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	Rudolf Steiner - Wikipedia, the free encyclopedia
		Steiner's father was a huntsman in the service of Count Hoyos in Geras, and later became a telegraph operator and stationmaster on the Southern Austrian Railway.
		Steiner argued that increased autonomy for the three spheres would not eliminate their mutual influence, but would cause that influence to be exerted in a more healthy and legitimate manner, because the increased separation would prevent any one of the three spheres from dominating.
		Steiner also occasionally averred that this consciousness of the spirit was not so much related to the content of his statements, where he tells readers the characteristics of this or that spiritual being (or something similar) that he says he has perceived.
	en.wikipedia.org /wiki/Rudolf_Steiner (4515 words)

	Jakob Steiner - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-29)
		After Steiner's publication (1832) of his Systematische Entwickelungen he received, through Jacobi's exertions, who was then professor at Königsberg, an honorary degree of that university; and through the influence of G. Jacobi and of the brothers Alexander and Wilhelm von Humboldt a new chair of geometry was founded for him at Berlin (1834).
		He next gives by aid of these projective rows and pencils a new generation of conics and ruled quadric surfaces, which leads quicker and more directly than former methods into the inner nature of conics and reveals to us the organic connection of their innumerable properties and mysteries.
		Starting from simple elementary propositions, Steiner advances to the solution of problems which analytically require the calculus of variations, but which at the time altogether surpassed the powers of that calculus.
	www.pineville.us /project/wikipedia/index.php/Jakob_Steiner (654 words)

	Roman surface - Wikipedia, the free encyclopedia
		The Roman surface (so called because Jakob Steiner was in Rome when he thought of it) is a self-intersecting immersion of the real projective plane into three-dimensional space, with an unusually high degree of symmetry.
		One of the lobes of the Roman surface is seen frontally in Figure 5, and its bulbous -- balloon-like -- shape is evident.
		If the Roman surface were to be inscribed inside the tetrahedron with least possible volume, one would find that each edge of the tetrahedron is tangent to the Roman surface at a point, and that each of these six points happens to be a Whitney singularity.
	en.wikipedia.org /wiki/Roman_surface (1397 words)

	Rudolf Steiners course on warmth - a summary
		Steiner analyzes the so-called "warmth death" our universe is experiencing because of "unstoppable negative entropy" and draws his public's attention to the fact that the formulations belong to an age of history when science was not too developed and physicists considered only mechanical work and heat as the most basic factors of creation.
		Steiner goes on analyzing temperature difference between one cold and one hot zone and an intermediate region that serves as conductor, indicating that great care must be taken in the mathematical expression of this for the heat distribution is not uniform.
		Steiner continues by indicating to his audience to conceive of the periodic arrangement of elements in octaves, so that their combination and interrelated assorted phenomena may be conceived as an outer reflection of an inner cosmic music.
	hem.passagen.se /thebee/SCIENCE/warmsumm.htm (3265 words)

	Cubic surfaces
		Other results on cubic surfaces were proved by Clebsch which included: there exists a covariant of order nine which intersects the cubic surface in exactly 27 lines; and every smooth cubic surface can be represented in the plane using four plane cubic surfaces through six points and vice-versa.
		Karl Geiser's great uncle was Steiner so he set out on his mathematical career already having links to one of the important figures in the development of the theory of cubic surfaces.
		Starting from the construction of a cubic surface given by a straight line, three groups of three points on a line, and six other points, Le Paige was led to the construction of a cubic surface given by a line, three points on a line and twelve other points.
	www-history.mcs.st-and.ac.uk /history/HistTopics/Cubic_surfaces.html (1002 words)

	Isosurface Tutorial: Steiner Surfaces (Site not responding. Last check: 2007-10-29)
		Steiner surfaces are representations of the projective plane, discovered by J. Steiner.
		For some reason, this surface didn't look anything remotely like what it was supposed to when I used a conventional isosurface so I've actually used a parametric isosurface to generate it.
		This surface is isomorphic to Whitney's Umbrella, but has the two pinch points close to the origin.
	www.econym.demon.co.uk /isotut/steiner.htm (355 words)

	Steiner's Roman Surface -- The 4th Dimension (Site not responding. Last check: 2007-10-29)
		The Roman surface, also known as Steiner's Roman surface, was discovered in 1844 by Jacob Steiner, you have probably guessed he was in Rome when he solved the problem.
		While Steiner was working on the solution he was having some difficulty with solving a complicated polynomial, so he asked his friend Karl Weierstrass to see if solution was given in any mathematical tables, luckily there was.
		The Roman surface is one of three known possible surfaces obtained by gluing the edge of a disc to the edge of a Möbius strip, the the other two such surfaces are the crosscap and Boy's surface.
	www.uta.edu /optics/sudduth/4d/nonorientable/steiners_roman_surface/steiners_roman_surface.htm (393 words)

	Encyclopedia: Steiner
		Steiner is a German surname that is derived from the word Stein, meaning stone.
		George Steiner, prominent literary critic, was born on April 23, 1929 in Paris to Viennese parents.
		Scott Steiner Scott Rechsteiner (born July 29, 1962), better known as Scott Steiner is a professional wrestler who previously worked for World Wrestling Entertainment on the RAW brand.
	www.nationmaster.com /encyclopedia/Steiner (495 words)

	Jakob Steiner
		Jakob Steiner (18 March 1796-April 1 1863) was a Swiss mathematician.
		He was born in the village of Utzendorf, Canton of Bern.
		Steiner's papers were collected and published in two volumes (Gesammelte Werke, 1881-1882) by the Berlin Academy.
	www.brainyencyclopedia.com /encyclopedia/j/ja/jakob_steiner.html (640 words)

	Steiner, Jakob (Site not responding. Last check: 2007-10-29)
		He discovered the Steiner surface (also called the Roman surface), which has a double infinity of conic sections on it, and the Steiner theorem.
		Steiner was born at Utzenstorf, near Bern, and did not learn to read and write until the age of 14.
		The Steiner theorem states that two pencils (collections of geometric objects) by which a conic is projected from two of its points are projectively related.
	cartage.org.lb /en/themes/Biographies/MainBiographies/S/Steiner/1.html (202 words)

	Steiner Surfaces
		The geometric properties of Steiner surfaces, including the degree and the number and type of singularities, such as pinch points, double lines, and triple points, depend on the projection from five ambient dimensions to three.
		A classification of Steiner surfaces was known in the XIX century in the case where the coordinates and projective transformations are allowed to be complex.
		In fact, this description is just a fancy way to state the original definition of a Steiner surface: the image of three rational functions of two variables, where the numerators are any three quadratic polynomials, and the denominators are equal quadratics.
	www.ipfw.edu /math/Coffman/steinersurface.html (2078 words)

	Mathematics ofSteiner's Roman Surface-- The 4th Dimension (Site not responding. Last check: 2007-10-29)
		The coloring of the surface is such that one side of the surface is blue while the other side is green.
		Of course the answer is that this is an effect of being nonorientable, because you cannot assign definite side to this surface.
		The analytical description of the Roman surface was first developed by Jacob Steiner while he was visiting his friend Karl Weierstrass in Rome.
	www.coolphysics.com /4d/nonorientable/steiners_roman_surface/math/mathematics.htm (356 words)

	JAKOB STEINER
		Steiner was one of the greatest of all geometers.
		Steiner remained at the University of Berlin until his last years, which he spent in his native Switzerland.
		Steiner's Ellipse Problem (that the ellipse circumscribing triangle ABC and having minimal area is the Steiner circum-ellipse, and that the ellipse inscribed in ABC and having maximal area is the Steiner in-ellipse; the two ellipses have as center the centroid of ABC)
	faculty.evansville.edu /ck6/bstud/steiner.html (776 words)

	Veronese surface (Site not responding. Last check: 2007-10-29)
		Giuseppe Veronese first introduced rational surfaces as projections of the related, so-called Veronese surface, a normal surface in 1/2 (n+1)(n+2)-1-dimensional projective space, where n is the degree of the resulting rational surfaces in three-dimensional projective space.
		The Steiner surface itself had already been discovered by Jakob Steiner during a visit to Rome in 1844; therefore the Steiner surface is sometimes also referred to as Roman surface.
		In this paper rational triangular Bézier patches of degree n are related to Veronese varieties, where the latter ones are shown to be singular subvarieties of certain algebraic varieties, and the obtained results are illustrated and extended for the low-degree cases n=2,3,4.
	www.univ-valenciennes.fr /macs/caomacs/veronese.htm (163 words)

	Adam Coffman Deposit #11 (Site not responding. Last check: 2007-10-29)
		Steiner's Roman surface is one representation of the real projective plane, and it intersects itself along three line segments.
		Any surface defined by homogeneous quadratic rational functions like this is called a "Steiner surface," and the Roman surface is one of 10 types, as classified by Coffman, Schwartz, and Stanton.
		One two-parameter equation for the Roman surface is:
	curvebank.calstatela.edu /romansurfaces/romansurfaces.htm (502 words)

	Numerical Simulation of Magnetic Flux Concentrations, by Oskar Steiner (Site not responding. Last check: 2007-10-29)
		For the interpretation and physical understanding of high resolution observations of solar surface magnetism as being made with the VTT telescope at Teneriffe, an intensive program of numerical simulations is carried out at the Kiepenheuer-Institut, using various computational platforms in house and at the High-Performance Computing-Center Stuttgart.
		In an attempt to understand the formation of small scale magnetic flux concentrations in the solar photosphere we have also followed the evolution of a homogeneous and dispersed vertical magnetic field that is initially superimposed on an evolved state of nonstationary convection.
		Also shown is the optical depth unity surface which is depressed at the position of the flux sheet due to the partial evacuation of the sheet.
	www.kis.uni-freiburg.de /~steiner (2022 words)

	Search Results for Steiner
		Steiner was a major influence on Schroter who spent most of his life working on geometry.
		Schlafli and Steiner were also with them, Schlafli's main task being to act as their interpreter but he studied mathematics with Dirichlet as his tutor.
		Steiner in 1832 studied notions of synthetic geometry which were to eventually become part of the study of transformation groups.
	www-history.mcs.st-and.ac.uk /Search/historysearch.cgi?SUGGESTION=Steiner&CONTEXT=1 (1825 words)

	Cool Klein Bottle (Site not responding. Last check: 2007-10-29)
		Examples of surfaces in this class are Dini’s surface, the Pseudosphere (both of which are developed directly from the Tractrix) and Kuen’s surface (for images of the surfaces mentioned in this essay, see http://www.uib.no/People/nfytn/mathgal.htm).
		A Minimal surface is naively the surface with minimal area spanning the space between its assigned edges.
		Surfaces in this class are Steiner’s Roman surface, the cross-cap, and Boy’s surface (a surface, interestingly enough, without singularities).
	www.sff.net /people/asaro/klein.html (809 words)

	Final Fantasy: Solid 2, The Plant Episode by d_Galloway
		Steiner: Terrorists blew up a tanker of crude oil off the shore of Galbadia, severely damaging the local wildlife and cotaminating the water supply.
		Steiner: A group of terrorists have seized control of the Big Shell and are threatening to destroy the entire place.
		Steiner: They have several hostages inside, including the head of an environmental protection agency, and the most important hostage of all: James Deling.
	www.rpgclassics.com /fanfics/tempfics/MGS2ff2.shtml (3830 words)

	Introduction to surf version 0.91: Features
		The direction of the normal vector given by the gradient of the surface equation defines one side of the surface which is regarded as outside.
		Ambient light is a constant which represents the light a point on the surface receives from the whole environment (the sky, the floor, the lawn...) but not from the light sources.
		The surfaces on fl and white images often don't look very impressive; often it is hard to detect the edges of a surface.
	enriques.mathematik.uni-mainz.de /surf/manual-3.html (2534 words)

	Mathematics 535 (Fall 2002) Information (Site not responding. Last check: 2007-10-29)
		The Veronese surface is the image of the projective plane using the 6 degree 2 monomials.
		The green lines are contained in the surface (any line hitting a hypersurface in P^3 more times than its degree is wholly contained in the surface), and are the singular locus of the variety, that is points where the gradient of the defining polynomial vanishes.
		These are all related to examples of Steiner surfaces, which are the images of the real projective plane in P^3 given by 4 independent homogeneous polynomials of degree 2 in 3 variables.
	www.math.rutgers.edu /courses/535/535-f02/Movie3.html (611 words)

	Curves & Surfaces > Surfaces > Special Surfaces > Topology (Site not responding. Last check: 2007-10-29)
		Examples of such surfaces are the Möbius strip, the Klein bottle, and Steiner's Roman surface.
		A surface is said to be complete if its geodesics can be extended to the entire real number line, remaining always on the surface.
		One example of a complete surface is one sheet of the cone, minus its vertex.
	math.hmc.edu /faculty/gu/curves_and_surfaces/surfaces/_topology.html (223 words)

	Steiner's Roman Surface (Site not responding. Last check: 2007-10-29)
		Steiner's Roman surface is one realization of a mathematical object known as the real projective plane.
		Note that, like the Möbius strip and the Klein bottle, Steiner's Roman surface is non-orientable.
		, the tangent plane to the surface is parameterized by:
	www.math.hmc.edu /faculty/gu/curves_and_surfaces/surfaces/roman.html (187 words)

	ppp.html
		Body:In October of 2000 the author discovered that a Steiner surface could be obtained from a sphere parametric by squaring it's SO(3) representation.
		It was by chance in going over old surface models that the time differential of the Pseudosphere was plotted as a 3d parametric and found to be a sphere.
		This new projective plane would seem to show that such parametric surfaces are possible and may even have a simple representation like that of a standing wave on a sphere.It is the differential nature of the Escher set geometry in a polynomial frame of reference that makes such geometry relatively simple in nature.
	www.homestead.com /tftn/ppp.html (755 words)

	David E. Breen: Abstract of MS Thesis (Site not responding. Last check: 2007-10-29)
		Steiner surface smooth-shading is implemented in a subroutine in The Clockworks' ray-tracer, BART.
		With the knowledge that a Steiner patch can be represented as a quadratic triangular Bernstein-Bezier patch, a method for calculating the Bernstein-Bezier control points of a Steiner patch was produced.
		The first three control points of the Steiner patch are on the approximated surface.
	www.cacr.caltech.edu /~david/Abstracts/ms-abs.html (368 words)

	Search Results for Steiner surface - Encyclopædia Britannica
		Austrian-born scientist, editor, and founder of anthroposophy, a movement based on the notion that there is a spiritual world comprehensible to pure thought but accessible only to the highest...
		The combination of a hard surface and a soft interior is greatly...
		The earth's surface and crust are constantly evolving through a process called the rock cycle.
	www.britannica.com /search?query=Steiner+surface (467 words)

	Steiner surface -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-29)
		More particularly, they are linear projections of a six-dimensional embedding called the (Click link for more info and facts about Veronese surface) Veronese surface, which is the image of an ordinary 2-sphere centered at the origin under the map
		There are ten different types, including the (Click link for more info and facts about Roman surface) Roman surface and (Click link for more info and facts about cross-cap) cross-cap.
		They are named after (Click link for more info and facts about Jakob Steiner) Jakob Steiner, who discovered them.
	www.absoluteastronomy.com /encyclopedia/s/st/steiner_surface.htm (96 words)

	Adam Coffman --- Graphics Gallery
		Steiner surface, with its six pinch points, three double lines, and one triple point.
		The Euclidean volume and surface area of the three objects are also compared, but other Roman surfaces related to this one by a linear transformation will of course have different values for volume and area.
		The "Bishop invariant" at these points starts with value 0, when the surface is a sphere, and increases, with upper bound 1/2, until the surface flattens into an elliptical disc in the yz-plane.
	www.ipfw.edu /math/Coffman/pov/gallery.html (612 words)

	[No title]
		For example, section 523 of the Chelsea edition has a table of 23 kinds of cubic surfaces, classified according to their singularities and giving the class and number of lines on the surface in each case.
		(Salmon uses the terminology "reciprocal surface" but is not referring to this transformation; instead, he means what we would call the dual surface.) If one of the coefficients a,b,c,d vanishes, say a, then the cubic is reducible and has a line of singularlties at least.
		Assuming I have guessed correctly, the class of this surface should be 4 and it should have exactly 9 lines on it, according to the table.
	www.math.niu.edu /~rusin/known-math/99/cubic_surf (1005 words)

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