Octahedron
In geometry, an octahedron is a polyhedron with eight faces, twelve edges, six vertices. The term is most used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. A regular octahedron is the dual polyhedron of a cube, it is a rectified tetrahedron. It is a square bipyramid in any of three orthogonal orientations, it is a triangular antiprism in any of four orientations. An octahedron is the three-dimensional case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan metric. If the edge length of a regular octahedron is a, the radius of a circumscribed sphere is r u = a 2 2 ≈ 0.707 ⋅ a and the radius of an inscribed sphere is r i = a 6 6 ≈ 0.408 ⋅ a while the midradius, which touches the middle of each edge, is r m = a 2 = 0.5 ⋅ a The octahedron has four special orthogonal projections, centered, on an edge, vertex and normal to a face. The second and third correspond to A2 Coxeter planes.
The octahedron can be represented as a spherical tiling, projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not lengths. Straight lines on the sphere are projected as circular arcs on the plane. An octahedron with edge length √2 can be placed with its center at the origin and its vertices on the coordinate axes. In an x–y–z Cartesian coordinate system, the octahedron with center coordinates and radius r is the set of all points such that | x − a | + | y − b | + | z − c | = r; the surface area A and the volume V of a regular octahedron of edge length a are: A = 2 3 a 2 ≈ 3.464 a 2 V = 1 3 2 a 3 ≈ 0.471 a 3 Thus the volume is four times that of a regular tetrahedron with the same edge length, while the surface area is twice. If an octahedron has been stretched so that it obeys the equation | x x m | + | y y m | + | z z m | = 1, the formulas for the surface area and volume expand to become A = 4 x m y m z m × 1 x m 2 + 1 y m 2 + 1 z m 2, V = 4 3 x m y m z m.
Additionally the inertia tensor of the stretched octahedron is I =. These reduce to the equations for the regular octahedron when x m = y m = z m = a 2 2. {\displaystyle x_=y_=z_=
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
Regular polyhedron
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used. A regular polyhedron is identified by its Schläfli symbol of the form, where n is the number of sides of each face and m the number of faces meeting at each vertex. There are 5 finite convex regular polyhedra, four regular star polyhedra, making nine regular polyhedra in all. In addition, there are five regular compounds of the regular polyhedra. There are five convex regular polyhedra, known as the Platonic solids, four regular star polyhedra, the Kepler–Poinsot polyhedra, five regular compounds of regular polyhedra: The property of having a similar arrangement of faces around each vertex can be replaced by any of the following equivalent conditions in the definition: The vertices of the polyhedron all lie on a sphere. All the dihedral angles of the polyhedron are equal All the vertex figures of the polyhedron are regular polygons.
All the solid angles of the polyhedron are congruent. A regular polyhedron has all of three related spheres which share its centre: An insphere, tangent to all faces. An intersphere or midsphere, tangent to all edges. A circumsphere, tangent to all vertices; the regular polyhedra are the most symmetrical of all the polyhedra. They lie in just three symmetry groups, which are named after them: Tetrahedral Octahedral Icosahedral Any shapes with icosahedral or octahedral symmetry will contain tetrahedral symmetry; the five Platonic solids have an Euler characteristic of 2. This reflects that the surface is a topological 2-sphere, so is true, for example,of any polyhedron, star-shaped with respect to some interior point; the sum of the distances from any point in the interior of a regular polyhedron to the sides is independent of the location of the point. However, the converse does not hold, not for tetrahedra. In a dual pair of polyhedra, the vertices of one polyhedron correspond to the faces of the other, vice versa.
The regular polyhedra show this duality as follows: The tetrahedron is self-dual, i.e. it pairs with itself. The cube and octahedron are dual to each other; the icosahedron and dodecahedron are dual to each other. The small stellated dodecahedron and great dodecahedron are dual to each other; the great stellated dodecahedron and great icosahedron are dual to each other. The Schläfli symbol of the dual is just the original written backwards. Stones carved in shapes resembling clusters of spheres or knobs have been found in Scotland and may be as much as 4,000 years old; some of these stones show not only the symmetries of the five Platonic solids, but some of the relations of duality amongst them. Examples of these stones are on display in the John Evans room of the Ashmolean Museum at Oxford University. Why these objects were made, or how their creators gained the inspiration for them, is a mystery. There is doubt regarding the mathematical interpretation of these objects, as many have non-platonic forms, only one has been found to be a true icosahedron, as opposed to a reinterpretation of the icosahedron dual, the dodecahedron.
It is possible that the Etruscans preceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery near Padua in the late 19th century of a dodecahedron made of soapstone, dating back more than 2,500 years. The earliest known written records of the regular convex solids originated from Classical Greece; when these solids were all discovered and by whom is not known, but Theaetetus, was the first to give a mathematical description of all five. H. S. M. Coxeter credits Plato with having made models of them, mentions that one of the earlier Pythagoreans, Timaeus of Locri, used all five in a correspondence between the polyhedra and the nature of the universe as it was perceived – this correspondence is recorded in Plato's dialogue Timaeus. Euclid's reference to Plato led to their common description as the Platonic solids. One might characterise the Greek definition as follows: A regular polygon is a planar figure with all edges equal and all corners equal A regular polyhedron is a solid figure with all faces being congruent regular polygons, the same number arranged all alike around each vertex.
This definition rules out, for example, the square pyramid, or the shape formed by joining two tetrahedra together. This concept of a regular polyhedron would remain unchallenged for 2000 years. Regular star polygons such as the pentagram were known to the ancient Greeks – the pentagram was used by the Pythagoreans as their secret sign, but they did not use them to construct polyhedra, it was not until the early 17th century that Johannes Kepler realised that pentagrams could be used as the faces of regular star polyhedra. Some of these star polyhedra may have been discovered
Volume
Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance or shape occupies or contains. Volume is quantified numerically using the SI derived unit, the cubic metre; the volume of a container is understood to be the capacity of the container. Three dimensional mathematical shapes are assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, circular shapes can be calculated using arithmetic formulas. Volumes of complicated shapes can be calculated with integral calculus if a formula exists for the shape's boundary. One-dimensional figures and two-dimensional shapes are assigned zero volume in the three-dimensional space; the volume of a solid can be determined by fluid displacement. Displacement of liquid can be used to determine the volume of a gas; the combined volume of two substances is greater than the volume of just one of the substances. However, sometimes one substance dissolves in the other and in such cases the combined volume is not additive.
In differential geometry, volume is expressed by means of the volume form, is an important global Riemannian invariant. In thermodynamics, volume is a fundamental parameter, is a conjugate variable to pressure. Any unit of length gives a corresponding unit of volume: the volume of a cube whose sides have the given length. For example, a cubic centimetre is the volume of a cube. In the International System of Units, the standard unit of volume is the cubic metre; the metric system includes the litre as a unit of volume, where one litre is the volume of a 10-centimetre cube. Thus 1 litre = 3 = 1000 cubic centimetres = 0.001 cubic metres. Small amounts of liquid are measured in millilitres, where 1 millilitre = 0.001 litres = 1 cubic centimetre. In the same way, large amounts can be measured in megalitres, where 1 million litres = 1000 cubic metres = 1 megalitre. Various other traditional units of volume are in use, including the cubic inch, the cubic foot, the cubic yard, the cubic mile, the teaspoon, the tablespoon, the fluid ounce, the fluid dram, the gill, the pint, the quart, the gallon, the minim, the barrel, the cord, the peck, the bushel, the hogshead, the acre-foot and the board foot.
Capacity is defined by the Oxford English Dictionary as "the measure applied to the content of a vessel, to liquids, grain, or the like, which take the shape of that which holds them". Capacity is not identical in meaning to volume, though related. Units of capacity are the SI litre and its derived units, Imperial units such as gill, pint and others. Units of volume are the cubes of units of length. In SI the units of volume and capacity are related: one litre is 1 cubic decimetre, the capacity of a cube with a 10 cm side. In other systems the conversion is not trivial; the density of an object is defined as the ratio of the mass to the volume. The inverse of density is specific volume, defined as volume divided by mass. Specific volume is a concept important in thermodynamics where the volume of a working fluid is an important parameter of a system being studied; the volumetric flow rate in fluid dynamics is the volume of fluid which passes through a given surface per unit time. In calculus, a branch of mathematics, the volume of a region D in R3 is given by a triple integral of the constant function f = 1 and is written as: ∭ D 1 d x d y d z.
The volume integral in cylindrical coordinates is ∭ D r d r d θ d z, the volume integral in spherical coordinates has the form ∭ D ρ 2 sin ϕ d ρ d θ d ϕ. The above formulas can be used to show that the volumes of a cone and cylinder of the same radius and height are in the ratio 1: 2: 3, as follows. Let the radius be r and the height be h the volume of cone is 1 3 π r 2 h = 1 3 π r 2 = × 1, the volume of the sphere
Stereographic projection
In geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane. The projection is defined except at one point: the projection point. Where it is defined, the mapping is bijective, it is conformal. It is neither isometric nor area-preserving: that is, it preserves neither distances nor the areas of figures. Intuitively the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises; because the sphere and the plane appear in many areas of mathematics and its applications, so does the stereographic projection. In practice, the projection is carried out by computer or by hand using a special kind of graph paper called a stereographic net, shortened to stereonet, or Wulff net; the stereographic projection was known to Hipparchus and earlier to the Egyptians. It was known as the planisphere projection. Planisphaerium by Ptolemy is the oldest surviving document. One of its most important uses was the representation of celestial charts.
The term planisphere is still used to refer to such charts. In the 16th and 17th century, the equatorial aspect of the stereographic projection was used for maps of the Eastern and Western Hemispheres, it is believed that the map created in 1507 by Gualterius Lud was in stereographic projection, as were the maps of Jean Roze, Rumold Mercator, many others. In star charts this equatorial aspect had been utilised by the ancient astronomers like Ptolemy. François d'Aguilon gave the stereographic projection its current name in his 1613 work Opticorum libri sex philosophis juxta ac mathematicis utiles. In 1695, Edmond Halley, motivated by his interest in star charts, published the first mathematical proof that this map is conformal, he used the established tools of calculus, invented by his friend Isaac Newton. The unit sphere in three-dimensional space R3 is the set of points such that x2 + y2 + z2 = 1. Let N = be the "north pole", let M be the rest of the sphere; the plane z = 0 runs through the center of the sphere.
For any point P on M, there is a unique line through N and P, this line intersects the plane z = 0 in one point P′. Define the stereographic projection of P to be this point P′ in the plane. In Cartesian coordinates on the sphere and on the plane, the projection and its inverse are given by the formulas =, =. In spherical coordinates on the sphere and polar coordinates on the plane, the projection and its inverse are = =, =. {\displaystyle {\begin&=\left=\left(\c
Vertex (geometry)
In geometry, a vertex is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices; the vertex of an angle is the point where two rays begin or meet, where two line segments join or meet, where two lines intersect, or any appropriate combination of rays and lines that result in two straight "sides" meeting at one place. A vertex is a corner point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection of edges, faces or facets of the object. In a polygon, a vertex is called "convex" if the internal angle of the polygon, that is, the angle formed by the two edges at the vertex, with the polygon inside the angle, is less than π radians. More a vertex of a polyhedron or polytope is convex if the intersection of the polyhedron or polytope with a sufficiently small sphere centered at the vertex is convex, concave otherwise. Polytope vertices are related to vertices of graphs, in that the 1-skeleton of a polytope is a graph, the vertices of which correspond to the vertices of the polytope, in that a graph can be viewed as a 1-dimensional simplicial complex the vertices of which are the graph's vertices.
However, in graph theory, vertices may have fewer than two incident edges, not allowed for geometric vertices. There is a connection between geometric vertices and the vertices of a curve, its points of extreme curvature: in some sense the vertices of a polygon are points of infinite curvature, if a polygon is approximated by a smooth curve there will be a point of extreme curvature near each polygon vertex. However, a smooth curve approximation to a polygon will have additional vertices, at the points where its curvature is minimal. A vertex of a plane tiling or tessellation is a point. More a tessellation can be viewed as a kind of topological cell complex, as can the faces of a polyhedron or polytope. A polygon vertex xi of a simple polygon P is a principal polygon vertex if the diagonal intersects the boundary of P only at x and x. There are two types of principal vertices: mouths. A principal vertex xi of a simple polygon P is called an ear if the diagonal that bridges xi lies in P. According to the two ears theorem, every simple polygon has at least two ears.
A principal vertex xi of a simple polygon P is called a mouth if the diagonal lies outside the boundary of P. Any convex polyhedron's surface has Euler characteristic V − E + F = 2, where V is the number of vertices, E is the number of edges, F is the number of faces; this equation is known as Euler's polyhedron formula. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces. For example, a cube has 12 edges and 6 faces, hence 8 vertices. In computer graphics, objects are represented as triangulated polyhedra in which the object vertices are associated not only with three spatial coordinates but with other graphical information necessary to render the object such as colors, reflectance properties and surface normal. Weisstein, Eric W. "Polygon Vertex". MathWorld. Weisstein, Eric W. "Polyhedron Vertex". MathWorld. Weisstein, Eric W. "Principal Vertex". MathWorld
Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century, he was born in London, received his BA and PhD from Cambridge, but lived in Canada from age 29. He was always called Donald, from his third name MacDonald, he was most noted for his work on higher-dimensional geometries. He was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra. In his youth, Coxeter composed music and was an accomplished pianist at the age of 10, he felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on "Mathematics and Music" in the Canadian Music Journal. Coxeter went up to Cambridge in 1926 to read mathematics. There he earned his BA in 1928, his doctorate in 1931. In 1932 he went to Princeton University for a year as a Rockefeller Fellow, where he worked with Hermann Weyl, Oswald Veblen, Solomon Lefschetz.
Returning to Trinity for a year, he attended Ludwig Wittgenstein's seminars on the philosophy of mathematics. In 1934 he spent a further year at Princeton as a Procter Fellow. In 1936 Coxeter moved to the University of Toronto. In 1938 he and P. Du Val, H. T. Flather, John Flinders Petrie published The Fifty-Nine Icosahedra with University of Toronto Press. In 1940 Coxeter edited the eleventh edition of Mathematical Recreations and Essays published by W. W. Rouse Ball in 1892, he was elevated to professor in 1948. Coxeter was elected a Fellow of the Royal Society of Canada in 1948 and a Fellow of the Royal Society in 1950, he met M. C. Escher in 1954 and the two became lifelong friends, he inspired some of the innovations of Buckminster Fuller. Coxeter, M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra, he published twelve books. Since 1978, the Canadian Mathematical Society have awarded the Coxeter–James Prize in his honor, he was made a Fellow of the Royal Society in 1950 and in 1997 he was awarded their Sylvester Medal.
In 1990, he became a Foreign Member of the American Academy of Arts and Sciences and in 1997 was made a Companion of the Order of Canada. In 1973 he received the Jeffery–Williams Prize. 1940: Regular and Semi-Regular Polytopes I, Mathematische Zeitschrift 46: 380-407, MR 2,10 doi:10.1007/BF01181449 1942: Non-Euclidean Geometry, University of Toronto Press, MAA. 1954: "Uniform Polyhedra", Philosophical Transactions of the Royal Society A 246: 401–50 doi:10.1098/rsta.1954.0003 1949: The Real Projective Plane 1957: Generators and Relations for Discrete Groups 1980: Second edition, Springer-Verlag ISBN 0-387-09212-9 1961: Introduction to Geometry 1963: Regular Polytopes, Macmillan Company 1967: Geometry Revisited 1970: Twisted honeycombs 1973: Regular Polytopes, Dover edition, ISBN 0-486-61480-8 1974: Projective Geometry 1974: Regular Complex Polytopes, Cambridge University Press 1981:, Zero-Symmetric Graphs, Academic Press. 1985: Regular and Semi-Regular Polytopes II, Mathematische Zeitschrift 188: 559–591 1987 Projective Geometry ISBN 978-0-387-40623-7 1988: Regular and Semi-Regular Polytopes III, Mathematische Zeitschrift 200: 3–45 1995: F. Arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivić Weiss, editors: Kaleidoscopes — Selected Writings of H.
S. M. Coxeter. John Wiley and Sons ISBN 0-471-01003-0 1999: The Beauty of Geometry: Twelve Essays, Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 Davis, Chandler; the Coxeter Legacy: Reflections and Projections. Providence, R. I.: American Mathematical Society. ISBN 978-0-8218-3722-1. OCLC 62282754. Roberts, Siobhan. King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry. New York: Walker & Company. ISBN 978-0-8027-1499-2. OCLC 71436884. Archival papers held at University of Toronto Archives and Records Management Services Harold Scott MacDonald Coxeter at the Mathematics Genealogy Project H. S. M. Coxeter, Erich W. Ellers, Branko Grünbaum, Peter McMullen, Asia Ivic Weiss Notices of the AMS: Volume 50, Number 10. Www.donaldcoxeter.com www.math.yorku.ca/dcoxeter webpages dedicated to him Jaron's World: Shapes in Other Dimensions, Discover mag. Apr 2007 The Mathematics in the Art of M. C. Escher video of a lecture by H. S. M. Coxeter, April 28, 2000