Point groups in three dimensions
In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O, the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O itself is a subgroup of the Euclidean group E of all isometries. Symmetry groups of objects are isometry groups. Accordingly, analysis of isometry groups is analysis of possible symmetries. All isometries of a bounded 3D object have one or more common fixed points. We choose the origin as one of them; the symmetry group of an object is sometimes called full symmetry group, as opposed to its rotation group or proper symmetry group, the intersection of its full symmetry group and the rotation group SO of the 3D space itself. The rotation group of an object is equal to its full symmetry group if and only if the object is chiral; the point groups in three dimensions are used in chemistry to describe the symmetries of a molecule and of molecular orbitals forming covalent bonds, in this context they are called molecular point groups.
Finite Coxeter groups are a special set of point groups generated purely by a set of reflectional mirrors passing through the same point. A rank n Coxeter group is represented by a Coxeter -- Dynkin diagram. Coxeter notation offers a bracketed notation equivalent to the Coxeter diagram, with markup symbols for rotational and other subsymmetry point groups. SO is a subgroup of E +, which consists of i.e. isometries preserving orientation. O is the direct product of SO and the group generated by inversion: O = SO × Thus there is a 1-to-1 correspondence between all direct isometries and all indirect isometries, through inversion. There is a 1-to-1 correspondence between all groups of direct isometries H in O and all groups K of isometries in O that contain inversion: K = H × H = K ∩ SOFor instance, if H is C2 K is C2h, or if H is C3 K is S6. If a group of direct isometries H has a subgroup L of index 2 apart from the corresponding group containing inversion there is a corresponding group that contains indirect isometries but no inversion: M = L ∪ where isometry is identified with A.
An example would be C4 for H and S4 for M. Thus M is obtained from H by inverting the isometries in H ∖ L; this group M is as abstract group isomorphic with H. Conversely, for all isometry groups that contain indirect isometries but no inversion we can obtain a rotation group by inverting the indirect isometries; this is clarifying when see below. In 2D the cyclic group of k-fold rotations Ck is for every positive integer k a normal subgroup of O and SO. Accordingly, in 3D, for every axis the cyclic group of k-fold rotations about that axis is a normal subgroup of the group of all rotations about that axis. Since any subgroup of index two is normal, the group of rotations is normal both in the group obtained by adding reflections in planes through the axis and in the group obtained by adding a reflection plane perpendicular to the axis; the isometries of R3 that leave the origin fixed, forming the group O, can be categorized as follows: SO: identity rotation about an axis through the origin by an angle not equal to 180° rotation about an axis through the origin by an angle of 180° the same with inversion, i.e. respectively: inversion rotation about an axis by an angle not equal to 180°, combined with reflection in the plane through the origin perpendicular to the axis reflection in a plane through the originThe 4th and 5th in particular, in a wider sense the 6th are called improper rotations.
See the similar overview including translations. When comparing the symmetry type of two objects, the origin is chosen for each separately, i.e. they need not have the same center. Moreover, two objects are considered to be of the same symmetry type if their symmetry groups are conjugate subgroups of O. For example, two 3D objects have the same symmetry type: if both have mirror symmetry, but with respect to a different mirror plane if both have 3-fold rotational symmetry, but with respect to a different axis. In the case of multiple mirror planes and/or axes of rotation, two symmetry groups are of the same symmetry type if and only if there is a rotation mapping the whole structure of the first symmetry group to that of the second; the conjugacy definition would allow a mirror image of the structure, but this is not needed, the structure itself is achiral. For example, if a symmetry group contains a 3-fold axis of rotation, it contains rotations in two opposite directions. There are many infinite isometry groups.
We may create non-cyclical abelian groups by adding more rotations around the same axis. There are non-abelian groups generated by rotations around different axes; these are free groups. They will be infinite. All the infinite groups mentioned so far are not closed as topological subgroups of O. We now discuss
Baseball is a bat-and-ball game played between two opposing teams who take turns batting and fielding. The game proceeds when a player on the fielding team, called the pitcher, throws a ball which a player on the batting team tries to hit with a bat; the objectives of the offensive team are to hit the ball into the field of play, to run the bases—having its runners advance counter-clockwise around four bases to score what are called "runs". The objective of the defensive team is to prevent batters from becoming runners, to prevent runners' advance around the bases. A run is scored when a runner advances around the bases in order and touches home plate; the team that scores the most runs by the end of the game is the winner. The first objective of the batting team is to have a player reach first base safely. A player on the batting team who reaches first base without being called "out" can attempt to advance to subsequent bases as a runner, either or during teammates' turns batting; the fielding team tries to prevent runs by getting batters or runners "out", which forces them out of the field of play.
Both the pitcher and fielders have methods of getting the batting team's players out. The opposing teams switch forth between batting and fielding. One turn batting for each team constitutes an inning. A game is composed of nine innings, the team with the greater number of runs at the end of the game wins. If scores are tied at the end of nine innings, extra innings are played. Baseball has no game clock. Baseball evolved from older bat-and-ball games being played in England by the mid-18th century; this game was brought by immigrants to North America. By the late 19th century, baseball was recognized as the national sport of the United States. Baseball is popular in North America and parts of Central and South America, the Caribbean, East Asia in Japan and South Korea. In the United States and Canada, professional Major League Baseball teams are divided into the National League and American League, each with three divisions: East and Central; the MLB champion is determined by playoffs. The top level of play is split in Japan between the Central and Pacific Leagues and in Cuba between the West League and East League.
The World Baseball Classic, organized by the World Baseball Softball Confederation, is the major international competition of the sport and attracts the top national teams from around the world. A baseball game is played between two teams, each composed of nine players, that take turns playing offense and defense. A pair of turns, one at bat and one in the field, by each team constitutes an inning. A game consists of nine innings. One team—customarily the visiting team—bats in the top, or first half, of every inning; the other team -- customarily the home team -- bats in second half, of every inning. The goal of the game is to score more points than the other team; the players on the team at bat attempt to score runs by circling or completing a tour of the four bases set at the corners of the square-shaped baseball diamond. A player bats at home plate and must proceed counterclockwise to first base, second base, third base, back home to score a run; the team in the field attempts to prevent runs from scoring and record outs, which remove opposing players from offensive action until their turn in their team's batting order comes up again.
When three outs are recorded, the teams switch roles for the next half-inning. If the score of the game is tied after nine innings, extra innings are played to resolve the contest. Many amateur games unorganized ones, involve different numbers of players and innings; the game is played on a field whose primary boundaries, the foul lines, extend forward from home plate at 45-degree angles. The 90-degree area within the foul lines is referred to as fair territory; the part of the field enclosed by the bases and several yards beyond them is the infield. In the middle of the infield is a raised pitcher's mound, with a rectangular rubber plate at its center; the outer boundary of the outfield is demarcated by a raised fence, which may be of any material and height. The fair territory between home plate and the outfield boundary is baseball's field of play, though significant events can take place in foul territory, as well. There are three basic tools of baseball: the ball, the bat, the glove or mitt: The baseball is about the size of an adult's fist, around 9 inches in circumference.
It wound in yarn and covered in white cowhide, with red stitching. The bat is a hitting tool, traditionally made of a solid piece of wood. Other materials are now used for nonprofessional games, it is a hard round stick, about 2.5 inches in diameter at the hitting end, tapering to a narrower handle and culminating in a knob. Bats used by adults are around 34 inches long, not longer than 42 inches; the glove or mitt is a fielding tool, made of padded leather with webbing between the fingers. As an aid in catching and holding onto the ball, it takes various shapes to meet the specific needs of differ
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object can always be rotated around an infinite number of imaginary lines called rotation axes. If the axis passes through the body's center of mass, the body is said to rotate upon itself, or spin. A rotation about an external point, e.g. the Earth about the Sun, is called a revolution or orbital revolution when it is produced by gravity. The axis is called a pole. Mathematically, a rotation is a rigid body movement which, unlike a translation, keeps a point fixed; this definition applies to rotations within both two and three dimensions All rigid body movements are rotations, translations, or combinations of the two. A rotation is a progressive radial orientation to a common point; that common point lies within the axis of that motion. The axis is 90 degrees perpendicular to the plane of the motion. If the axis of the rotation lies external of the body in question the body is said to orbit. There is no fundamental difference between a “rotation” and an “orbit” and or "spin".
The key distinction is where the axis of the rotation lies, either within or outside of a body in question. This distinction can be demonstrated for "non rigid" bodies. If a rotation around a point or axis is followed by a second rotation around the same point/axis, a third rotation results; the reverse of a rotation is a rotation. Thus, the rotations around a point/axis form a group. However, a rotation around a point or axis and a rotation around a different point/axis may result in something other than a rotation, e.g. a translation. Rotations around the x, y and z axes are called principal rotations. Rotation around any axis can be performed by taking a rotation around the x axis, followed by a rotation around the y axis, followed by a rotation around the z axis; that is to say, any spatial rotation can be decomposed into a combination of principal rotations. In flight dynamics, the principal rotations are known as yaw and roll; this terminology is used in computer graphics. In astronomy, rotation is a observed phenomenon.
Stars and similar bodies all spin around on their axes. The rotation rate of planets in the solar system was first measured by tracking visual features. Stellar rotation is measured by tracking active surface features; this rotation induces a centrifugal acceleration in the reference frame of the Earth which counteracts the effect of gravity the closer one is to the equator. One effect is that an object weighs less at the equator. Another is that the Earth is deformed into an oblate spheroid. Another consequence of the rotation of a planet is the phenomenon of precession. Like a gyroscope, the overall effect is a slight "wobble" in the movement of the axis of a planet; the tilt of the Earth's axis to its orbital plane is 23.44 degrees, but this angle changes slowly. While revolution is used as a synonym for rotation, in many fields astronomy and related fields, revolution referred to as orbital revolution for clarity, is used when one body moves around another while rotation is used to mean the movement around an axis.
Moons revolve around their planet, planets revolve about their star. The motion of the components of galaxies is complex, but it includes a rotation component. Most planets in our solar system, including Earth, spin in the same direction; the exceptions are Uranus. Uranus rotates nearly on its side relative to its orbit. Current speculation is that Uranus started off with a typical prograde orientation and was knocked on its side by a large impact early in its history. Venus may be thought of as rotating backwards; the dwarf planet Pluto is anomalous in other ways. The speed of rotation is given by period; the time-rate of change of angular frequency is angular acceleration, caused by torque. The ratio of the two is given by the moment of inertia; the angular velocity vector describes the direction of the axis of rotation. The torque is an axial vector; the physics of the rotation around a fixed axis is mathematically described with the axis–angle representation of rotations. According to the right-hand rule, the direction away from the observer is associated with clockwise rotation and the direction towards the observer with counterclockwise rotation, like a screw.
The laws of physics are believed to be invariant under any fixed rotation. In modern physical cosmology, the cosmological principle is the notion that the distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale, since the forces are expected to act uniformly throughout the universe and have no preferred direction, should, produce no observable irregularities in the large scale structuring over the course of evolution of the matter field, laid down by the Big Bang. In particular, for a system which behaves the same regardless of how it is oriented in space, its Lagrangian is rotationally invariant. According to Noether's theorem, if the action (the integral over ti
An n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base. An n-gonal bipyramid has 2n triangle faces, 3n edges, 2 + n vertices; the referenced n-gon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the two pyramid halves. A right bipyramid has two points below the centroid of its base. Nonright bipyramids are called oblique bipyramids. A regular bipyramid has a regular polygon internal face and is implied to be a right bipyramid. A right bipyramid can be represented as + P for internal polygon P, a regular n-bipyramid +. A concave bipyramid has a concave interior polygon; the face-transitive regular bipyramids are the dual polyhedra of the uniform prisms and will have isosceles triangle faces. A bipyramid can be projected on a sphere or globe as n spaced lines of longitude going from pole to pole, bisected by a line around the equator. Bipyramid faces, projected as spherical triangles, represent the fundamental domains in the dihedral symmetry Dnh.
Indeed, an n-tonal bipyramid can be seen as the Kleetope of the respective n-gonal dihedron. The volume of a bipyramid is V =2/3Bh where B is the area of the base and h the height from the base to the apex; this works for any location of the apex, provided that h is measured as the perpendicular distance from the plane which contains the base. The volume of a bipyramid whose base is a regular n-sided polygon with side length s and whose height is h is therefore: V = n 6 h s 2 cot π n. Only three kinds of bipyramids can have all edges of the same length: the triangular and pentagonal bipyramids; the tetragonal bipyramid with identical edges, or regular octahedron, counts among the Platonic solids, while the triangular and pentagonal bipyramids with identical edges count among the Johnson solids. If the base is regular and the line through the apexes intersects the base at its center, the symmetry group of the n-gonal bipyramid has dihedral symmetry Dnh of order 4n, except in the case of a regular octahedron, which has the larger octahedral symmetry group Oh of order 48, which has three versions of D4h as subgroups.
The rotation group is Dn of order 2n, except in the case of a regular octahedron, which has the larger symmetry group O of order 24, which has three versions of D4 as subgroups. The digonal faces of a spherical 2n-bipyramid represents the fundamental domains of dihedral symmetry in three dimensions: Dnh, order 4n; the reflection domains can be shown as alternately colored triangles as mirror images. An asymmetric right bipyramid joins two unequal height pyramids. An inverted form can have both pyramids on the same side. A regular n-gonal asymmetry right pyramid has order 2n; the dual polyhedron of an asymmetric bipyramid is a frustum. A scalenohedron is topologically identical to a 2n-bipyramid, but contains congruent scalene triangles. There are two types. In one type the 2n vertices around the center alternate in rings below the center. In the other type, the 2n vertices are on the same plane, but alternate in two radii; the first has 2-fold rotation axes mid-edge around the sides, reflection planes through the vertices, n-fold rotation symmetry on its axis, representing symmetry Dnd, order 2n.
In crystallography, 8-sided and 12-sided scalenohedra exist. All of these forms are isohedra; the second has order 2n. The smallest scalenohedron is topologically identical to the regular octahedron; the second type is a rhombic bipyramid. The first type has 6 vertices can be represented as, where z is a parameter between 0 and 1, creating a regular octahedron at z = 0, becoming a disphenoid with merged coplanar faces at z = 1. For z > 1, it becomes concave. Self-intersecting bipyramids exist with a star polygon central figure, defined by triangular faces connecting each polygon edge to these two points. A bipyramid has Coxeter diagram. Isohedral even-sided stars can be made with zig-zag offplane vertices, in-out isotoxal forms, or both, like this form: The dual of the rectification of each convex regular 4-polytopes is a cell-transitive 4-polytope with bipyramidal cells. In the following, the apex vertex of the bipyramid is A and an equator vertex is E; the distance between adjacent vertices on the equator EE = 1, the apex to equator edge is AE and the distance between the apices is AA.
The bipyramid 4-polytope will have VA vertices. It will have VE vertices. NAE bipyramids meet along each type AE edge. NEE bipyramids meet along each type EE edge. CAE is the cosine of the dihedral angle along an AE edge. CEE is the cosine of the dihedral angle along an EE edge; as cells must fit around an edge, NAA cos−1 ≤ 2π, NAE cos−1 ≤ 2π. * The rectified 16-cell is the regular 24-cell and vertices are all equivalent – octahedra are regular bipyramids. ** Given numerically due to more complex form. In general, a bipyramid can be seen as an n-polytope constructed with a -polytope in a hyperplane with two points in opposite directions, equal distance perpendicular from the hyperplane. If the -polytope is a regular polytope, it will have identical pyramids facets. An example is the 16-cell, an octahedral bipyramid, more an n-orthoplex is an -orth
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, they play an important role in group theory and chemistry; the notation for the dihedral group differs in abstract algebra. In geometry, Dn or Dihn refers to the symmetries of a group of order 2n. In abstract algebra, D2n refers to this same dihedral group; the geometric convention is used in this article. A regular polygon with n sides has 2 n different symmetries: n rotational symmetries and n reflection symmetries. We take n ≥ 3 here; the associated rotations and reflections make up the dihedral group D n. If n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is there are n/2 axes of symmetry connecting the midpoints of opposite sides and n / 2 axes of symmetry connecting opposite vertices. In either case, there are 2 n elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes.
The following picture shows the effect of the sixteen elements of D 8 on a stop sign: The first row shows the effect of the eight rotations, the second row shows the effect of the eight reflections, in each case acting on the stop sign with the orientation as shown at the top left. As with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, this gives the symmetries of a polygon the algebraic structure of a finite group; the following Cayley table shows the effect of composition in the group D3. R0 denotes the identity. For example, s2s1 = r1, because the reflection s1 followed by the reflection s2 results in a rotation of 120°; the order of elements denoting the composition is right to left, reflecting the convention that the element acts on the expression to its right. The composition operation is not commutative. In general, the group Dn has elements r0, …, rn−1 and s0, …, sn−1, with composition given by the following formulae: r i r j = r i + j, r i s j = s i + j, s i r j = s i − j, s i s j = r i − j.
In all cases and subtraction of subscripts are to be performed using modular arithmetic with modulus n. If we center the regular polygon at the origin elements of the dihedral group act as linear transformations of the plane; this lets us represent elements of Dn with composition being matrix multiplication. This is an example of a group representation. For example, the elements of the group D4 can be represented by the following eight matrices: r 0 =, r 1 =, r 2 =, r 3 =, s 0 =, s 1 =, s 2 =
The n-gonal trapezohedron, antibipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism. With a highest symmetry, its 2n faces are congruent kites; the faces are symmetrically staggered. The n-gon part of the name does not reference the faces here but arrangement of vertices around an axis of symmetry; the dual n-gonal antiprism has two actual n-gon faces. An n-gonal trapezohedron can be dissected into an n-gonal antiprism; these figures, sometimes called deltohedra, must not be confused with deltahedra, whose faces are equilateral triangles. In texts describing the crystal habits of minerals, the word trapezohedron is used for the polyhedron properly known as a deltoidal icositetrahedron; the symmetry group of an n-gonal trapezohedron is Dnd of order 4n, except in the case of a cube, which has the larger symmetry group Od of order 48, which has four versions of D3d as subgroups. The rotation group is Dn of order 2n, except in the case of a cube, which has the larger rotation group O of order 24, which has four versions of D3 as subgroups.
One degree of freedom within Dn symmetry changes the kites into congruent quadrilaterals with 3 edges lengths. In the limit, one edge of each quadrilateral goes to zero length, these become bipyramids. If the kites surrounding the two peaks are of different shapes, it can only have Cnv symmetry, order 2n; these can be called asymmetric trapezohedra. The dual is an unequal antiprism, with the bottom polygons of different radii. If it twisted and unequal its symmetry is reduced to cyclic symmetry, Cn symmetry, order n. A n-trapezohedron has 2n quadrilateral faces, with 2n+2 vertices. Two vertices are on the polar axis, the others are in two regular n-gonal rings of vertices. Special cases: n=2: A degenerate form, form a geometric tetrahedron with 6 vertices, 8 edges, 4 degenerate kite faces that are degenerated into triangles, its dual is a degenerate form of antiprism a tetrahedron. N=3: In the case of the dual of a triangular antiprism the kites are rhombi, hence these trapezohedra are zonohedra.
They are called rhombohedra. They are cubes scaled in the direction of a body diagonal, they are the parallelepipeds with congruent rhombic faces. A special case of a rhombohedron is one in which the rhombi which form the faces have angles of 60° and 120°, it can be decomposed into a regular octahedron. Since parallelepipeds can fill space, so can a combination of regular tetrahedra and regular octahedra. Crystal arrangements of atoms can repeat in space with hexagonal trapezohedral cells; the pentagonal trapezohedron is the only polyhedron other than the Platonic solids used as a die in roleplaying games such as Dungeons & Dragons. Having 10 sides, it can be used in repetition to generate any decimal-based uniform probability desired. Two dice of different colors are used for the two digits to represent numbers from 00 to 99. Self-intersecting trapezohedron exist with a star polygon central figure, defined by kite faces connecting each polygon edge to these two points. A p/q-trapezohedron has Coxeter-Dynkin diagram.
Diminished trapezohedron Rhombic dodecahedron Rhombic triacontahedron Bipyramid Conway polyhedron notation Anthony Pugh. Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7. Chapter 4: Duals of the Archimedean polyhedra and antiprisms Weisstein, Eric W. "Trapezohedron". MathWorld. Weisstein, Eric W. "Isohedron". MathWorld. Virtual Reality Polyhedra The Encyclopedia of Polyhedra VRML models <3> <4> <5> <6> <7> <8> <9> <10> Conway Notation for Polyhedra Try: "dAn", where n=3,4,5... example "dA5" is a pentagonal trapezohedron. Paper model tetragonal trapezohedron
A beach ball is an inflatable ball for beach and water games. Their large size and light weight take little effort to propel, they became popular in the beach-themed films of the 1960s starring Annette Funicello and Frankie Avalon. These movies include Beach Party, Muscle Beach Party, Beach Blanket Bingo and How to Stuff a Wild Bikini. Beach balls range from larger, they have a set of soft plastic panels, with two circular end panels, one with an oral inflation valve, intended to be inflated by mouth or pump. A common design is vertical solid colored stripes alternating with white stripes. There are numerous other designs, which include but are not limited to beach balls in a single solid color, promotional beach balls with advertisements or company slogans, as globes or as Emojis; some manufacturers specify the size of their beach balls as the tip-to-tip length of a deflated ball, or the length of the panels before they have their ends cut and joined into a beach ball. Thus the actual diameter may be about 2 π of the nominal size.
Moreover, other sizes of beach balls exist. There are beach balls that have a diameter of 5 feet or 9 feet; the world's largest beach ball was made in London, England on May 30, 2017. It was carried by a barge on River Thames, it had a diameter of 65.5 feet with the word "Baywatch" written all over it. It was produced by Paramount Pictures to promote the 2017 movie Baywatch; the record was registered by Guinness and the certificates were given to the members of the film's cast. Beach ball sports include water volleyball. While they are much less expensive than the balls used in professional sports, they are much less durable as most of them are made of soft plastic. Giant beach balls may be tossed between crowd members at concerts and sporting events. Many graduates use beach balls as a prank during ceremonies, they are bounced around crowds at cricket and football games but confiscated and popped by security. Some security personnel at these events might inspect the beach ball's interior after tearing it, most searching for illegal items that might be transported inside the beach ball.
The guards may do this so that the beach ball cannot end up on the field and obstruct or distract players, as happened in August 1999, in a baseball game between the Cleveland Indians and the Los Angeles Angels, where the distraction caused by a beach ball on the field resulted in the Angels' defeat. Their light weight and stability make beach balls ideal for trained seals to balance on their noses, which has become an iconic scene. Beach balls are a popular prop used in swimsuit photography and to promote or represent beach-themed events or locations. Balloon Inflatable List of inflatable manufactured goods