C Standard^ 1Q73*B*3F*J Space Groups and Lattice Complexes
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The Conway Space-Filling
Symmetry: Art and Science Buenos Aires Congress, 2007 THE CONWAY SPACE-FILLING MICHAEL LONGUET-HIGGINS Name: Michael S. Longuet-Higgins, Mathematician and Oceanographer, (b. Lenham, England, 1925). Address: Institute for Nonlinear Science, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92037-0402, U.S.A. Email: [email protected] Fields of interest: Fluid dynamics, ocean waves and currents, geophysics, underwater sound, projective geometry, polyhedra, mathematical toys. Awards: Rayleigh Prize for Mathematics, Cambridge University, 1950; Hon.D. Tech., Technical University of Denmark, 1979; Hon. LL.D., University of Glasgow, Scotland, 1979; Fellow of the American Geophysical Union, 1981; Sverdrup Gold Medal of the American Meteorological Society, 1983; International Coastal Engineering Award of the American Society of Civil Engineers, 1984; Oceanography Award of the Society for Underwater Technology, 1990; Honorary Fellow of the Acoustical Society of America, 2002. Publications: “Uniform polyhedra” (with H.S.M. Coxeter and J.C.P. Miller (1954). Phil. Trans. R. Soc. Lond. A 246, 401-450. “Some Mathematical Toys” (film), (1963). British Association Meeting, Aberdeen, Scotland; “Clifford’s chain and its analogues, in relation to the higher polytopes,” (1972). Proc. R. Soc. Lond. A 330, 443-466. “Inversive properties of the plane n-line, and a symmetric figure of 2x5 points on a quadric,” (1976). J. Lond. Math. Soc. 12, 206-212; Part II (with C.F. Parry) (1979) J. Lond. Math. Soc. 19, 541-560; “Nested triacontahedral shells, or How to grow a quasi-crystal,” (2003) Math. Intelligencer 25, 25-43. Abstract: A remarkable new space-filling, with an unusual symmetry, was recently discovered by John H. -
A Parallelepiped Based Approach
3D Modelling Using Geometric Constraints: A Parallelepiped Based Approach Marta Wilczkowiak, Edmond Boyer, and Peter Sturm MOVI–GRAVIR–INRIA Rhˆone-Alpes, 38330 Montbonnot, France, [email protected], http://www.inrialpes.fr/movi/people/Surname Abstract. In this paper, efficient and generic tools for calibration and 3D reconstruction are presented. These tools exploit geometric con- straints frequently present in man-made environments and allow cam- era calibration as well as scene structure to be estimated with a small amount of user interactions and little a priori knowledge. The proposed approach is based on primitives that naturally characterize rigidity con- straints: parallelepipeds. It has been shown previously that the intrinsic metric characteristics of a parallelepiped are dual to the intrinsic charac- teristics of a perspective camera. Here, we generalize this idea by taking into account additional redundancies between multiple images of multiple parallelepipeds. We propose a method for the estimation of camera and scene parameters that bears strongsimilarities with some self-calibration approaches. Takinginto account prior knowledgeon scene primitives or cameras, leads to simpler equations than for standard self-calibration, and is expected to improve results, as well as to allow structure and mo- tion recovery in situations that are otherwise under-constrained. These principles are illustrated by experimental calibration results and several reconstructions from uncalibrated images. 1 Introduction This paper is about using partial information on camera parameters and scene structure, to simplify and enhance structure from motion and (self-) calibration. We are especially interested in reconstructing man-made environments for which constraints on the scene structure are usually easy to provide. -
Area, Volume and Surface Area
The Improving Mathematics Education in Schools (TIMES) Project MEASUREMENT AND GEOMETRY Module 11 AREA, VOLUME AND SURFACE AREA A guide for teachers - Years 8–10 June 2011 YEARS 810 Area, Volume and Surface Area (Measurement and Geometry: Module 11) For teachers of Primary and Secondary Mathematics 510 Cover design, Layout design and Typesetting by Claire Ho The Improving Mathematics Education in Schools (TIMES) Project 2009‑2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations. The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations. © The University of Melbourne on behalf of the international Centre of Excellence for Education in Mathematics (ICE‑EM), the education division of the Australian Mathematical Sciences Institute (AMSI), 2010 (except where otherwise indicated). This work is licensed under the Creative Commons Attribution‑NonCommercial‑NoDerivs 3.0 Unported License. http://creativecommons.org/licenses/by‑nc‑nd/3.0/ The Improving Mathematics Education in Schools (TIMES) Project MEASUREMENT AND GEOMETRY Module 11 AREA, VOLUME AND SURFACE AREA A guide for teachers - Years 8–10 June 2011 Peter Brown Michael Evans David Hunt Janine McIntosh Bill Pender Jacqui Ramagge YEARS 810 {4} A guide for teachers AREA, VOLUME AND SURFACE AREA ASSUMED KNOWLEDGE • Knowledge of the areas of rectangles, triangles, circles and composite figures. • The definitions of a parallelogram and a rhombus. • Familiarity with the basic properties of parallel lines. • Familiarity with the volume of a rectangular prism. • Basic knowledge of congruence and similarity. • Since some formulas will be involved, the students will need some experience with substitution and also with the distributive law. -
Evaporationin Icparticles
The JapaneseAssociationJapanese Association for Crystal Growth (JACG){JACG) Small Metallic ParticlesProduced by Evaporation in lnert Gas at Low Pressure Size distributions, crystal morphologyand crystal structures KazuoKimoto physiosLaboratory, Department of General Educatien, Nago)ra University Various experimental results ef the studies on fine 1. Introduction particles produced by evaperation and subsequent conden$ation in inert gas at low pressure are reviewed. Small particles of metals and semi-metals can A brief historical survey is given and experimental be produced by evaporation and subsequent arrangements for the production of the particles are condensation in the free space of an inert gas at described. The structure of the stnoke, the qualitative low pressure, a very simple technique, recently particle size clistributions, small particle statistios and "gas often referred to as evaporation technique"i) the crystallographic aspccts ofthe particles are consid- ered in some detail. Emphasis is laid on the crystal (GET). When the pressure of an inert is in the inorphology and the related crystal structures ef the gas range from about one to several tens of Torr, the particles efsome 24 elements. size of the particles produced by GET is in thc range from several to several thousand nm, de- pending on the materials evaporated, the naturc of the inert gas and various other evaporation conditions. One of the most characteristic fea- tures of the particles thus produced is that the particles have, generally speaking, very well- defined crystal habits when the particle size is in the range from about ten to several hundred nm. The crystal morphology and the relevant crystal structures of these particles greatly interested sDme invcstigators in Japan, 4nd encouraged them to study these propenies by means of elec- tron microscopy and electron difliraction. -
Crystallography Ll Lattice N-Dimensional, Infinite, Periodic Array of Points, Each of Which Has Identical Surroundings
crystallography ll Lattice n-dimensional, infinite, periodic array of points, each of which has identical surroundings. use this as test for lattice points A2 ("bcc") structure lattice points Lattice n-dimensional, infinite, periodic array of points, each of which has identical surroundings. use this as test for lattice points CsCl structure lattice points Choosing unit cells in a lattice Want very small unit cell - least complicated, fewer atoms Prefer cell with 90° or 120°angles - visualization & geometrical calculations easier Choose cell which reflects symmetry of lattice & crystal structure Choosing unit cells in a lattice Sometimes, a good unit cell has more than one lattice point 2-D example: Primitive cell (one lattice pt./cell) has End-centered cell (two strange angle lattice pts./cell) has 90° angle Choosing unit cells in a lattice Sometimes, a good unit cell has more than one lattice point 3-D example: body-centered cubic (bcc, or I cubic) (two lattice pts./cell) The primitive unit cell is not a cube 14 Bravais lattices Allowed centering types: P I F C primitive body-centered face-centered C end-centered Primitive R - rhombohedral rhombohedral cell (trigonal) centering of trigonal cell 14 Bravais lattices Combine P, I, F, C (A, B), R centering with 7 crystal systems Some combinations don't work, some don't give new lattices - C tetragonal C-centering destroys cubic = P tetragonal symmetry 14 Bravais lattices Only 14 possible (Bravais, 1848) System Allowed centering Triclinic P (primitive) Monoclinic P, I (innerzentiert) Orthorhombic P, I, F (flächenzentiert), A (end centered) Tetragonal P, I Cubic P, I, F Hexagonal P Trigonal P, R (rhombohedral centered) Choosing unit cells in a lattice Unit cell shape must be: 2-D - parallelogram (4 sides) 3-D - parallelepiped (6 faces) Not a unit cell: Choosing unit cells in a lattice Unit cell shape must be: 2-D - parallelogram (4 sides) 3-D - parallelepiped (6 faces) Not a unit cell: correct cell Stereographic projections Show or represent 3-D object in 2-D Procedure: 1. -
Unit 6 Visualising Solid Shapes(Final)
• 3D shapes/objects are those which do not lie completely in a plane. • 3D objects have different views from different positions. • A solid is a polyhedron if it is made up of only polygonal faces, the faces meet at edges which are line segments and the edges meet at a point called vertex. • Euler’s formula for any polyhedron is, F + V – E = 2 Where F stands for number of faces, V for number of vertices and E for number of edges. • Types of polyhedrons: (a) Convex polyhedron A convex polyhedron is one in which all faces make it convex. e.g. (1) (2) (3) (4) 12/04/18 (1) and (2) are convex polyhedrons whereas (3) and (4) are non convex polyhedron. (b) Regular polyhedra or platonic solids: A polyhedron is regular if its faces are congruent regular polygons and the same number of faces meet at each vertex. For example, a cube is a platonic solid because all six of its faces are congruent squares. There are five such solids– tetrahedron, cube, octahedron, dodecahedron and icosahedron. e.g. • A prism is a polyhedron whose bottom and top faces (known as bases) are congruent polygons and faces known as lateral faces are parallelograms (when the side faces are rectangles, the shape is known as right prism). • A pyramid is a polyhedron whose base is a polygon and lateral faces are triangles. • A map depicts the location of a particular object/place in relation to other objects/places. The front, top and side of a figure are shown. Use centimetre cubes to build the figure. -
Cross Product Review
12.4 Cross Product Review: The dot product of uuuu123, , and v vvv 123, , is u v uvuvuv 112233 uv u u u u v u v cos or cos uv u and v are orthogonal if and only if u v 0 u uv uv compvu projvuv v v vv projvu cross product u v uv23 uv 32 i uv 13 uv 31 j uv 12 uv 21 k u v is orthogonal to both u and v. u v u v sin Geometric description of the cross product of the vectors u and v The cross product of two vectors is a vector! • u x v is perpendicular to u and v • The length of u x v is u v u v sin • The direction is given by the right hand side rule Right hand rule Place your 4 fingers in the direction of the first vector, curl them in the direction of the second vector, Your thumb will point in the direction of the cross product Algebraic description of the cross product of the vectors u and v The cross product of uu1, u 2 , u 3 and v v 1, v 2 , v 3 is uv uv23 uvuv 3231,, uvuv 1312 uv 21 check (u v ) u 0 and ( u v ) v 0 (u v ) u uv23 uvuv 3231 , uvuv 1312 , uv 21 uuu 123 , , uvu231 uvu 321 uvu 312 uvu 132 uvu 123 uvu 213 0 similary: (u v ) v 0 length u v u v sin is a little messier : 2 2 2 2 2 2 2 2uv 2 2 2 uvuv sin22 uv 1 cos uv 1 uvuv 22 uv now need to show that u v2 u 2 v 2 u v2 (try it..) An easier way to remember the formula for the cross products is in terms of determinants: ab 12 2x2 determinant: ad bc 4 6 2 cd 34 3x3 determinants: An example Copy 1st 2 columns 1 6 2 sum of sum of 1 6 2 1 6 forward backward 3 1 3 3 1 3 3 1 diagonal diagonal 4 5 2 4 5 2 4 5 products products determinant = 2 72 30 8 15 36 40 59 19 recall: uv uv23 uvuv 3231, uvuv 1312 , uv 21 i j k i j k i j u1 u 2 u 3 u 1 u 2 now we claim that uvu1 u 2 u 3 v1 v 2 v 3 v 1 v 2 v1 v 2 v 3 iuv23 j uv 31 k uv 12 k uv 21 i uv 32 j uv 13 u v uv23 uv 32 i uv 13 uv 31 j uv 12 uv 21 k uv uv23 uvuv 3231,, uvuv 1312 uv 21 Example: Let u1, 2,1 and v 3,1, 2 Find u v. -
Uniform Panoploid Tetracombs
Uniform Panoploid Tetracombs George Olshevsky TETRACOMB is a four-dimensional tessellation. In any tessellation, the honeycells, which are the n-dimensional polytopes that tessellate the space, Amust by definition adjoin precisely along their facets, that is, their ( n!1)- dimensional elements, so that each facet belongs to exactly two honeycells. In the case of tetracombs, the honeycells are four-dimensional polytopes, or polychora, and their facets are polyhedra. For a tessellation to be uniform, the honeycells must all be uniform polytopes, and the vertices must be transitive on the symmetry group of the tessellation. Loosely speaking, therefore, the vertices must be “surrounded all alike” by the honeycells that meet there. If a tessellation is such that every point of its space not on a boundary between honeycells lies in the interior of exactly one honeycell, then it is panoploid. If one or more points of the space not on a boundary between honeycells lie inside more than one honeycell, the tessellation is polyploid. Tessellations may also be constructed that have “holes,” that is, regions that lie inside none of the honeycells; such tessellations are called holeycombs. It is possible for a polyploid tessellation to also be a holeycomb, but not for a panoploid tessellation, which must fill the entire space exactly once. Polyploid tessellations are also called starcombs or star-tessellations. Holeycombs usually arise when (n!1)-dimensional tessellations are themselves permitted to be honeycells; these take up the otherwise free facets that bound the “holes,” so that all the facets continue to belong to two honeycells. In this essay, as per its title, we are concerned with just the uniform panoploid tetracombs. -
41 Three Dimensional Shapes
41 Three Dimensional Shapes Space figures are the first shapes children perceive in their environment. In elementary school, children learn about the most basic classes of space fig- ures such as prisms, pyramids, cylinders, cones and spheres. This section concerns the definitions of these space figures and their properties. Planes and Lines in Space As a basis for studying space figures, first consider relationships among lines and planes in space. These relationships are helpful in analyzing and defining space figures. Two planes in space are either parallel as in Figure 41.1(a) or intersect as in Figure 41.1(b). Figure 41.1 The angle formed by two intersecting planes is called the dihedral angle. It is measured by measuring an angle whose sides line in the planes and are perpendicular to the line of intersection of the planes as shown in Figure 41.2. Figure 41.2 Some examples of dihedral angles and their measures are shown in Figure 41.3. 1 Figure 41.3 Two nonintersecting lines in space are parallel if they belong to a common plane. Two nonintersecting lines that do not belong to the same plane are called skew lines. If a line does not intersect a plane then it is said to be parallel to the plane. A line is said to be perpendicular to a plane at a point A if every line in the plane through A intersects the line at a right angle. Figures illustrating these terms are shown in Figure 41.4. Figure 41.4 Polyhedra To define a polyhedron, we need the terms ”simple closed surface” and ”polygonal region.” By a simple closed surface we mean any surface with- out holes and that encloses a hollow region-its interior. -
Space Groups and Lattice Complexes ; —
Mm 1W 18 072 if "^^^"^ Space Groups and Lattice Complexes ; — — Werner FISHER Mineralogisches Institut der Philipps-Universitat D-3550 Marburg/Lahn, German Federal Republic Hans BURZLAFF Lehrstuhl fiir Kristallographie der Friedrich-Alexander-Universitat Erlangen-Niirnberg D-8520 Erlangen, German Federal Republic Erwin HELLNER Fachbereich Geowissenschaften der Philipps-Universitat D-3550 Marburg/Lahn, German Federal Republic J. D. H. DONNAY Department de Geologic, Universite'de Montreal CP 6128 Montreal 101, Quebec, Canada Formerly, The Johns Hopkins University, Baltimore, Maryland Prepared under the auspices of the Commission on International Tables, International Union of Crystallography, and the Office of Standard Reference Data, National Bureau of Standards. U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secreiory NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Dnecior Issued May 1973 Library of Congress Catalog Number: 73-600099 National Bureau of Standards Monograph 134 Nat. Bur. Stand. (U.S.), Monogr. 134, 184 pages (May 1973) CODEN: NBSMA6 For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 (Order by SD Catalog No. C13,44:134). Price: $4.10, domestic postpaid; $3.75, GPO Bookstore Stock Number 0303-01143 Abstract The lattice complex is to the space group what the site set is to the point group - an assemblage of symmetry- related equivalent points. The symbolism introduced by Carl Hermann has been revised and extended. A total of 402 lattice complexes are derived from 67 Weissenberg com- plexes. The Tables list site sets and lattice complexes in standard and alternate representations. They answer the following questions: What are the co-ordinates of the points in a given lattice complex? In which space groups can a given lattice complex occur? What are the lattice complexes that can occur in a given space group? The higher the symmetry of the crystal structures is, the more useful the lattice- complex approach should be on the road to the ultimate goal of their classification. -
Study on Delaunay Tessellations of 1-Irregular Cuboids for 3D Mixed Element Meshes
Study on Delaunay tessellations of 1-irregular cuboids for 3D mixed element meshes David Contreras and Nancy Hitschfeld-Kahler Department of Computer Science, FCFM, University of Chile, Chile E-mails: dcontrer,[email protected] Abstract Mixed elements meshes based on the modified octree approach con- tain several co-spherical point configurations. While generating Delaunay tessellations to be used together with the finite volume method, it is not necessary to partition them into tetrahedra; co-spherical elements can be used as final elements. This paper presents a study of all co-spherical elements that appear while tessellating a 1-irregular cuboid (cuboid with at most one Steiner point on its edges) with different aspect ratio. Steiner points can be located at any position between the edge endpoints. When Steiner points are located at edge midpoints, 24 co-spherical elements appear while tessellating 1-irregular cubes. By inserting internal faces and edges to these new elements, this numberp is reduced to 13. When 1-irregular cuboids with aspect ratio equal to 2 are tessellated, 10 co- spherical elementsp are required. If 1-irregular cuboids have aspect ratio between 1 and 2, all the tessellations are adequate for the finite volume method. When Steiner points are located at any position, the study was done for a specific Steiner point distribution on a cube. 38 co-spherical elements were required to tessellate all the generated 1-irregular cubes. Statistics about the impact of each new element in the tessellations of 1-irregular cuboids are also included. This study was done by develop- ing an algorithm that construct Delaunay tessellations by starting from a Delaunay tetrahedral mesh built by Qhull. -
Mini Set 1 Activities
Mini Set 1 Activities Geometiles® is a product of TM www.geometiles.com Welcome to Geometiles®! Your Mini Set 1 contains 20 equilateral triangles and 12 regular pentagons. This booklet contains some problems and brainteasers for you to try. The puzzles are all at different levels, so there’s something for everyone. Puzzle 1 Make an equilateral triangle that is 1/9 yellow, 1/3 green and 5/9 purple. Now make an equilateral triangle that is 1/4 purple, 1/4 green, 3/16 orange and 5/16 yellow. Notes: In these problems, the student must first realize that he needs 9 and 16 triangles, respectively, to make each figure. (A multiple of 9 or 16 would also work, but this impossible due to the size of the set) Puzzle 2 Make a closed object out of all the 12 pentagons in your box such that any two adjacent tiles have different colors. Puzzle 3 Build a closed solid having 6 rhombic faces Hints: You can make a rhombus by joining two equilateral triangles. Think “stretched out cube”. Just as a cube is made of 6 square faces, with three faces joining at each corner, so too will your solid. The difference is that in your solid, the faces will be rhombic instead of square. Puzzle 4 A solid is strictly convex if a line segment joining any two of its points is entirely contained inside the solid (not on its surface or outside it). Note that a strictly convex solid may not have any coplanar faces. Here are some examples of solids that are NOT strictly convex.