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_=
Catoptrics
Catoptrics deals with the phenomena of reflected light and image-forming optical systems using mirrors. A catoptric system is called a catopter. Catoptrics is the title of two texts from ancient Greece: The Pseudo-Euclidean Catoptrics; this book is attributed to Euclid, although the contents are a mixture of work dating from Euclid's time together with work which dates to the Roman period. It has been argued that the book may have been compiled by the 4th century mathematician Theon of Alexandria; the book covers the mathematical theory of mirrors the images formed by plane and spherical concave mirrors. Hero's Catoptrics. Written by Hero of Alexandria, this work concerns the practical application of mirrors for visual effects. In the Middle Ages, this work was falsely ascribed to Ptolemy, it only survives in a Latin translation. The Latin translation of Alhazen's main work, Book of Optics, exerted a great influence on Western science: for example, on the work of Roger Bacon, who cites him by name.
His research in catoptrics centred on parabolic mirrors and spherical aberration. He made the observation that the ratio between the angle of incidence and refraction does not remain constant, investigated the magnifying power of a lens, his work on catoptrics contains the problem known as "Alhazen's problem". Alhazen's work influenced Averroes' writings on optics, his legacy was further advanced through the'reforming' of his Optics by Persian scientist Kamal al-Din al-Farisi in the latter's Kitab Tanqih al-Manazir; the first practical catoptric telescope was built by Isaac Newton as a solution to the problem of chromatic aberration exhibited in telescopes using lenses as objectives. Dioptrics Catadioptrics Optical telescope List of telescope types Image-forming optical system Fresnel lens Lighthouse lens Dr. Al Deek, Mahmoud, "Ibn Al-Haitham: Master of Optics, Mathematics and Medicine", Al Shindagah
Geometry
Geometry is a branch of mathematics concerned with questions of shape, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a practical way for dealing with lengths and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.
While geometry has evolved throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, planes, surfaces and curves, as well as the more advanced notions of manifolds and topology or metric. Geometry has applications to many fields, including art, physics, as well as to other branches of mathematics. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense; the mandatory educational curriculum of the majority of nations includes the study of points, planes, triangles, similarity, solid figures and analytic geometry. Euclidean geometry has applications in computer science and various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry, it has applications in physics, including in general relativity. Topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this means dealing with large-scale properties of spaces, such as connectedness and compactness.
Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues using techniques of real analysis. It has close connections to convex analysis and functional analysis and important applications in number theory. Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques, it has applications including cryptography and string theory. Discrete geometry is concerned with questions of relative position of simple geometric objects, such as points and circles, it shares many principles with combinatorics. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc; the earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles and volumes, which were developed to meet some practical need in surveying, construction and various crafts.
The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. Clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space; these geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.
Eudoxus developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom and proof. Although most of the contents of the Elements were known, Euclid arranged them into a single, coherent logical framework; the Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, gave remarkably accurate approximations of Pi.
He studied the sp
Convex uniform honeycomb
In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells. Twenty-eight such honeycombs are known: the familiar cubic honeycomb and 7 truncations thereof, they can be considered the three-dimensional analogue to the uniform tilings of the plane. The Voronoi diagram of any lattice forms a convex uniform honeycomb in which the cells are zonohedra. 1900: Thorold Gosset enumerated the list of semiregular convex polytopes with regular cells in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions, including one regular cubic honeycomb, two semiregular forms with tetrahedra and octahedra. 1905: Alfredo Andreini enumerated 25 of these tessellations. 1991: Norman Johnson's manuscript Uniform Polytopes identified the list of 28. 1994: Branko Grünbaum, in his paper Uniform tilings of 3-space independently enumerated all 28, after discovering errors in Andreini's publication.
He found the 1905 paper, which listed 25, had 1 wrong, 4 being missing. Grünbaum states in this paper that Norman Johnson deserves priority for achieving the same enumeration in 1991, he mentions that I. Alexeyev of Russia had contacted him regarding a putative enumeration of these forms, but that Grünbaum was unable to verify this at the time. 2006: George Olshevsky, in his manuscript Uniform Panoploid Tetracombs, along with repeating the derived list of 11 convex uniform tilings, 28 convex uniform honeycombs, expands a further derived list of 143 convex uniform tetracombs. Only 14 of the convex uniform polyhedra appear in these patterns: three of the five Platonic solids, six of the thirteen Archimedean solids, five of the infinite family of prisms; this set can be called the regular and semiregular honeycombs. It has been called the Archimedean honeycombs by analogy with the convex uniform polyhedra called Archimedean solids. Conway has suggested naming the set as the Architectonic tessellations and the dual honeycombs as the Catoptric tessellations.
The individual honeycombs are listed with names given to them by Norman Johnson. For cross-referencing, they are given with list indices from Andreini, Johnson, Grünbaum. Coxeter uses δ4 for a cubic honeycomb, hδ4 for an alternated cubic honeycomb, qδ4 for a quarter cubic honeycomb, with subscripts for other forms based on the ring patterns of the Coxeter diagram; the fundamental infinite Coxeter groups for 3-space are: The C ~ 3, The B ~ 3, alternated cubic, The A ~ 3 cyclic group, or, There is a correspondence between all three families. Removing one mirror from C ~ 3 produces B ~ 3, removing one mirror from B ~ 3 produces A ~ 3; this allows multiple constructions of the same honeycombs. If cells are colored based on unique positions within each Wythoff construction, these different symmetries can be shown. In addition there are 5 special honeycombs which don't have pure reflectional symmetry and are constructed from reflectional forms with elongation and gyration operations; the total unique honeycombs above are 18.
The prismatic stacks from infinite Coxeter groups for 3-space are: The C ~ 2 × I ~ 1, prismatic group, The H ~ 2 × I ~ 1, prismatic group, The A ~ 2 × I ~ 1, prismatic group, The I ~ 1 × I ~ 1 × I ~ 1, prismatic group, In addition there is one special elongated form of the triangular prismatic honeycomb. The total unique prismatic honeycombs above are 10. Combining these counts, 18 and 10 gives us the total 28 uniform honeycombs; the regular cubic honeycomb, represented by Schläfli symbol, offers seven unique derived uniform honeycombs via truncation operations. (One redundant form, the runcinated cubic honeycomb, is included for completeness though identica
Cubic honeycomb
The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation in Euclidean 3-space, made up of cubic cells. It has 4 cubes around every edge, 8 cubes around each vertex, its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol. John Horton Conway calls this honeycomb a cubille. A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps, it is an example of the more general mathematical tiling or tessellation in any number of dimensions. Honeycombs are constructed in ordinary Euclidean space, like the convex uniform honeycombs, they may be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space; the Cartesian coordinates of the vertices are: for all integral values: i,j,k, with edges parallel to the axes and with an edge length of 1. It is part of a multidimensional family of hypercube honeycombs, with Schläfli symbols of the form, starting with the square tiling, in the plane.
It is one of 28 uniform honeycombs using convex uniform polyhedral cells. Simple cubic lattices can be distorted into lower symmetries, represented by lower crystal systems: There is a large number of uniform colorings, derived from different symmetries; these include: The cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements. The highest symmetry form projects into a triangular tiling. A square symmetry projection forms a square tiling, it is related to the regular 4-polytope tesseract, Schläfli symbol, which exists in 4-space, only has 3 cubes around each edge. It's related to the order-5 cubic honeycomb, Schläfli symbol, of hyperbolic space with 5 cubes around each edge, it is in a sequence of honeycomb with octahedral vertex figures. It in a sequence of regular polytopes and honeycombs with cubic cells. The, Coxeter group generates 15 permutations of uniform tessellations, 9 with distinct geometry including the alternated cubic honeycomb; the expanded cubic honeycomb is geometrically identical to the cubic honeycomb.
The, Coxeter group generates 9 permutations of uniform tessellations, 4 with distinct geometry including the alternated cubic honeycomb. This honeycomb is one of five distinct uniform honeycombs constructed by the A ~ 3 Coxeter group; the symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams: The rectified cubic honeycomb or rectified cubic cellulation is a uniform space-filling tessellation in Euclidean 3-space. It is composed of octahedra and cuboctahedra in a ratio of 1:1. John Horton Conway calls this honeycomb a cuboctahedrille, its dual an oblate octahedrille; the rectified cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements. There are four uniform colorings for the cells of this honeycomb with reflective symmetry, listed by their Coxeter group, Wythoff construction name, the Coxeter diagram below; this honeycomb can be divided on trihexagonal tiling planes, using the hexagon centers of the cuboctahedra, creating two triangular cupolae.
This scaliform honeycomb is represented by Coxeter diagram, symbol s3, with coxeter notation symmetry.. The truncated cubic honeycomb or truncated cubic cellulation is a uniform space-filling tessellation in Euclidean 3-space, it is composed of truncated cubes and octahedra in a ratio of 1:1. John Horton Conway calls this honeycomb a truncated cubille, its dual pyramidille; the truncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements. There is a second uniform coloring by reflectional symmetry of the Coxeter groups, the second seen with alternately colored truncated cubic cells; the alternated bitruncated cubic honeycomb or bisnub cubic honeycomb can be creating regular icosahedron from the truncated octahedra with irregular tetrahedral cells created in the gaps. There are three constructions from three related Coxeter diagrams:, and; these have symmetry, + respectively. The first and last symmetry can be doubled as and +; this honeycomb is represented in the boron atoms of the α-rhombihedral crystal.
The centers of the icosahedra are located at the fcc positions of the lattice. The cantellated cubic honeycomb or cantellated cubic cellulation is a uniform space-filling tessellation in Euclidean 3-space, it is composed of rhombicuboctahedra and cubes in a ratio of 1:1:3. John Horton Conway calls this honeycomb a 2-RCO-trille, its dual quarter oblate octahedrille; the cantellated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements. There is a second uniform colorings by reflectional symmetry of the Coxeter groups, the second seen with alternately colored rhombicuboctahedral cells; the dual of the cantellated cubic honeycomb is called a quarter oblate octahedrille, a catoptric tessellation with Coxeter diagram, containing faces from two of four hyperplanes of the cubic fundamental domain. It has irregular triangle bipyramid cells which can be seen as 1/12 of a cube, made from the cube center, 2 face centers, 2 vertices; the cantitruncated cubic honeycomb or cantitruncated cubic cellulation is a uniform space-filling tessellation in Euclidean 3-space, made up of truncated cuboctahedra, truncated octah
John Horton Conway
John Horton Conway is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has contributed to many branches of recreational mathematics, notably the invention of the cellular automaton called the Game of Life. Conway spent the first half of his long career at the University of Cambridge, in England, the second half at Princeton University in New Jersey, where he now holds the title Professor Emeritus. Conway was born in the son of Cyril Horton Conway and Agnes Boyce, he became interested in mathematics at a early age. By the age of eleven his ambition was to become a mathematician. After leaving sixth form, Conway entered Caius College, Cambridge to study mathematics. Conway, a "terribly introverted adolescent" in school, interpreted his admission to Cambridge as an opportunity to transform himself into a new person: an "extrovert", he was awarded his Bachelor of Arts degree in 1959 and began to undertake research in number theory supervised by Harold Davenport.
Having solved the open problem posed by Davenport on writing numbers as the sums of fifth powers, Conway began to become interested in infinite ordinals. It appears that his interest in games began during his years studying the Cambridge Mathematical Tripos, where he became an avid backgammon player, spending hours playing the game in the common room, he was awarded his doctorate in 1964 and was appointed as College Fellow and Lecturer in Mathematics at the University of Cambridge. After leaving Cambridge in 1986, he took up the appointment to the John von Neumann Chair of Mathematics at Princeton University. Conway is known for the invention of the Game of Life, one of the early examples of a cellular automaton, his initial experiments in that field were done with pen and paper, long before personal computers existed. Since the game was introduced by Martin Gardner in Scientific American in 1970, it has spawned hundreds of computer programs, web sites, articles, it is a staple of recreational mathematics.
There is an extensive wiki devoted to cataloging the various aspects of the game. From the earliest days it has been a favorite in computer labs, both for its theoretical interest and as a practical exercise in programming and data display. At times Conway has said he hates the Game of Life–largely because it has come to overshadow some of the other deeper and more important things he has done; the game did help launch a new branch of mathematics, the field of cellular automata. The Game of Life is now known to be Turing complete. Conway's career is intertwined with mathematics popularizer and Scientific American columnist Martin Gardner; when Gardner featured Conway's Game of Life in his Mathematical Games column in October 1970, it became the most read of all his columns and made Conway an instant celebrity. Gardner and Conway had first corresponded in the late 1950s, over the years Gardner had written about recreational aspects of Conway's work. For instance, he discussed Conway's game of Sprouts and his angel and devil problem.
In the September 1976 column he reviewed Conway's book On Numbers and Games and introduced the public to Conway's surreal numbers. Conferences called Gathering 4 Gardner are held every two years to celebrate the legacy of Martin Gardner, Conway himself has been a featured speaker at these events, discussing various aspects of recreational mathematics. Conway is known for his contributions to combinatorial game theory, a theory of partisan games; this he developed with Elwyn Berlekamp and Richard Guy, with them co-authored the book Winning Ways for your Mathematical Plays. He wrote the book On Numbers and Games which lays out the mathematical foundations of CGT, he is one of the inventors of sprouts, as well as philosopher's football. He developed detailed analyses of many other games and puzzles, such as the Soma cube, peg solitaire, Conway's soldiers, he came up with the angel problem, solved in 2006. He invented a new system of numbers, the surreal numbers, which are related to certain games and have been the subject of a mathematical novel by Donald Knuth.
He invented a nomenclature for exceedingly large numbers, the Conway chained arrow notation. Much of this is discussed in the 0th part of ONAG. In the mid-1960s with Michael Guy, son of Richard Guy, Conway established that there are sixty-four convex uniform polychora excluding two infinite sets of prismatic forms, they discovered the grand antiprism in the only non-Wythoffian uniform polychoron. Conway has suggested a system of notation dedicated to describing polyhedra called Conway polyhedron notation. In the theory of tessellations, he devised the Conway criterion which describes rules for deciding if a prototile will tile the plane, he investigated lattices in higher dimensions, was the first to determine the symmetry group of the Leech lattice. In knot theory, Conway formulated a new variation of the Alexander polynomial and produced a new invariant now called the Conway polynomial. After lying dormant for more than a decade, this concept became central to work in the 1980s on the novel knot polynomials.
Conway further developed tangle theory and invented a system of notation for tabulating knots, nowadays known as Conway notation, while correcting a number of errors in the 19th century knot tables and extending them to include all but four of the non-alternating primes with 11 crossings. See Topology Proceedings 7 118, he was the primary author of the ATLAS of Finite Groups giving prope
Tetragonal disphenoid honeycomb
The tetragonal disphenoid tetrahedral honeycomb is a space-filling tessellation in Euclidean 3-space made up of identical tetragonal disphenoidal cells. Cells are face-transitive with 4 identical isosceles triangle faces. John Horton Conway calls it an oblate tetrahedrille or shortened to obtetrahedrille. A cell can be seen as 1/12 of a translational cube, with its vertices centered on two faces and two edges. Four of its edges belong to 6 cells, two edges belong to 4 cells; the tetrahedral disphenoid honeycomb is the dual of the uniform bitruncated cubic honeycomb. Its vertices form the A*3 / D*3 lattice, known as the Body-Centered Cubic lattice; this honeycomb's vertex figure is a tetrakis cube: 24 disphenoids meet at each vertex. The union of these 24 disphenoids forms a rhombic dodecahedron; each edge of the tessellation is surrounded by either four or six disphenoids, according to whether it forms the base or one of the sides of its adjacent isosceles triangle faces respectively. When an edge forms the base of its adjacent isosceles triangles, is surrounded by four disphenoids, they form an irregular octahedron.
When an edge forms one of the two equal sides of its adjacent isosceles triangle faces, the six disphenoids surrounding the edge form a special type of parallelepiped called a trigonal trapezohedron. An orientation of the tetragonal disphenoid honeycomb can be obtained by starting with a cubic honeycomb, subdividing it at the planes x = y, x = z, y = z squashing it along the main diagonal until the distance between the points and becomes the same as the distance between the points and; the hexakis cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. John Horton Conway calls it a pyramidille. Cells can be seen in a translational cube, using 4 vertices on one face, the cube center. Edges are colored by, it can be seen as a cubic honeycomb with each cube subdivided by a center point into 6 square pyramid cells. There are two types of planes of faces: one as a square tiling, flattened triangular tiling with half of the triangles removed as holes, it is dual to the truncated cubic honeycomb with octahedral and truncated cubic cells: If the square pyramids of the pyramidille are joined on their bases, another honeycomb is created with identical vertices and edges, called a square bipyramidal honeycomb, or the dual of the rectified cubic honeycomb.
It is analogous to the 2-dimensional tetrakis square tiling: The square bipyramidal honeycomb is a uniform space-filling tessellation in Euclidean 3-space. John Horton Conway calls it an oblate octahedrille or shortened to oboctahedrille. A cell can be seen positioned within a translational cube, with 4 vertices mid-edge and 2 vertices in opposite faces. Edges are labeled by the number of cells around the edge, it can be seen as a cubic honeycomb with each cube subdivided by a center point into 6 square pyramid cells. The original cubic honeycomb walls are removed, joining pairs of square pyramids into square bipyramids, its vertex and edge framework is identical to the hexakis cubic honeycomb. There is one type of plane with faces: a flattended triangular tiling with half of the triangles as holes; these cut face-diagonally through the original cubes. There are square tiling plane that exist as nonface holes passing through the centers of the octahedral cells, it is dual to the rectified cubic honeycomb with octahedral and cuboctahedral cells: The phyllic disphenoidal honeycomb is a uniform space-filling tessellation in Euclidean 3-space.
John Horton Conway calls this an Eighth pyramidille. A cell can be see as 1/48 of a translational cube with vertices positioned: one corner, one edge center, one face center, the cube center; the edge colors and labels specify. It is dual to the omnitruncated cubic honeycomb: Architectonic and catoptric tessellation Cubic honeycomb space frame Triakis truncated tetrahedral honeycomb Gibb, William, "Paper patterns: solid shapes from metric paper", Mathematics in School, 19: 2–4, reprinted in Pritchard, Chris, ed; the Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching, Cambridge University Press, pp. 363–366, ISBN 0-521-53162-4. Senechal, Marjorie, "Which tetrahedra fill space?", Mathematics Magazine, Mathematical Association of America, 54: 227–243, doi:10.2307/2689983, JSTOR 2689983. Conway, John H.. "21. Naming Archimedean and Catalan Polyhedra and Tilings"; the Symmetries of Things. A K Peters, Ltd. pp. 292–298. ISBN 978-1-56881-220-5