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

Polyhedron

In geometry, a polyhedron is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from as poly - + - hedron. A convex polyhedron is the convex hull of finitely many points on the same plane. Cubes and pyramids are examples of convex polyhedra. A polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions. Convex polyhedra are well-defined, with several equivalent standard definitions. However, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of "polyhedron" have been given within particular contexts, some more rigorous than others, there is not universal agreement over which of these to choose; some of these definitions exclude shapes that have been counted as polyhedra or include shapes that are not considered as valid polyhedra. As Branko Grünbaum observed, "The Original Sin in the theory of polyhedra goes back to Euclid, through Kepler, Poinsot and many others... at each stage... the writers failed to define what are the polyhedra".

There is general agreement that a polyhedron is a solid or surface that can be described by its vertices, edges and sometimes by its three-dimensional interior volume. One can distinguish among these different definitions according to whether they describe the polyhedron as a solid, whether they describe it as a surface, or whether they describe it more abstractly based on its incidence geometry. A common and somewhat naive definition of a polyhedron is that it is a solid whose boundary can be covered by finitely many planes or that it is a solid formed as the union of finitely many convex polyhedra. Natural refinements of this definition require the solid to be bounded, to have a connected interior, also to have a connected boundary; the faces of such a polyhedron can be defined as the connected components of the parts of the boundary within each of the planes that cover it, the edges and vertices as the line segments and points where the faces meet. However, the polyhedra defined in this way do not include the self-crossing star polyhedra, their faces may not form simple polygons, some edges may belong to more than two faces.

Definitions based on the idea of a bounding surface rather than a solid are common. For instance, O'Rourke defines a polyhedron as a union of convex polygons, arranged in space so that the intersection of any two polygons is a shared vertex or edge or the empty set and so that their union is a manifold. If a planar part of such a surface is not itself a convex polygon, O'Rourke requires it to be subdivided into smaller convex polygons, with flat dihedral angles between them. Somewhat more Grünbaum defines an acoptic polyhedron to be a collection of simple polygons that form an embedded manifold, with each vertex incident to at least three edges and each two faces intersecting only in shared vertices and edges of each. Cromwell gives a similar definition but without the restriction of three edges per vertex. Again, this type of definition does not encompass the self-crossing polyhedra. Similar notions form the basis of topological definitions of polyhedra, as subdivisions of a topological manifold into topological disks whose pairwise intersections are required to be points, topological arcs, or the empty set.

However, there exist topological polyhedra. One modern approach is based on the theory of abstract polyhedra; these can be defined as ordered sets whose elements are the vertices and faces of a polyhedron. A vertex or edge element is less than an edge or face element when the vertex or edge is part of the edge or face. Additionally, one may include a special bottom element of this partial order and a top element representing the whole polyhedron. If the sections of the partial order between elements three levels apart have the same structure as the abstract representation of a polygon these ordered sets carry the same information as a topological polyhedron. However, these requirements are relaxed, to instead require only that sections between elements two levels apart have the same structure as the abstract representation of a line segment. Geometric polyhedra, defined in other ways, can be described abstractly in this way, but it is possible to use abstract polyhedra as the basis of a definition of geometric polyhedra.

A realization of an abstract polyhedron is taken to be a mapping from the vertices of the abstract polyhedron to geometric points, such that the points of each face are coplanar. A geometric polyhedron can be defined as a realization of an abstract polyhedron. Realizations that forgo the requirement of planarity, that impose additional requirements of symmetry, or that map the vertices to higher dimensional spaces have been considered. Unlike the solid-based and surface-based definitions, this works well for star polyhedra. However, without additional restrictions, this definition allows degenerate or unfaithful polyhedra (for instance, by mapp

Faceting

In geometry, faceting is the process of removing parts of a polygon, polyhedron or polytope, without creating any new vertices. New edges of a faceted polyhedron may be created along internal space diagonals. A faceted polyhedron will have two faces on each edge and creates new polyhedra or compounds of polyhedra. Faceting is the dual process to stellation. For every stellation of some convex polytope, there exists a dual faceting of the dual polytope. For example, a regular pentagon has one symmetry faceting, the pentagram, the regular hexagon has two symmetric facetings, one as a polygon, one as a compound of two triangles; the regular icosahedron can be faceted into three regular Kepler–Poinsot polyhedra: small stellated dodecahedron, great dodecahedron, great icosahedron. They all have 30 edges; the regular dodecahedron can be faceted into one regular Kepler–Poinsot polyhedron, three uniform star polyhedra, three regular polyhedral compound. The uniform stars and compound of five cubes are constructed by face diagonals.

The excavated dodecahedron is a facetting with star hexagon faces. Faceting has not been studied as extensively as stellation. In 1568 Wenzel Jamnitzer published his book Perspectiva Corporum Regularium, showing many stellations and facetings of polyhedra. In 1619, Kepler described a regular compound of two tetrahedra which fits inside a cube, which he called the Stella octangula. In 1858, Bertrand derived the regular star polyhedra by faceting the regular convex icosahedron and dodecahedron. In 1974, Bridge enumerated the more straightforward facetings of the regular polyhedra, including those of the dodecahedron. In 2006, Inchbald described the basic theory of faceting diagrams for polyhedra. For a given vertex, the diagram shows all the possible edges and facets which may be used to form facetings of the original hull, it is dual to the dual polyhedron's stellation diagram, which shows all the possible edges and vertices for some face plane of the original core. Bertrand, J. Note sur la théorie des polyèdres réguliers, Comptes rendus des séances de l'Académie des Sciences, 46, pp. 79–82.

Bridge, N. J. Facetting the dodecahedron, Acta crystallographica A30, pp. 548–552. Inchbald, G. Facetting diagrams, The mathematical gazette, 90, pp. 253–261. Alan Holden, Shapes and Symmetry. New York: Dover, 1991. P.94 Weisstein, Eric W. "Faceting". MathWorld. Olshevsky, George. "Faceting". Glossary for Hyperspace. Archived from the original on 4 February 2007

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

Francisco Santos Leal

Francisco Santos Leal is a Spanish mathematician at the University of Cantabria, known for finding a counterexample to the Hirsch conjecture in polyhedral combinatorics. In 2015 he won the Fulkerson Prize for this research. Santos was born in Spain, he earned a licenciate in mathematics from the University of Cantabria in 1991, a master's degree in pure mathematics from Joseph Fourier University in Grenoble, France in the same year. He returned to Cantabria for his doctorate, which he finished in 1995, with a thesis on the combinatorial geometry of algebraic curves and Delaunay triangulations supervised by Tomás Recio, he has a second licenciate, in physics, from Cantabria in 1996. After postdoctoral studies at the University of Oxford he returned to Cantabria as a faculty member in 1997, was promoted to full professor in 2008. From 2009 to 2013 he has been vice-dean of the Faculty of Sciences at Cantabria; as well as being honored by the Fulkerson Prize in 2015 for a counter-example of the Hirsch conjecture, he was a semiplenary speaker at the 2006 International Congress of Mathematicians.

Santos is an Editor-in-Chief of the Electronic Journal of Combinatorics. Home page Google scholar profile

Polygon

In elementary geometry, a polygon is a plane figure, described by a finite number of straight line segments connected to form a closed polygonal chain or polygonal circuit. The solid plane region, the bounding circuit, or the two together, may be called a polygon; the segments of a polygonal circuit are called its edges or sides, the points where two edges meet are the polygon's vertices or corners. The interior of a solid polygon is sometimes called its body. An n-gon is a polygon with n sides. A simple polygon is one. Mathematicians are concerned only with the bounding polygonal chains of simple polygons and they define a polygon accordingly. A polygonal boundary may be allowed to cross over itself, creating star polygons and other self-intersecting polygons. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. There are many more generalizations of polygons defined for different purposes; the word polygon derives from the Greek adjective πολύς "much", "many" and γωνία "corner" or "angle".

It has been suggested. Polygons are classified by the number of sides. See the table below. Polygons may be characterized by their convexity or type of non-convexity: Convex: any line drawn through the polygon meets its boundary twice; as a consequence, all its interior angles are less than 180°. Equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. Non-convex: a line may be found which meets its boundary more than twice. Equivalently, there exists a line segment between two boundary points that passes outside the polygon. Simple: the boundary of the polygon does not cross itself. All convex polygons are simple. Concave. Non-convex and simple. There is at least one interior angle greater than 180°. Star-shaped: the whole interior is visible from at least one point, without crossing any edge; the polygon must be simple, may be convex or concave. All convex polygons are star-shaped. Self-intersecting: the boundary of the polygon crosses itself.

The term complex is sometimes used in contrast to simple, but this usage risks confusion with the idea of a complex polygon as one which exists in the complex Hilbert plane consisting of two complex dimensions. Star polygon: a polygon which self-intersects in a regular way. A polygon can not be both star-shaped. Equiangular: all corner angles are equal. Cyclic: all corners lie on a single circle, called the circumcircle. Isogonal or vertex-transitive: all corners lie within the same symmetry orbit; the polygon is cyclic and equiangular. Equilateral: all edges are of the same length; the polygon need not be convex. Tangential: all sides are tangent to an inscribed circle. Isotoxal or edge-transitive: all sides lie within the same symmetry orbit; the polygon is equilateral and tangential. Regular: the polygon is both isogonal and isotoxal. Equivalently, it is both equilateral, or both equilateral and equiangular. A non-convex regular polygon is called a regular star polygon. Rectilinear: the polygon's sides meet at right angles, i.e. all its interior angles are 90 or 270 degrees.

Monotone with respect to a given line L: every line orthogonal to L intersects the polygon not more than twice. Euclidean geometry is assumed throughout. Any polygon has as many corners; each corner has several angles. The two most important ones are: Interior angle – The sum of the interior angles of a simple n-gon is π radians or × 180 degrees; this is because any simple n-gon can be considered to be made up of triangles, each of which has an angle sum of π radians or 180 degrees. The measure of any interior angle of a convex regular n-gon is 180 − 360 n degrees; the interior angles of regular star polygons were first studied by Poinsot, in the same paper in which he describes the four regular star polyhedra: for a regular p q -gon, each interior angle is π p radians or 180 p degrees. Exterior angle – The exterior angle is the supplementary angle to the interior angle. Tracing around a convex n-gon, the angle "turned" at a corner is external angle. Tracing all the way around the polygon makes one full turn, so the sum of the exterior angles must be 360°.

This argument can be generalized to concave simple polygons, if external angles that turn in the opposite direction are subtracted from the total turned. Tracing around an n-gon in general, the sum of the exterior angles can be any integer multiple d of 360°, e.g. 720° for a pentagram and 0° for an angular "eight" or antiparallelogram, where d is the density or starriness of the polygon. See orbit. In this section, the vertices of the polygon under consideration are taken to be, ( x 1