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 About us   |   Why use us?   |   Reviews   |   PR   |   Contact us    # Topic: Hypersphere ###### Related Topics Sphere Euclidean space Simplex Euclidean metric Metric space Banach space Riemann sphere Dimension Hypercube Ellipsoid Tesseract Euclidean geometry Astrodynamics Oblate spheroid Hyperbola

 Hypersphere - Wikipedia, the free encyclopedia In mathematics, a hypersphere is a sphere which has dimension 3 or higher. The interior of a hypersphere, that is the set of all points whose distance from the centre is less than ; the ratio of the volume of the hypersphere to its circumscribed hypercube decreases monotonically as the dimension increases. en.wikipedia.org /wiki/Hypersphere   (419 words)

 The hypersphere We cannot directly visualize a hypersphere for the very reason that it is a 4-dimensional object. A hypersphere is in essence an array of spheres: the outer slices (the 'poles') are solid spheres and are smallest. For added insight into the mystery of the hypersphere that is to come in future sections, set aside the anticipation of the current moment and read an article written on the topic of the hypercoin. www.geocities.com /jsfhome/Think4d/Hyprsphr/hsphere.html   (602 words)

 hypersphere Just as the shadow cast by a sphere is a circle, the shadow cast by a hypersphere is a sphere, and just as the intersection of a sphere with a plane is a circle, the intersection of a hypersphere with a hyperplane is a sphere. is the equation of a hypersphere, where w is measured along a fourth dimension at right angles to the x-, y-, and z-axes. The apparent pattern of 2(pi) radians in a circle and 4pi steradians in a sphere does not continue with 8pi hypersteradians because the n-volume, n-area, and number of n-radians of an n-sphere are all related to gamma function and the way it can cancel out powers of pi halfway between integers. www.daviddarling.info /encyclopedia/H/hypersphere.html   (287 words)

 HyperSphere   (Site not responding. Last check: 2007-11-06) HyperSphere Analytics has achieved what statisticians and business graphics professionals have been trying to achieve for decades. HyperSphere Analysis is a simple, understandable a way of graphing tables of numbers, comparing more than just two or three variables or attributes at a time. Traditional methods of charting numbers show one, two, or sometimes three variables at a time, but a HyperSphere is a single chart that shows the relationship of thousands of data points and up to 64 different variables. www.hypersphereanalytics.com   (137 words)

 Fractal of the Day (FotD) by Jim Muth The four-dimensional hypersphere -- the locus of a point in four-dimensional hyperspace which is at a constant distance from a fixed point -- in other words, a four-dimensional ball -- perhaps the simplest hyperfigure. All two-dimensional slices of a hypersphere of any number of dimensions appear as circles; all three-dimensional slices are spheres. The surface of the 4-D hypersphere, as well as the surface of all 4-D figures is three-dimensional -- in the hypersphere, it is a space of constant curvature. home.att.net /~Fractals_2/FotD_01-10-23.html   (495 words)

 The HyperSphere The mathematical objects that live on the sphere in four dimensional space -- the hypersphere -- are both beautiful and interesting. The surface of the hypersphere is three dimensional -- we could walk around on a hypersphere and not know the difference from our own space. The difficulty with viewing an object on a hypersphere is not being able to see it -- it is, after all, a three dimensional object living in a three dimensional space -- but being able to see all of it. www.swiss.ai.mit.edu /~rfrankel/fourd/FourDArt.html   (3236 words)

 Hypersphere   (Site not responding. Last check: 2007-11-06) That is, through each point on the hypersphere (the surface, remember), passes one horizontal circle and one vertical/chromatic circle (that's what I'm going to call the fourth direction, since I'm representing it by color). The point we're imagining going around the circle is sliding downward and blue-ward, always riding on the surface of the hypersphere, always riding on the surface of the appropriately-colored 3d sphere. Once we have those, we can say we're done, because the hair-combing on the hypersphere as a whole will be the vector sum of the horizontal hair and the vertical/chromatic hair that come from each point. web.meson.org /hypersphere   (2792 words)

 General Relativity Then the radius of a spherical shell inside the hypersphere would equal Rsinθ and an infinitesimal step away from this sphere would equal Rdθ; where θ runs from 0 at the origin to π at the antipode. Even on a hypersphere, circumference and radius are related in the usual manner. Were the universe a hypersphere, its circumference would have to be at least 156 billion light years. hypertextbook.com /physics/modern/general-relativity/index.shtml   (748 words)

 Circle, Sphere, Hypersphere, Fourth Spatial Dimension   (Site not responding. Last check: 2007-11-06) Because the circle and the spherical surface are embedded in this metric (when two or one of the terms equal zero) and we know that the circumference of a circle and a spherical surface increases faster than its radius for the simple reason that the circumference C = 2πr. But since we're representing the hypersphere by using one less spatial dimension than it really has, the best we can do is to show the two spheres contiguous at only one position, like a pair of soccer balls touching each other. Just as the route around a sphere with a constant radius r is 2πr and halfway around (to the opposite pole) is πr, then a route all the way around the hypersphere is likewise 2πr and the length of the path to the opposite side is πr. users.adelphia.net /~44mrf/hierarchy(1).html   (3312 words)

 [No title]   (Site not responding. Last check: 2007-11-06) For practical purposes the example discussed next and shown in Pov-ray is for a simple hypersphere and a tangent plane parallel with one of the coordinate axes. Following the same model of a sphere and its tangent plane being projected into two dimensions, a hypersphere and its “hyper-plane,” while they cannot be directly visualized in our world, can be seen as a projection into three dimensions. A hypersphere passing through the third dimension would exist as initially a single point, but would rapidly expand into larger and larger spheres until the circumference of the slice is the circumference of the hypersphere. www.ms.uky.edu /~lee/visual05/gallery/Honors301paper.doc   (1374 words)

 3-sphere - Wikipedia, the free encyclopedia Whereas a 2-sphere is a smooth 2-dimensional surface, a 3-sphere is an object with three dimensions, also known as 3-manifold. In an entirely analogous manner one can define higher-dimensional spheres called hyperspheres or n-spheres. Some people refer to a 3-sphere as a glome from the Latin word glomus meaning ball. en.wikipedia.org /wiki/3-sphere   (2137 words)

 Math Awareness Month 2000: Images of the Fourth Dimension (Essays/SciAm) The hypersphere is a far more complex object than the hypercube, and I shall not describe it in detail. When the hypersphere is rotated, the toruses appear to swell up and sweep past one another. Projected motion of toruses as the hypersphere is rotated, analogous to the projected motion of latitude lines during the rotation of a sphere. www.mathaware.org /mam/00/master/essays/SciAm/SA03.html   (511 words)

 deMon-KS3p5 User Manual for transition state search Starting from equilibrium geometry (zero value of the hypersphere radius R) and giving increment of R, a step-by-step walking uphill process along the MEP [minimum energy path] is carried out. Accounting for the constraints in a straightforward manner allows us to calculate gradients on a hypersphere and to use an efficient quasi-Newton-type algorithm for energy minimization at the given R. In a saddle point vicinity the refining procedure of TS [transition state] parameters is performed. Since it is the value of the hypersphere radius, which determines the optimized structure, the step size can be varied in an arbitrary range. www.chem.yorku.ca /profs/renef/TS.html   (1627 words)

 Hypersphere: Facts and details from Encyclopedia Topic   (Site not responding. Last check: 2007-11-06) A hypersphere is a higher-dimensional analogue of a sphere sphere quick summary: The above hypersphere in n-dimensional Euclidean space is an example of an (n−1)-manifold Manifold quick summary: In mathematics, a manifold m is a type of space, characterized in one of two equivalent ways:... www.absoluteastronomy.com /encyclopedia/h/hy/hypersphere.htm   (359 words)

 ePrintsUQ - A Generalisation Of The Delogne-Kasa Method For Fitting Hypersphere In this paper, we examine the problem of fitting a hypersphere to a set of noisy measurements of points on its surface. We perform a statistical analysis of the estimate of the hypersphere's centre. We find that the mean exists when the number of sample points is greater than M+1, where M is the dimension of the hypersphere. eprint.uq.edu.au /archive/00002008   (306 words)