number of Touching Pairs in a Finite Packing of Translates of a Convex Body
Journal of Combinatorial Theory, Series A 98, 19 00 (00) doi:10.1006/jcta.001.304, available online at http://www.idealibrary.com on NOTE On the Maximum Number of Touching Pairs in a Finite Packing of
Class 29: Reciprocal Space 3: Ewald sphere, Simple Cubic, FCC and BCC in Reciprocal Space
Class 29: Reciprocal Space 3: Ewald sphere, Simple Cubic, FCC and BCC in Reciprocal Space We have seen that diffraction occurs when, in reciprocal space, Let us now plot this information. Let us designate
On Minimal Tilings with Convex Cells Each Containing a Unit Ball
On Minimal Tilings with Convex Cells Each Containing a Unit Ball Károly Bezdek Abstract We investigate the following problem that one can regard as a very close relative of the densest sphere packing problem.
Intermediate Math Circles March 11, 2009 Sequences and Series
1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Intermediate Math Circles March 11, 009 Sequences and Series Tower of Hanoi The Tower of Hanoi is a game
Tomato Packing and Lettuce-Based Crypto
Tomato Packing and Lettuce-Based Crypto A. S. Mosunov and L. A. Ruiz-Lopez University of Waterloo Joint PM & C&O Colloquium March 30th, 2017 Spring in Waterloo Love is everywhere... Picture from https://avatanplus.com/files/resources/mid/
Prestress stability. Lecture VI. Session on Granular Matter Institut Henri Poincaré. R. Connelly Cornell University Department of Mathematics
Prestress stability Lecture VI Session on Granular Matter Institut Henri Poincaré R. Connelly Cornell University Department of Mathematics 1 Potential functions How is the stability of a structure determined
Lecture 4! ü Review on atom/ion size! ü Crystal structure (Chap 4 of Nesseʼs book)!
Lecture 4! ü Review on atom/ion size! ü Crystal structure (Chap 4 of Nesseʼs book)! 15 C 4+ 42 Si 4+ Size of atoms! Hefferan and O Brien, 2010; Earth Materials Force balance! Crystal structure (Chap. 4)!
Analogs of Steiner s porism and Soddy s hexlet in higher dimensions via spherical codes
Analogs of Steiner s porism and Soddy s hexlet in higher dimensions via spherical codes Oleg R. Musin arxiv:1801.08278v2 [math.mg] 2 Feb 2018 Abstract In this paper we consider generalizations to higher
Density bounds for outer parallel domains of unit ball packings
Density bounds for outer parallel domains of unit ball packings Károly Bezdek and Zsolt Lángi arxiv:1409.4508v1 [math.mg] 16 Sep 014 September 17, 014 Abstract We give upper bounds for the density of unit
Packing Congruent Bricks into a Cube
Journal for Geometry and Graphics Volume 5 (2001), No. 1, 1 11. Packing Congruent Bricks into a Cube Ákos G. Horváth, István Prok Department of Geometry, Budapest University of Technology and Economics
DISCRETE MINIMAL ENERGY PROBLEMS
DISCRETE MINIMAL ENERGY PROBLEMS Lecture II E. B. Saff Center for Constructive Approximation Vanderbilt University University of Crete, Heraklion May, 2017 Recall Notation ω N = {x 1,..., x N } A, A compact,
Combinatorial Generalizations of Jung s Theorem
Discrete Comput Geom (013) 49:478 484 DOI 10.1007/s00454-013-9491-3 Combinatorial Generalizations of Jung s Theorem Arseniy V. Akopyan Received: 15 October 011 / Revised: 11 February 013 / Accepted: 11
The Densest Packing of 13 Congruent Circles in a Circle
Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 44 (003), No., 431-440. The Densest Packing of 13 Congruent Circles in a Circle Ferenc Fodor Feherviz u. 6, IV/13 H-000 Szentendre,
Ceramic Bonding. CaF 2 : large SiC: small
Recall ceramic bonding: - Mixed ionic and covalent. - % ionic character ( f ) increases with difference in electronegativity Large vs small ionic bond character: Ceramic Bonding CaF : large SiC: small
A shape which can be divided by a straight line, so that each part is the mirror image of the other has reflective symmetry.
SYMMETRY There are two kinds of symmetry: a) reflective symmetry b) rotational symmetry a) Reflective Symmetry A shape which can be divided by a straight line, so that each part is the image of the other
Activity 5&6: Metals and Hexagonal Close-Packing
Chemistry 150 Name(s): Activity 5&6: Metals and Hexagonal Close-Packing Metals are chemicals characterized by high thermal and electrical conductivity, malleability and ductility. Atoms are the smallest
Lecture 1 Maths for Computer Science. Denis TRYSTRAM Lecture notes MoSIG1. sept. 2017
Lecture 1 Maths for Computer Science Denis TRYSTRAM Lecture notes MoSIG1 sept. 2017 1 / 21 Context The main idea of this preliminary lecture is to show how to obtain some results in Mathematics (in the
Rational Numbers and Exponents
Rational and Exponents Math 7 Topic 4 Math 7 Topic 5 Math 8 - Topic 1 4-2: Adding Integers 4-3: Adding Rational 4-4: Subtracting Integers 4-5: Subtracting Rational 4-6: Distance on a Number Line 5-1: Multiplying
Part II. Geometry and Groups. Year
Part II Year 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2014 Paper 4, Section I 3F 49 Define the limit set Λ(G) of a Kleinian group G. Assuming that G has no finite orbit in H 3 S 2, and that Λ(G),
Math Tool: Dot Paper. Reproducible page, for classroom use only Triumph Learning, LLC
Math Tool: Dot Paper A Reproducible page, for classroom use only. 0 Triumph Learning, LLC CC_Mth_G_TM_PDF.indd /0/ : PM Math Tool: Coordinate Grid y 7 0 9 7 0 9 7 0 7 9 0 7 9 0 7 x Reproducible page, for
PACKERS PUZZLE. Teacher s Guide Getting Started. Kai Chung Tam Macau, People s Republic of China
Teacher s Guide Getting Started Kai Chung Tam Macau, People s Republic of China Purpose In this two-day lesson, students consider ways to estimate the number of spheres that will fit within a container.
Structure of Materials Prof. Anandh Subramaniam Department of Material Science and Engineering Indian Institute of Technology, Kanpur
Structure of Materials Prof. Anandh Subramaniam Department of Material Science and Engineering Indian Institute of Technology, Kanpur Lecture - 5 Geometry of Crystals: Symmetry, Lattices The next question
MATH GRADE 8 PLD Standard Below Proficient Approaching Proficient Proficient Highly Proficient
MATH GRADE 8 PLD Standard Below Proficient Approaching Proficient Proficient Highly Proficient The Level 1 student is below proficient The Level 2 student is approaching The Level 3 student is proficient
Practice Problems Set II
P1. For the HCP crystal structure, (a) show that the ideal c/a ratio is 1.633; (b) show that the atomic packing factor for HCP is 0.74. (a) A sketch of one-third of an HCP unit cell is shown below. Consider
Optimal configurations for classical interacting particles
Optimal configurations for classical interacting particles Edoardo Mainini DIME, Università degli Studi di Genova, & Facoltà di Matematica, Università di Vienna Joint research with Ulisse Stefanelli, Paolo
COUNTING AND SYMMETRY
COUNTING AND SYMMETRY INTERMEDIATE GROUP NOVEMBER 6, 2016 Warm Up Problem 1. Examine the figure below. (1) How many lines of symmetry does the figure have? (2) How many symmetries does the figure have?
Crystal Models. Figure 1.1 Section of a three-dimensional lattice.
Crystal Models The Solid-State Structure of Metals and Ionic Compounds Objectives Understand the concept of the unit cell in crystalline solids. Construct models of unit cells for several metallic and
Write your Name, Registration Number, Test Centre, Test Code and the Number of this booklet in the appropriate places on the answersheet.
2009 Booklet No. Test Code : SIA Forenoon Questions : 30 Time : 2 hours Write your Name, Registration Number, Test Centre, Test Code and the Number of this booklet in the appropriate places on the answersheet.
The 123 Theorem and its extensions
The 123 Theorem and its extensions Noga Alon and Raphael Yuster Department of Mathematics Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv, Israel Abstract It is shown
POLYOMINOES WITH MINIMUM SITE-PERIMETER AND FULL SET ACHIEVEMENT GAMES. Nándor Sieben
POLYOMINOES WITH MINIMUM SITE-PERIMETER AN FULL SET ACHIEVEMENT GAMES Nándor Sieben Abstract The site-perimeter of a polyomino is the number of empty cells connected to the polyomino by an edge. A formula
Interesting Surfaces in Nil Space *
Proceedings of the 8 th International Conference on Applied Informatics Eger, Hungary, January 27 30, 2010. Vol. 1. pp. 185 192. Interesting Surfaces in Nil Space * Benedek Schultz a, Jenő Szirmai b a
Mathematical Research Letters 1, (1994) AVERAGE KISSING NUMBERS FOR NON-CONGRUENT SPHERE PACKINGS. Greg Kuperberg and Oded Schramm
Mathematical Research Letters 1, 339 344 (1994) AVERAGE KISSING NUMBERS FOR NON-CONGRUENT SPHERE PACKINGS Greg Kuperberg and Oded Schramm 1. Introduction Let P be a packing of n (round) balls in R 3. (A
12-neighbour packings of unit balls in E 3
12-neighbour packings of unit balls in E 3 Károly Böröczky Department of Geometry Eötvös Loránd University Pázmány Péter sétány 1/c H-1117 Budapest Hungary László Szabó Institute of Informatics and Economics
1. Find the domain of the following functions. Write your answer using interval notation. (9 pts.)
MATH- Sample Eam Spring 7. Find the domain of the following functions. Write your answer using interval notation. (9 pts.) a. 9 f ( ) b. g ( ) 9 8 8. Write the equation of the circle in standard form given
arxiv: v1 [math.ho] 12 Dec 2016
![arxiv: v1 [math.ho] 12 Dec 2016](/thumbs/94/118989193.jpg "arxiv: v1 [math.ho] 12 Dec 2016")
General Two-variable Functions on the Slide-rule Istvan Szalkai University of Pannonia, Veszprém, Hungary szalkai@almos.uni-pannon.hu arxiv:6.03955v [math.ho] Dec 06 March 4, 09 Abstract We discuss the
INTRODUCTION. Fig. 1.1
1 INTRODUCTION 1.1 SYMMETRY: AN INTRODUCTION In nature, when we see the fascinating world of plants, flowers, birds, architectural buildings (Lotus Temple of Delhi, Taj Mahal, Ashoka Pillar, Rastrapati
DISCRETE MINIMAL ENERGY PROBLEMS
DISCRETE MINIMAL ENERGY PROBLEMS Lecture III E. B. Saff Center for Constructive Approximation Vanderbilt University University of Crete, Heraklion May, 2017 Random configurations ΩN = {X1, X2,..., XN }:
Advanced Ceramics for Strategic Applications Prof. H. S. Maiti Department of Mechanical Engineering Indian Institute of Technology, Kharagpur
Advanced Ceramics for Strategic Applications Prof. H. S. Maiti Department of Mechanical Engineering Indian Institute of Technology, Kharagpur Lecture -3 Crystal Structure Having made some introductory
A Stability Property of the Densest Circle Packing. (Received 9 December 1987)
Monatshefte f/it (1988) Maihemalik 9 by Springer-Verlag 1988 A Stability Property of the Densest Circle Packing By Imre Bfirfiny 1, Budapest, and N. P. Dolbilin, Moscow (Received 9 December 1987) Abstract.
On Distance Sets in the Triangular Lattice
On Distance Sets in the Triangular Lattice Tanbir Ahmed Department of Computer Science and Software Engineering Concordia University, Montréal, Canada ta ahmed@cs.concordia.ca David Jacob Wildstrom Department
THE UNIT DISTANCE PROBLEM ON SPHERES
THE UNIT DISTANCE PROBLEM ON SPHERES KONRAD J. SWANEPOEL AND PAVEL VALTR Abstract. For any D > 1 and for any n 2 we construct a set of n points on a sphere in R 3 of diameter D determining at least cn
Examples: u = is a vector in 2. is a vector in 5.
3 Vectors and vector equations We'll carefully define vectors, algebraic operations on vectors and geometric interpretations of these operations, in terms of displacements These ideas will eventually give
A Note on Projecting the Cubic Lattice
A Note on Projecting the Cubic Lattice N J A Sloane (a), Vinay A Vaishampayan, AT&T Shannon Labs, 180 Park Avenue, Florham Park, NJ 07932-0971, USA and Sueli I R Costa, University of Campinas, Campinas,
Crystals, Quasicrystals and Shape
Crystals, Quasicrystals and Shape by Azura Newman Table of contents 1) Introduction 2)Diffraction Theory 3)Crystals 4)Parametric Density 5) Surface Energy and Wulff-Shape 6) Quasicrystals 7) The Bound
SELECTIVELY BALANCING UNIT VECTORS AART BLOKHUIS AND HAO CHEN
SELECTIVELY BALANCING UNIT VECTORS AART BLOKHUIS AND HAO CHEN Abstract. A set U of unit vectors is selectively balancing if one can find two disjoint subsets U + and U, not both empty, such that the Euclidean
CS 372: Computational Geometry Lecture 14 Geometric Approximation Algorithms
CS 372: Computational Geometry Lecture 14 Geometric Approximation Algorithms Antoine Vigneron King Abdullah University of Science and Technology December 5, 2012 Antoine Vigneron (KAUST) CS 372 Lecture
Geol /19/06 Labs 5 & 6 Crystal Chemistry Ionic Coordination and Mineral Structures
Geol 2311 9/19/0 Labs 5 & Crystal Chemistry Ionic Coordination and Mineral Structures Handout Oral Mineral Tray Report Samples Ionic Coordination Exercise Investigating Mineral Structures using XtalDraw
Hawai`i Post-Secondary Math Survey -- Content Items
Hawai`i Post-Secondary Math Survey -- Content Items Section I: Number sense and numerical operations -- Several numerical operations skills are listed below. Please rank each on a scale from 1 (not essential)
Academic Vocabulary CONTENT BUILDER FOR THE PLC MATH GRADE 8
Academic Vocabulary CONTENT BUILDER FOR THE PLC MATH GRADE 8 Real Number Relationships 8.2 Number and operations. The student applies mathematical process standards to represent and use real numbers in
A Pair in a Crowd of Unit Balls
Europ. J. Combinatorics (2001) 22, 1083 1092 doi:10.1006/eujc.2001.0547 Available online at http://www.idealibrary.com on A Pair in a Crowd of Unit Balls K. HOSONO, H. MAEHARA AND K. MATSUDA Let F be a
PRACTICE TEST ANSWER KEY & SCORING GUIDELINES GRADE 8 MATHEMATICS
Ohio s State Tests PRACTICE TEST ANSWER KEY & SCORING GUIDELINES GRADE 8 MATHEMATICS Table of Contents Questions 1 25: Content Summary and Answer Key... iii Question 1: Question and Scoring Guidelines...
Experiment 7: Understanding Crystal Structures
Experiment 7: Understanding Crystal Structures To do well in this laboratory experiment you need to be familiar with the concepts of lattice, crystal structure, unit cell, coordination number, the different
To gain a working knowledge of unit cells, packing efficiency, coordination numbers and their relationship to each other.
Solid State Modeling PURPOSE A B To explore some simple solid state structures. To explore unit cell stoichiometry. GOALS To visualize the three-dimensional arrangement of atoms and ions in common solid
The Densest Geodesic Ball Packing by a Type of Nil Lattices
Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 48 (7), No., 383-397. The Densest Geodesic Ball Packing by a Type of Nil Lattices Jenő Szirmai Budapest University of Technology
Randomness in Number Theory *
Randomness in Number Theory * Peter Sarnak Asia Pacific Mathematics Newsletter Number Theory Probability Theory Mahler (953): Whole numbers Random objects Prime numbers Points in space Arithmetic operations
MATH Calculus IV Spring 2014 Three Versions of the Divergence Theorem
MATH 2443 008 Calculus IV pring 2014 Three Versions of the Divergence Theorem In this note we will establish versions of the Divergence Theorem which enable us to give it formulations of div, grad, and
BOUNDS FOR SOLID ANGLES OF LATTICES OF RANK THREE
BOUNDS FOR SOLID ANGLES OF LATTICES OF RANK THREE LENNY FUKSHANSKY AND SINAI ROBINS Abstract. We find sharp absolute consts C and C with the following property: every well-rounded lattice of rank 3 in
Ohio s State Tests ITEM RELEASE SPRING 2017 GRADE 8 MATHEMATICS
Ohio s State Tests ITEM RELEASE SPRING 2017 GRADE 8 MATHEMATICS Table of Contents Questions 1 22: Content Summary and Answer Key... iii Question 1: Question and Scoring Guidelines... 1 Question 1: Sample
GCSE Mathematics Specification (8300/2H)
NEW SPECIMEN PAPERS PUBLISHED JUNE 2015 GCSE Mathematics Specification (8300/2H) Paper 2 Higher tier H Date Morning 1 hour 30 minutes Materials For this paper you must have: a calculator mathematical instruments.
Distinct difference configurations and Wireless Sensor Networks
Distinct difference configurations and Wireless Sensor Networks Simon R. Blackburn Joint work with: Tuvi Etzion, Keith M. Martin, Maura B. Paterson Royal Holloway, University of London 12th May 2009 S.R.
SOLID STATE 18. Reciprocal Space
SOLID STATE 8 Reciprocal Space Wave vectors and the concept of K-space can simplify the explanation of several properties of the solid state. They will be introduced to provide more information on diffraction
Note on the chromatic number of the space
Note on the chromatic number of the space Radoš Radoičić Géza Tóth Abstract The chromatic number of the space is the minimum number of colors needed to color the points of the space so that every two points
BMO Round 2 Problem 3 Generalisation and Bounds
BMO 2007 2008 Round 2 Problem 3 Generalisation and Bounds Joseph Myers February 2008 1 Introduction Problem 3 (by Paul Jefferys) is: 3. Adrian has drawn a circle in the xy-plane whose radius is a positive
1 Convexity, concavity and quasi-concavity. (SB )
UNIVERSITY OF MARYLAND ECON 600 Summer 2010 Lecture Two: Unconstrained Optimization 1 Convexity, concavity and quasi-concavity. (SB 21.1-21.3.) For any two points, x, y R n, we can trace out the line of
Item 6. Pi and the Universe. Gives Applications to the Geometry of our Universe. 5 Total Pages
Item 6 Pi and the Universe Gives Applications to the Geometry of our Universe 5 Total Pages 1 Pi and the Universe How Geometry Proves the Correct Ratio for Pi 201/64, as a decimal 3.140625 For our geometry
Review: Directed Models (Bayes Nets)
X Review: Directed Models (Bayes Nets) Lecture 3: Undirected Graphical Models Sam Roweis January 2, 24 Semantics: x y z if z d-separates x and y d-separation: z d-separates x from y if along every undirected
2008 Euclid Contest. Solutions. Canadian Mathematics Competition. Tuesday, April 15, c 2008 Centre for Education in Mathematics and Computing
Canadian Mathematics Competition An activity of the Centre for Education in Mathematics and Computing, University of Waterloo, Waterloo, Ontario 008 Euclid Contest Tuesday, April 5, 008 Solutions c 008
Name: MATH 3195 :: Fall 2011 :: Exam 2. No document, no calculator, 1h00. Explanations and justifications are expected for full credit.
Name: MATH 3195 :: Fall 2011 :: Exam 2 No document, no calculator, 1h00. Explanations and justifications are expected for full credit. 1. ( 4 pts) Say which matrix is in row echelon form and which is not.
43.1 Vector Fields and their properties
Module 15 : Vector fields, Gradient, Divergence and Curl Lecture 43 : Vector fields and their properties [Section 43.1] Objectives In this section you will learn the following : Concept of Vector field.
Numeracy, Including Rational numbers and Square roots
Numeracy, Including Rational numbers and Square roots Objective No Daily Topic Key Idea The first 18 pages are review and have been added to ensure a smooth transition into the WNCP Math 9 curriculum.
MCR3U 1.1 Relations and Functions Date:
MCR3U 1.1 Relations and Functions Date: Relation: a relationship between sets of information. ie height and time of a ball in the air. In relations, the pairs of time and heights are "ordered"; ie ordered
MATH 118, LECTURES 13 & 14: POLAR EQUATIONS
MATH 118, LECTURES 13 & 1: POLAR EQUATIONS 1 Polar Equations We now know how to equate Cartesian coordinates with polar coordinates, so that we can represents points in either form and understand what
Distinct Difference Configurations
Distinct Difference Configurations Simon R. Blackburn Joint work with: Tuvi Etzion, Keith M. Martin, Maura B. Paterson Royal Holloway, University of London 16th July 009 S.R. Blackburn (RHUL) Distinct
SOLID STATE CHEMISTRY
SOLID STATE CHEMISTRY Crystal Structure Solids are divided into 2 categories: I. Crystalline possesses rigid and long-range order; its atoms, molecules or ions occupy specific positions, e.g. ice II. Amorphous
Highland Park Public School District
435 Mansfield Street Highland Park, New Jersey 08904 Curriculum Guide Mathematics 8 Version 1 Slated for Revision Final Review 6/30/2017 Mastery Skills Students will be able to understand, explain, and
Crystal Structure. Dr Bindu Krishnan
Solid State Physics-1 Crystal Structure Dr Bindu Krishnan CRYSTAL LATTICE What is crystal (space) lattice? In crystallography, only the geometrical properties of the crystal are of interest, therefore
SAMPLE EXAMINATION PAPER FORM 3
THE G C SCHOOL OF CAREERS MATHEMATICS DEPARTMENT SCHOOL YEAR 2017-2018 SAMPLE EXAMINATION PAPER FORM 3 Name: INFORMATION TO CANDIDATES Full marks may be obtained for answers to ALL questions. This paper
Kinematics. Chapter Multi-Body Systems
Chapter 2 Kinematics This chapter first introduces multi-body systems in conceptual terms. It then describes the concept of a Euclidean frame in the material world, following the concept of a Euclidean
COLLISIONS OF BILLIARD BALLS AND FOLDINGS. Krzysztof Burdzy University of Washington
University of Washington Collaborators and advisers Collaborators: Jayadev Athreya and Mauricio Duarte Work in progress. KB and Mauricio Duarte. A lower bound for the number of elastic collisions. arxiv:1803.00979
Improving Dense Packings of Equal Disks in a Square
Improving Dense Packings of Equal Disks in a Square David W. Boll Jerry Donovan Ronald L. Graham Boris D. Lubachevsky Hewlett-Packard Hewlett-Packard University of California Lucent Technologies 00 st
The effect of plasticity in crumpling of thin sheets: Supplementary Information
The effect of plasticity in crumpling of thin sheets: Supplementary Information T. Tallinen, J. A. Åström and J. Timonen Video S1. The video shows crumpling of an elastic sheet with a width to thickness
2. Which numbers below are perfect squares? Explain how you know. b) 0.004
Grade 9 Math Final Eam Review Unit 1 Outcomes Determine the square root of positive rational numbers that are perfect squares. o Determine whether or not a given rational number is a square number and
Date: Math 7 Final Exam Review Name:
Specific Outcome Achievement Indicators Chapter Got it?? the Solve a given problem involving the addition of two or more decimal numbers. addition, subtraction, multiplication and division of decimals
2.1 Definition. Let n be a positive integer. An n-dimensional vector is an ordered list of n real numbers.
2 VECTORS, POINTS, and LINEAR ALGEBRA. At first glance, vectors seem to be very simple. It is easy enough to draw vector arrows, and the operations (vector addition, dot product, etc.) are also easy to
Examples of Manifolds
Eamples of Manifolds Eample 1 Open Subset of IR n Anyopensubset, O, ofir n isamanifoldofdimension n. OnepossibleatlasisA = { O,ϕ id, whereϕ id istheidentitymap. Thatis, ϕ id =. Of course one possible choice
Name Period. Date: Topic: 9-2 Circles. Standard: G-GPE.1. Objective:
Name Period Date: Topic: 9-2 Circles Essential Question: If the coefficients of the x 2 and y 2 terms in the equation for a circle were different, how would that change the shape of the graph of the equation?
MENSURATION. Mensuration is the measurement of lines, areas, and volumes. Before, you start this pack, you need to know the following facts.
MENSURATION Mensuration is the measurement of lines, areas, and volumes. Before, you start this pack, you need to know the following facts. When you see kilo, it indicates 000 in length, mass and capacity.
Review of Last Class 1
Review of Last Class 1 X-Ray diffraction of crystals: the Bragg formulation Condition of diffraction peak: 2dd sin θθ = nnλλ Review of Last Class 2 X-Ray diffraction of crystals: the Von Laue formulation
Spring Lake Middle School- Accelerated Math 7 Curriculum Map Updated: January 2018
Domain Standard Learning Targets Resources Ratios and Proportional Relationships 7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured
Geographers Perspectives on the World
What is Geography? Geography is not just about city and country names Geography is not just about population and growth Geography is not just about rivers and mountains Geography is a broad field that
Solids. properties & structure
Solids properties & structure Determining Crystal Structure crystalline solids have a very regular geometric arrangement of their particles the arrangement of the particles and distances between them is
Algebra 2 End of Course Review
1 Natural numbers are not just whole numbers. Integers are all whole numbers both positive and negative. Rational or fractional numbers are all fractional types and are considered ratios because they divide
Lesson 8: Why Stay with Whole Numbers?
Student Outcomes Students use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Students create functions that
Introductory Chemistry: A Foundation, 6 th Ed. Introductory Chemistry, 6 th Ed. Basic Chemistry, 6 th Ed.
Introductory Chemistry: A Foundation, 6 th Ed. Introductory Chemistry, 6 th Ed. Basic Chemistry, 6 th Ed. by Steven S. Zumdahl & Donald J. DeCoste University of Illinois Chapter 12 Chemical Bonding Structure
Section 4.1. the k-th coordinate function of f. We call the system of equations . =.
The Calculus of Functions of Several Variables Section 4. Geometry, Limits, and Continuity In this chapter we will treat the general case of a function mapping R m to R n. Since the cases m = and n = have
Theoretical Computer Science
Theoretical Computer Science 406 008) 3 4 Contents lists available at ScienceDirect Theoretical Computer Science journal homepage: www.elsevier.com/locate/tcs Discrete sets with minimal moment of inertia
Cambridge International Examinations Cambridge International General Certificate of Secondary Education
Cambridge International Examinations Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/11 Paper 1 (Core) October/November 2017 45 minutes Candidates
PreCalc 11 Chapter 1 Review Pack v1
Period: Date: PreCalc 11 Chapter 1 Review Pack v1 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Determine the first 4 terms of an arithmetic sequence,
Limits at. x means that x gets larger and larger without a bound. Oktay Olmez and Serhan Varma Calculus Lecture 2 1 / 1
Limits at x means that x gets larger and larger without a bound. Oktay Olmez and Serhan Varma Calculus Lecture 2 1 / 1 Limits at x means that x gets larger and larger without a bound. x means that x gets