DOCSLIB.ORG

7 Space Groups and Point Groups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
7 Space Groups and Point Groups 1 7 Space groups and point groups In certain cases the simple form may consist of merely one face (monohedron), two parallel faces (pinacoid), or two intersecting faces (dihedron). Shubnikov [1]. In the plane or in two dimensions there are 17 groups, in space or in three dimensions there are 230 groups. The number of groups increases rapidly when adding on dimension. The permutations in space under the constraint of periodic translation give the 230 groups. From the allowed symmetry operations - thus under the constraint of periodic translation - comes the 32 point groups. Instead of 10 in 2D. A particular combination of the variables like x,y,z corresponds to coordinates for a point in space. If we add them up like x+y+z they represent a surface. In order to keep an identity of each surface you may go cyclic (cosπ[x+y+z]) which has shown to be of extreme power in mathematics or natural science. Another way is to go exponential as below and surfaces keep again their identity and collaborate to give finite shapes. We describe the fundamental permutations of variables under the constraint of the allowed symmetry –the 32 point groups – as mathematical functions using the exponential scale. These general permutations and also some special permutations (that correspond to the special positions in a group) describe the 47 simple forms which crystals may take [1]. In these simple forms, all faces are equal to one another. Each term of permutation x,y,z describes the {hkl} faces of the simple form of the crystal of the corresponding mineral. Mathematically we use the concept of how planes build symmetries. Which is normal as periodic structures – like crystals – can be described as built of planes. In the permutations below for a position of type x,x,x in crystallographic literature we use x,y,z in the calculations in the Cartesian system of Mathematica. For a position of type x,y,z in the crystallographic literature we use here x,2y,3z, whenever needed. This is particularly useful for the cubic system as will come in the next chapter. The point groups except the cubic In our exponential expressions for planes to describe various forms it is possible to simplify, and certainly so in the use of hyperbolic functions for the centric cases. But we have preferred to use the elaborated expressions that are to be compared with the space group co-ordinates of equivalent positions. The names are after Shubnikov[1], and also from Bob’s rockshop[2]. 2 7.1 The triclinic, monoclinic and orthorhombic groups Below we give the groups with symbols, numbers and equivalent positions after the crystallographic literature[3]. Numbers in brackets are the Shubnikov numbers for the simple forms. P1 is group no 1 and has the co-ordinates x,y,z. We formulate the equation 7.1.1. And the figure is in 7.1.1. P1 No.1 x,y,z; x+ y+z e = 100 7.1.1 P 1 No.2 x,y,z; -x,-y,-z; This group has a centre of symmetry, as is obvious from figure 7.1.2. Its equation is in 7.1.2. x+ y+z ! x! y!z e + e = 100 7.1.2 The figures are called monohedron resp pinacoid(Shub 32). Figure 7.1.1 Monohedron Figure 7.1.2 Pinacoid P2 No.3 x,y,z; -x,y,-z; x+ y+z ! x+ y!z e + e = 100 7.1.3 The form is called dihedron, one plane is continually bent to another. 3 Figure 7.1.3a Dihedron b Different projection Pm No.6 x,y,z; x,-y,z; x+ y+z x! y+z e + e = 100 7.1.4 This form in figure 7.1.4 is also called dihedron(Shub 31). Bending again, but in another direction. Figure 7.1.4a Dihedron b Different projection P2/m No.10 x,y,z; -x,-y,-z; x,-y,z; -x,y,-z; x+ y+z ! x! y!z x! y+z ! x+ y!z e + e + e + e = 1000 7.1.5 The shape in figure 7.1.5 is an orthorhombic prism (Shub 15). Figure 7.1.5a Orthorhombic prism b Different projection 4 P222 No.16 x,y,z; -x,-y,z; x,-y,-z; -x,y,-z; Below is also the equation for a regular tetrahedron given in 7.1.6 to be compared with the next simple form 7.1.7. x+ y+z ! x! y+z x! y!z ! x+ y!z e + e + e + e = 1000 7.1.6 x+ 2y+3z !x! 2y+3z x!2 y! 3z !x+2 y! 3z e + e + e + e = 1000 7.1.7 The shape of this simple form in figure 7.1.7 is an orthorhombic tetrahedron (Shub 22). Figure 7.1.6 Regular tetrahedron Figure 7.1.7 Orthorhombic tetrahedron Pmm2 No.25 x,y,z; -x,-y,z; x,-y,z; -x,y,z; x+ 2y+3z !x! 2y+3z x!2 y+ 3z !x+2 y+ 3z e + e + e + e = 1000 7.1.8 The name of this simple form as shown in figure 7.1.8 is orthorhombic pyramid (Shub 1). Figure 7.1.8 Orthorhombic pyramid. Pmmm No.47 x,y,z; -x,-y,z; x,-y,-z; -x,y,-z; -x,-y,-z; x,y,-z; -x,y,z; x,-y,z; x 2y 3z x 2y 3z x 2 y 3z x 2 y 3z e + + + e! ! + + e ! ! + e! + ! + ! (x+2 y+3z) ! (!x!2 y+3z) ! (x! 2y+3z) !(!x + 2y+3z) e + e + e + e = 1000 7.1.9 5 The name is orthorhombic bipyramid (Shub 8), which is shown in figure 7.1.9. Figure 7.1.9a Orthorhombic bipyramid. b Different projection 7.2 Tetragonal groups P4 No.75 x,y,z; -x,-y,z; y,-x,z; -y,x,z; x+ 2y+3z !x! 2y+3z 2x! y+ 3z !2x+ y+ 3z 5 e + e + e + e = 10 7.2.1 The name is tetragonal pyramid(Shub 4) and shown in figure 7.2.1. Figure 7.2.1a Tetragonal pyramid b Different projection. P 4 No. 81 x,y,z; -x,-y,z; y,-x,-z; -y,x,-z; x+ 2y+3z !x! 2y+3z 2x! y! 3z !2x+ y! 3z 5 e + e + e + e = 10 7.2.2 The name is tetragonal tetrahedron(Shub 25)and shown in figure 7.2.2. Figure7.2.2a Tetragonal tetrahedron. b Different projection. 6 P4/m No.83 x,y,z; -x,-y,z; x,y,-z; -x,-y,-z; -y,x,z; y,-x,z; -y,x,-z; y,-x,-z; x 2y 3z x 2y 3z x 2y 3z x 2 y 3z e + + + e! ! + + e + ! + e! ! ! 7.2.3 !2x+ y+ 3z 2x! y+3z !2x+ y! 3z 2x! y!3z 6 + e + e + e + e = 10 The name is tetragonal bipyramid (Shub 11) and shown in figure 7.2.3. Figure 7.2.3a Tetragonal bipyramid. b Different projection. P422 No.89 x,y,z; -x,-y,z; -x,y,-z; x,-y,-z; -y,-x,-z; y,x,-z; -y,x,-z; y,-x,-z; x 2y 3z x 2y 3z x 2y 3z x 2 y 3z e + + + e! ! + + e! + ! + e ! ! 7.2.4 !2x! y! 3z 2x+ y!3z 2x! y+3z !2x + y+3z 6 + e + e + e + e = 10 The name is the tetragonal trapezohedron (Shub 26) and shown in figure 7.2.4. Figure 7.2.4a Tetragonal trapezohedron. b Different projection. 7 P4mm No.99 x,y,z; -x,-y,z; -x,y,z; x,-y,z; y,x,z; -y,-x,z; -y,x,z; y,-x,z; x 2y 3z x 2y 3z x 2y 3z x 2 y 3z e + + + e! ! + + e! + + + e ! + 7.2.5 2x+ y+3z !2x! y+3z !2x+ y+ 3z 2x! y+3z 10 + e + e + e + e = 10 The name is ditetragonal pyramid(Shub 5) and shown in figure 7.2.5. Figure 7.2.5a Ditetragonal pyramid. b Different projection. P 4 m2 No.115 x,y,z; -x,-y,z; -x,y,z; x,-y,z; -y,x,-z; y,-x,-z; y,x,-z; -y,-x,-z; x 2y 3z x 2y 3z x 2y 3z x 2 y 3z e + + + e! ! + + e! + + + e ! + 7.2.6 !2x+ y! 3z 2x! y!3z 2x+ y!3z !2x ! y!3z 10 + e + e + e + e = 10 The name is tetragonal scalenohedron (Shub 33) shown in figure 7.2.6. Figure 7.2.6a Tetragonal scalenohedron b Different projection. 8 There is a second mineral form to this symmetry called the tetragonal disphenoid with only four faces so we use the special position in 4j. We arrive at the equation in 2x+ z !2x +z 2y! z !2y !z 10 e + e + e + e = 10 and we see the disphenoid in figure 7.2.6c. The only difference from the tetragonal tetrahedron in P 4 above is its orientation. That give it mirror planes parallel to the axes in the co-ordinate system chosen. Figure 7.2.6c Disphenoid. P4/mmm No.123 x,y,z; -x,-y,z; x,y,-z; -x,-y,-z; -x,y,z; x,-y,z; -x,y,-z; x,-y,-z; -y,x,z; y,-x,z; -y,x,-z; y,-x,-z; y,x,z; -y,-x,z; y,x,-z; -y,-x,-z; ex+ 2y+3z + e!x! 2y+3z + ex+ 2y! 3z + e!x!2 y! 3z x 2 y 3z x 2 y 3z x 2 y 3z x 2y 3z +e! + + + e ! + + e! + ! + e ! ! 7.2.7 +e!2x+ y+ 3z + e2x! y+3z + e!2x+ y! 3z + e2x! y!3z 2x+ y+3z !2x! y+3z 2x+ y!3z !2x! y!3z 10 + e + e + e + e = 10 The name is ditetragonal bipyramid (Shub 12) and shown in figure 7.2.7.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Hexagonal-System.Pdf
    94 CRYSTALLOGRAPHY the indices of the face in question. The indices of a prism face like l(310) can be readily obtained in exactly the same manner as described under the Isometric System, Art. 84. p. 75. 111. HEXAGONAL SYSTEM 119. The HEXAGONALSYSTEM includes all the forms which are referred to four axes, three equal horizontal axes in a common plane intersecting at angles of 60") and a fourth, vertical axis, at right angles to them. Two sections are here included, each embracing a number of distinct classes related among themselves. They are called the Hexagonal Division and the Trigonal (or Rhombohedral) Division. The symmetry of the former, about the vertical axis, belongs to the hexagonal type, that of the latter to the trigonal type. Miller (1852) referred all the forms of the hexagonal system to three equal axes arallel to the faces of the fundamental rhombohedron, and hence intersecting at e ual angr)es, not 90". This method (further explained in Art. 169) had the disadvantage ofyailing to bring out the relationship between the normal hexagonal and tetragonal types, both characterized by a principal axis of symmetry, which (on the system adopted in this book) is the vertical crystallo raphic axis. It further gave different symbols to faces which are crystallo- graphic$y identical. It is more natural to employ the three rhombohedra1 axes for tri- gonal forms only, aa done by Groth (1905), who includes these groups in a Trigonal Syslem; but this also has some disadvantages. The indices commonly used in describing hexagonal forms are known as the Miller-Bravais indices, since the were adopted by Bravais for use with the four axes from the scheme used by Miller in tKe other crystal systems.
    [Show full text]
  • 10.1. CRYSTALLOGRAPHIC and NONCRYSTALLOGRAPHIC POINT GROUPS Table 10.1.1.2
    International Tables for Crystallography (2006). Vol. A, Section 10.1.2, pp. 763–795. 10.1. CRYSTALLOGRAPHIC AND NONCRYSTALLOGRAPHIC POINT GROUPS Table 10.1.1.2. The 32 three-dimensional crystallographic point groups, arranged according to crystal system (cf. Chapter 2.1) Full Hermann–Mauguin (left) and Schoenflies symbols (right). Dashed lines separate point groups with different Laue classes within one crystal system. Crystal system General Monoclinic (top) symbol Triclinic Orthorhombic (bottom) Tetragonal Trigonal Hexagonal Cubic n 1 C1 2 C2 4 C4 3 C3 6 C6 23 T n 1 Ci m 2 Cs 4 S4 3 C3i 6 3=mC3h –– n=m 2=mC2h 4=mC4h ––6=mC6h 2=m3 Th n22 222 D2 422 D4 32 D3 622 D6 432 O nmm mm2 C2v 4mm C4v 3mC3v 6mm C6v –– n2m ––42mD2d 32=mD3d 62mD3h 43mTd n=m 2=m 2=m 2=m 2=m 2=mD2h 4=m 2=m 2=mD4h ––6=m 2=m 2=mD6h 4=m 32=mOh 10.1.2. Crystallographic point groups the a axis points down the page and the b axis runs horizontally from left to right. For the five trigonal point groups, the c axis is 10.1.2.1. Description of point groups normal to the page only for the description with ‘hexagonal axes’; if In crystallography, point groups usually are described described with ‘rhombohedral axes’, the direction [111] is normal (i) by means of their Hermann–Mauguin or Schoenflies symbols; and the positive a axis slopes towards the observer. The (ii) by means of their stereographic projections; conventional coordinate systems used for the various crystal (iii) by means of the matrix representations of their symmetry systems are listed in Table 2.1.2.1 and illustrated in Figs.
    [Show full text]
  • Hexagon - Wikipedia
    12/2/2018 Hexagon - Wikipedia Hexagon In geometry, a hexagon (from Greek ἕξ hex, "six" and γωνία, gonía, "corner, angle") is a six sided polygon or 6-gon. The total Regular hexagon of the internal angles of any hexagon is 7 20°. Contents Regular hexagon Parameters Symmetry A2 and G2 groups Related polygons and tilings Hexagonal structures A regular hexagon Tesselations by hexagons Type Regular polygon Hexagon inscribed in a conic section Cyclic hexagon Edges and 6 vertices Hexagon tangential to a conic section Schläfli {6}, t{3} Equilateral triangles on the sides of an arbitrary hexagon symbol Skew hexagon Petrie polygons Coxeter diagram Convex equilateral hexagon Polyhedra with hexagons Symmetry Dihedral (D6), order Hexagons: natural and human-made group 2×6 See also Internal 120° References angle External links (degrees) Dual Self polygon https://en.wikipedia.org/wiki/Hexagon 1/18 12/2/2018 Hexagon - Wikipedia Regular hexagon Properties Convex, cyclic, equilateral, isogonal, A regular hexagon has Schläfli symbol {6}[1] and can also be constructed as a truncated equilateral triangle, t{3}, which isotoxal alternates two types of edges. A regular hexagon is defined as a hexagon that is both equilateral and equiangular. It is bicentric, meaning that it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle). The common length of the sides equals the radius of the circumscribed circle, which equals times the apothem (radius of the inscribed circle). All internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries (rotational symmetry of order six) and 6 reflection symmetries (six lines of symmetry), making up the dihedral group D6 .
    [Show full text]
  • Do What You Can. These Activities Are for the Week. If You Feel Like Y
    Week beginning 18/05/2020 Maths Week 5 Remember – Do what you can. These activities are for the week. If you feel like you are up for the extra challenge, BBC Bitesize have new daily lessons for English and Maths on their website. These are specific for each year group. You also have the online White Rose Maths that have daily Maths lessons and the new Maths Shed resource. Both of these are excellent resources and you will be able to extend this work on shape through them. The following is learning you would have been doing at school with us. Starters – try doing one every day: These starters are objectives that you have learnt in Year 5. • Revise times tables and all related division facts. • Pick a number between 0 and 20 or 20 and 100. Now double that number. Now try halving that number. Use partitioning to help you. Repeat with 10 other numbers. • Write equivalent fractions and decimals for: 25%, 50%, 75%, 33%, 89%, 90%, 2%, 8% and 16%. • Find ½, ¼ and 1/10 of: 20, 36, 40, 60, 66 and 72. • Multiply these numbers by 10,100 and 1000: 6, 17, 26, 59, 66 and 101. Can you remember the rules? Write them down. You can use the squares below to move the digits if you need to. Shape Week 2 L.O. To identify the properties of prisms and pyramids. Knowledge: A pyramid is named after the shape of the base as you can see from the above. The number of triangular faces the shape has, depends upon how many edges the polygon shape at the bottom has.
    [Show full text]
  • Reteach 10-1 Introduction to Three-Dimensional Figures a Polyhedron Is a Three-Dimensional Figure Whose Faces Are Polygons
    MSM07G7_RESBK_CH10_003–010.pe 2/14/06 11:27 AM Page 6 Name Date Class LESSON Reteach 10-1 Introduction to Three-Dimensional Figures A polyhedron is a three-dimensional figure whose faces are polygons. There are two types of polyhedrons. Prisms Pyramids A prism has 2 bases that are congruent A pyramid has one base that is a and parallel polygons. The other faces polygon. Its faces are triangles that are parallelograms or rectangles. meet at one vertex. Prisms and pyramids are named by their bases. 1. Look at the figure at the right. Its base is a square . Its faces are triane, . It is called a square pyramid. 2. Look at the figure at the right. It has 2 congruent and parallel bases that are triane, . Its faces are triane, . It is called a pentagonal prism. If a three-dimensional figure has a face that is curved, it is not a polyhedron. not a polyhedron Is each figure a polyhedron? 3. 4. 5. triane, no triane, Copyright © by Holt, Rinehart and Winston. All rights reserved. 6 Holt Mathematics MSM07G7_RESBK_Ch10_045-055.pe 2/14/06 11:50 AM Page 45 LESSON Practice A LESSON Practice B 10-1 Introduction to Three-Dimensional Figures 10-1 Introduction to Three-Dimensional Figures Identify the base of each prism or pyramid. Then choose the Identify the base or bases of each figure. Then name the figure. name of the prism or pyramid from the box. 1. 2. 3. rectangular prism square pyramid triangular prism pentagonal prism square prism triangular pyramid hexagonal prism rectangular pyramid hexagonal pyramid pentagonal pyramid octagonal prism 1.
    [Show full text]
  • A Standardized Japanese Nomenclature for Crystal Forms
    MINERALOGICAL JOURNAL, VOL. 4, No. 4, pp. 291-298, FEB., 1565 A STANDARDIZED JAPANESE NOMENCLATURE FOR CRYSTAL FORMS J. D. H. DONNAY and HIROSHI TAKEDA The Johns Hopkins University, Baltimore, Maryland, U. S. A. with the help of many Japanese crystallographers ABSTRACT The system of the crystal-form names according to the proposal of the Fedorov Institute is explained, and adequate Japanese equivalents are given. Introduction Efforts have been made in the past to standardize the names of the crystal forms. Particularly noteworthy were the proposals of the Fedorov Institute (Boldyrev 1925, 1936) and of Rogers (1935), both of which are modifications of the Groth nomenclature. Recently the Committee on Nomenclature of the French Society of Mineralogy and Crystallography have re-examined the problem (see Donnay and Curien 1959) and have essentially proposed the sys tem of the Fedorov Institute as a basis for possible international agreement. In view of these developments, it was thought that it would be useful to see if the Japanese nomenclature of crystal forms could be adapted to the Federov-Institute system. The present paper first explains the system of form names, and attempts to give adequate Japanese equivalents, for non-cubic and then for cubic forms. Then the English names are listed with syno nyms, together with the various Japanese terms that have been used. When there is more than one Japanese name, they are listed in a tentative order of preference, arrived at after many inquiries and consultations. Finally a key to the crystal forms in the 32 point 292 A Standardized Japanese Nomenclature for Crystal Forms groups is given.
    [Show full text]
  • 2Lines, Shapes and Solids
    Lines, shapes Chapter2and solids What Australian you will learn curriculum 2A Angles at a point MEASUREMENT AND GEOMETRY (Consolidating) Geometric reasoning 2B Parallel lines (Consolidating) Establish properties of quadrilaterals using congruent triangles 2C Triangles (Consolidating) and angle properties, and16x16 solve related numerical problems32x32 using 2D Quadrilaterals reasoning (ACMNA202) Polygons (Extending) 2E Solids and Euler’s rule UNCORRECTED(Extending) SAMPLE PAGES Uncorrected 3rd sample pages • Cambridge University Press © Greenwood et al., 2015 • 978-1-107-56885-3 • Ph 03 8671 1400 Online resources • Chapter pre-test • Videos of all worked examples • Interactive widgets • Interactive walkthroughs • Downloadable HOTsheets • Access to HOTmaths Australian Curriculum courses The geometry of honey The cells in honeycomb made Instead, the geometry of the cells by bees are hexagonal in shape, allows the cells to t neatly together but each cell is not exactly a to form a very ef cient geometrical hexagonal prism. Each cell is construction. The structure allows actually a dodecahedron (12-faced for a minimum surface area for a polyhedron) with 6 rectangular sides given volume, maximising the use of (giving the hexagonal appearance) space in which to store honey. and 3 faces at each end. Each of the UNCORRECTEDend faces is a rhombus and forms angles of 120° with the other sides of the honeycomb cell. The shape of each cell is not just a random arrangementSAMPLE of quadrilateral faces. PAGES Uncorrected 3rd sample pages • Cambridge University Press © Greenwood et al., 2015 • 978-1-107-56885-3 • Ph 03 8671 1400 52 Chapter 2 Lines, shapes and solids 2A Angles at a point CONSOLIDATING For more than 2000 years, geometry has been based on the work of Euclid, the Greek mathematician who lived in Egypt in about 300 BCE.
    [Show full text]
  • Chapter 17 Mensuration of Pyramid
    344 Applied Math Mensuration of Pyramid Chapter 17 Mensuration of Pyramid 17.1 Pyramid: A pyramid is a solid whose base is a plane polygon and sides are triangles that meet in a common vertex. The triangular sides are called lateral faces. The common vertex is also called Apex. A pyramid is named according to the shape of its base. If the base is a triangle, square, hexagon etc. the pyramid is called as a triangular pyramid, a square pyramid, a hexagonal pyramid etc. respectively. Altitude (or height): The altitude of a pyramid is the perpendicular distance from the vertex to the base. Axis: The axis of a pyramid is the distance from the vertex to the centre of the base. 17.2 Right or Regular Pyramid: A pyramid whose base is a regular polygon and congruent isosceles triangles as lateral faces. In a regular pyramid the axis is perpendicular to the base. Thus in a regular pyramid th axis and the altitude are identical. Slant Height: The slant height of a regular pyramid is the length of the median through the apex of any lateral face. In the Fig 17.1 OG is the slant height. It is denoted by . Fig. 17.1 Fig. 17.1 Lateral edge: It is the common side where the two, faces meet. In the OA is the lateral edge. 17.3 The surface area and Volume of a Regular Pyramid: If a is the side of the polygon base, h is the height and is the slant height of a regular pyramid, then (i) Lateral surface area = Sum of the triangular sides forming the pyramid, which are all equal in areas = n (area of one triangle of base a and slant height ) 345 Applied Math Mensuration of Pyramid 1 = n a 2 1 = (n a) x 2 1 = Perimeter of the base x slant height 2 (ii) Total surface area = Lateral surface area + area of the base (iii) Volume: The volume of a pyramid may be easily derived from the volume of a cube.
    [Show full text]