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Topic: Finite geometry

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	Finite Element Modeling in Comparative Biomechanics
		Plate finite element: A plate finite element is similar in shape to the plate geometry element except the element has thickness associated with it, material properties, and an underlying structural behavioral model that accounts for the stiffness of the physical structure represented by the element.
		For the simplest shaped finite elements, these points are located at the vertices of the polygon (triangle or quadrilateral shape) or polyhedron (tetrahedral or hexahedral shape) that represent the element geometry.
		The finite elements are assigned material properties such as elasticity, thermal conductivity, etc. to account for the material behavior associated with the physical phenomenon being modeled.
	www.biomesh.org /glossary.htm (1457 words)

	PlanetMath: finite plane
		A finite plane (synonym linear space) is the finite (discrete) analogue of planes in more familiar geometries.
		Another kind of finite plane is an affine plane, which can be obtained from a projective plane by removing one line (and all the points on it).
		This is version 15 of finite plane, born on 2002-10-07, modified 2005-05-22.
	planetmath.org /encyclopedia/FanoPlane.html (293 words)

	Finite Geometries?
		Steiner is known for a variety of contributions to geometry, including work on isoperimetric problems (what region in the plane has the largest area with a fixed perimeter?) and projective geometry (geometry where all lines meet).
		Fano's work in the area of finite geometry included the discussion of a 3-dimensional finite geometry which consisted of 15 points, 35 lines, and 15 planes where each line had 3 points on it and each plane had 7 points.
		In particular, the structure of finite geometries is often used in the branch of statistics devoted to the design of experiments.
	www.ams.org /featurecolumn/archive/finitegeometries.html (5303 words)

	Finite geometry - Wikipedia, the free encyclopedia
		Euclidean geometry, for example, is not finite, because a Euclidean line contains infinitely many points, in fact precisely the same number of points as there are real numbers.
		Both finite affine plane geometry and finite projective plane geometry may be described by fairly simple axioms.
		The last axiom ensures that the geometry is not empty, while the first two specify the nature of the geometry.
	en.wikipedia.org /wiki/Finite_geometry (955 words)

	Geometry Page G 8 (Site not responding. Last check: 2007-09-17)
		Finite or "miniature" geometries have only a few axioms and theorems and a finite number of elements; that is, a finite number of points or lines or "things to work with." The study of finite geometries provides an opportunity to study geometries with a simple structure.
		Historically, the first finite geometry was a three-dimensional geometry, each plane containing seven points and seven lines.
		This finite geometry was explored by Gino Fano in 1892.
	www.csc.vsc.edu /Math/Part1/G8doc2.html (174 words)

	Finitism in Geometry (Stanford Encyclopedia of Philosophy)
		Finitism is one of the foundational views of mathematics that is listed under the broader heading of constructivism.
		However, finitism goes one step further and argues that an indefinite outcome is not be accepted as an outcome, since, as all computational resources are finite, it could very well be that these resources have been used up before the outcome has been reached.
		Although such geometries can be very inspiring for a strict finitist proposal (as will be shown in section 2), their aim is not to provide an alternative for continuous infinite geometries.
	plato.stanford.edu /entries/geometry-finitism (4850 words)

	Square (geometry) - Wikipedia, the free encyclopedia
		In plane (Euclidean) geometry, a square is a polygon with four equal sides, four right angles, and parallel opposite sides.
		In spherical geometry, a square is a polygon whose edges are great circle arcs of equal distance, which meet at equal angles.
		In finite geometry, a subdivided p×p square, with p a prime number, provides a model for a finite geometry with p
	en.wikipedia.org /wiki/Square_(geometry) (564 words)

	Finite Geometry
		Gino Fano (1871–1952) is credited with being the first person to explore finite geometries beginning in 1892.
		Here, we obtain a finite geometry by restricting the system to one of the planes.
		In the diagram model on the left, points are defined by the seven dots and lines by the six straight segments and one curved segment.
	www.mnstate.edu /peil/geometry/C1AxiomSystem/finitegeometry.htm (720 words)

	Table of Infinite Regular Polyhedra
		At first it appears that finite and infinite polyhedra are very different sorts of beasts but it turns out that mathematically they can be looked at in very much the same way through the concept of surface "genus".
		Finite models exist in the normal infinite 3D space whereas infinite models can be thought of as existing in finite spaces which repeat in three dimensions.
		The yellow cells represent the flat plane tilings which are a sort of degenerate class of polyhedra which sit exactly on the border of the finite and infinite regular polyhedra.
	www.superliminal.com /geometry/infinite/infinite.htm (1134 words)

	51: Geometry
		Solid geometry is placed here (actually in 51M05) because it mirrors elementary plane geometry, but spherical geometry is primarily on the page for general convex geometry.
		Cabri-geometry is used for teaching secondary school geometry, but, equally important, is its use for university level instruction and as a tool by mathematicians in their research work.
		A useful collection of Geometry Formulas and Facts is taken from the CRC Standard Mathematical Tables and Formulas, and available at the The Geometry Center.
	www.math.niu.edu /~rusin/known-math/index/51-XX.html (828 words)

	Geometry of the 4x4 Square
		A pairing equivalent to that in the MOG was described (though without the rectangular arrays devised by Curtis) in a paper on the projective space PG(3,2) published in 1910.
		Some Constructions and Embeddings of the Tilde Geometry (pdf), by Antonio Pasini and Hendrick Van Maldeghem (a 2002 paper on "the flag transitive connected triple cover of the unique generalized quadrangle W(2) of order (2,2)").
		A 1994 AMS research announcement (pdf), by A. Ivanov and S. Shpectorov, on the role played by (generalized) tilde geometries in the study of various sporadic simple groups.
	m759.freeservers.com /geometry.html (1627 words)

	Elements of Finite Geometry
		Steven H. Cullinane - Finite Geometry of the Square and Cube
		These books supply background for the study of finite geometry, but they are not all limited to that topic.
		The Geometry of Incidence, by Harold L. Dorwart
	finitegeometry.org (408 words)

	Generalized Method of Decomposing Solid Geometry into Hexahedron Finite Elements
		Decomposing geometry in terms of a function of external features is novel in that it utilizes the relationship of the solid geometry outer surface connectivity with respect to the number of edges on a solid independant of an assessment of the solid's interior.
		Reducing external complex shape characteristics to a level compatible with generic geometry is interfaced with various shape recognition conditions for identifying local configurations such as: reentrant edges, multiple surface solid geometry in compacted transition regions, and various hybrid solids, such as the filletted trailing edge of a turbine blade.
		Rather than attempting to decompose geometry with one universal approach, the process is a series of steps compatible with the Jacobian transform of finite element geometry.
	www.andrew.cmu.edu /user/sowen/abstracts/Ho169.html (684 words)

	Good solid modeling, Bad FEA
		Solid CAD geometry almost always requires extensive modifications before it is suitable for meshing with finite elements.
		The CAD geometry for the fin and a portion of the base plate are exported to an FEA program as a solid and meshed using default parameters in the automesher.
		Another way around the problem of singularities is to recreate the geometry in the FEA program for use with shell elements as in the illustration Fins from shells.
	www.machinedesign.com /BDE/cadcam/bdecad3/bdecad3_4.html (2301 words)

	Geometry/Dynamics of the Universe
		But the number of stars, finite as it might be, is still large enough to light up the entire sky, i.e., the total amount of luminous matter in the Universe is too large to allow this escape.
		One possible finite geometry is donutspace or more properly known as the Euclidean 2-torus, is a flat square whose opposite sides are connected.
		The geometry may be flat or open, and therefore infinite in possible size (it continues to grow forever), but the amount of mass and time in our Universe is finite.
	abyss.uoregon.edu /~js/ast123/lectures/lec15.html (2323 words)

	Math Forum - Mathematics Teacher Bibliography: Finite Geometries (Site not responding. Last check: 2007-09-17)
		General Finite Geometries Steven H. Heath Finite systems in which parallelism is not unique.
		Developing A Finite Geometry Charles M. Bundrick, Robert C. Frazier, and Homer C. Gerber Details of the development of a model for a finite affine plane.
		Finite Planes and Latin Squares Truman Botts Developments in finite geometry.
	mathforum.org /mathed/mtbib/finite.geometries.html (156 words)

	Geometry Algorithm Journals
		Advances in Geometry is a new journal for publications in the broad area of geometry.
		This is an international journal which publishes original papers on algorithms that are either (i) theoretical papers addressing problems arising in practical areas or (ii) experimental papers that have general appeal because of their practical importance or techniques.
		This journal presents papers on algorithms that are inherently discrete and finite and that have some natural mathematical content, either in their objective or in their analysis.
	geometryalgorithms.com /books_journals.htm (1041 words)

	Introduction (Site not responding. Last check: 2007-09-17)
		Basic works are `Projective Geometries over Finite Fields', `Finite Projective Spaces of Three Dimensions' and `General Galois Geometries', the first two volumes being written by Hirschfeld [1979, 1985] and the third volume by Hirschfeld and Thas [1991]; in fact, it consists of a single work in three volumes.
		Deep results in finite algebraic geometry and number theory on estimates for the number of points on an algebraic curve were obtained by Hasse [1934] and by Weil [1948].
		Finally, the applications of finite projective geometry to the theory of error-correcting codes completed the deserved success of this beautiful part of the modern development of classical geometry.
	www1.elsevier.com /homepage/saj/504595/07a.htm (456 words)

	Finite-Geometry Models
		In this note we describe four of the most useful construction principles for constructing pictures of small incidence geometries which capture large parts of the abstract beauty of the geometries they depict.
		Given a small, highly symmetrical geometry with n points, look for the same number of points arranged into a highly symmetrical spatial object.
		A defense of the honour of an unjustly neglected little geometry or a combinatorial approach to the projective plane of order five.
	finitegeometry.org /sc/gen/Polster.html (438 words)

	The Diamond Theorem
		Among the 35 structures of the 840 4x4 arrays of tiles, orthogonality (in the sense of Latin-square orthogonality) corresponds to skewness of lines in the finite projective space PG(3,2).
		The proof uses a decomposition technique for functions into a finite field that might be of more general use.
		The underlying geometry of the 4x4 patterns is closely related to the Miracle Octad Generator of R. Curtis-- used in the construction of the Steiner system S(5,8,24)-- and hence is also related to the Leech lattice, which, as Walter Feit has remarked, "is a blown up version of S(5,8,24)."
	diamondtheorem.com (717 words)

	Finite Mode (Site not responding. Last check: 2007-09-17)
		Geometric elements can be created in Finite mode (with defined Start and End Points) or in Infinite Mode (without fixed Start and End Points).
		For example, if the FINITE Mode is active and you define a Line through two points, the line will start at the first point and end at the second.
		If the Finite mode is switched off (Infinite Mode) then the line will pass through the two points but will not be trimmed.
	www.ezcam.com /web/products/help/ezmill/menu_geometry/finite_mode.htm (131 words)

	Finite Geometry Web --- Online Notes
		In addition to the notes collected here, invaluable resources are people's theses in finite geometry.
		Finite Geometry and its Applications, Socrates Intensive Course, April 2000, lecture notes, hosted by Frank De Clerck.
		Galois Geometry and Generalized Polygons, Intensive Course, April 1998, lecture notes, hosted by Frank De Clerck.
	cage.ugent.be /~aoffer/fgw/notes.html (306 words)

	My Netscape FAQ
		Optimal normal bases in finite fields were introduced at the University of Waterloo by Mullin et al.\ (1989), and are used in practical hardware implementation of public-key cryptosystems.
		The codes associated with finite geometries are members of the class of well known and frequently used generalized Reed-Muller codes.
		In cryptography, we investigate efficient arithmetic of finite fields which is crucial in practical implementation of public-key cryptosystems that are based on finite fields and elliptic curves over finite fields.
	www.math.clemson.edu /~sgao/WEB/research.html (4157 words)

	Minicourse (MC-DAG-2): Generalized polygons and semipartial geometries.
		The lecturers are members of the research group Incidence Geometry of the Department of Pure Mathematics and Computer Algebra at the University of Ghent (Belgium).
		He is also editor of the Proceedings "Advances in Finite Geometry and Designs" (co-editors J.W.P. Hirschfeld and D.R. Hughes) and "Finite Geometry and Combinatorics" (co-editors A. Beutelspacher, F. Buekenhout, J. Doyen, F. De Clerck and J.W.P. Hirschfeld).
		De Clerck was the first student in finite geometry of J.A. Thas.
	www.win.tue.nl /math/eidma/courses/minicourses/gent/gent.html (482 words)

	Finite Elements
		on the number of nodes or finite elements that can be used in a model.
		The truss and the cubic beam element are the most widely used finite elements for bar, beam, or column modeling.
		The rib element is a 3-node isoparametric element with quadratic displacement interpolation that can be used similar to the beam element (but takes account for the shear deformations) or in conjunction with surface elements for eccentric rib modeling.
	www.axisvm.com /finite_elements.htm (550 words)

	Finite Geometry (Site not responding. Last check: 2007-09-17)
		The general geometric setting is typically some projective or affine space (or perhaps a circle geometry of some type) over a finite field.
		Techniques used are typically a blend of finite field theory, elementary number theory, and finite group theory together with the usual counting arguments.
		In recent years I have been using various software packages, such as Cayley or Magma, to help construct relatively 'large' examples and to analyze those examples in the hope of recognizing patterns that can be generalized and eventually proven as theorems.
	www.math.udel.edu /~ebert/info/interests.html (165 words)

	index.gif
		is used as part of the modern geometry independent study course at Washington and Lee University as well as by interested parties outside WandL.
		these sites will be to stimulate the visualization of mathematical concepts, to present the interplay between classical and modern geometry, to develop and exercise proof writing skills, and to give the student a sense of the unity of mathematics by relating geometry to other fields.
		This web site on finite geometry is organized into problems and reading projects.
	home.wlu.edu /~mcraea/Finite_Geometry/MainPage/main_page.htm (150 words)

	Undergraduate Seminar: Arithmetic and Algebraic Geometry over Finite Fields
		Course description: Finite fields are somehow easy to think about (because they're finite) and yet provide a rich foundation over which to study arithmetic and algebraic geometry.
		We will start with the basic properties of finite fields and some of the interesting immediate consequences of these properties.
		As a consequence of this work, we'll get two proofs of the fact that the Brauer group of a finite field is trivial.
	www.math.columbia.edu /~xander/Fa06/Finite_Fields.html (466 words)

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