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Topic: Hypersphere


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  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)

  
 --> Antigravity.org - "Big Spin" Model of Gravity</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> The “Big Spin” model suggests that gravity is a result of rotation of the Universe’s <b>hypersphere</b> (not a Gödel's rotation). </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> I should stress that rotation of a <b>hypersphere</b> is quite different from the rotations we see in everyday life. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Where r is a vector from the center of the hyperspehere to some point of the <b>hypersphere</b> and k is a constant.</td></tr> <tr><td></td><td colspan=2><font color=gray>www.antigravity.org /BigSpinModelOfGravity.html</font>   (1272 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://mathforum.org/library/drmath/view/51942.html">Math Forum - Ask Dr. Math</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Date: 06/04/99 at 05:24:40 From: Doctor Mitteldorf Subject: Re: Volume of a <b>hypersphere</b> Dear Krishna, You can get the formula for a 3-d <a href="/topics/Sphere" title="Sphere" class=fl>sphere</a> by integrating the surface Area of a spherical shell from 0 out to R: Integral(4pi r^2 dr) = 4/3 pi r^3. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Now we have the hyper-area of the <b>hypersphere</b> of <a href="/topics/Radius" title="Radius" class=fl>radius</a> r, and we can integrate from r = 0 to r = R, to get 1/2 pi^2 R^4 as the volume of the 4-dimensional <b>hypersphere</b>. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> The volume of a 4-d <b>hypersphere</b> is less than 1/3 of the volume of the circumscribed <a href="/topics/Hypercube" title="Hypercube" class=fl>hypercube</a>.</td></tr> <tr><td></td><td colspan=2><font color=gray>mathforum.org /library/drmath/view/51942.html</font>   (283 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://users.adelphia.net/~44mrf/App3.html">Circle, Sphere, Hypersphere, Fourth Spatial Dimension</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> In order to understand why the two <a href="/topics/Sphere" title="Sphere" class=fl>sphere</a> configuration is a reasonable approximation of the four-space <b>hypersphere</b>, it helps to know how the <b>hypersphere</b> is generated from a 3D spherical surface. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> As an example, if the <b>hypersphere's</b> <a href="/topics/Radius" title="Radius" class=fl>radius </a>= r, then a <a href="/topics/Circle" title="Circle" class=fl>circle</a> all the way around it is 2πr, and the (necessarily curved) length from the north pole to any position on the equator is πr/2. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> In the <b>hypersphere</b> we extrapolated, half of it occurs when y assumes values that are exclusively positive or 0 and the other half of it occurs when the y values are exclusively negative or 0.</td></tr> <tr><td></td><td colspan=2><font color=gray>users.adelphia.net /~44mrf/App3.html</font>   (834 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><u>[No title]</u>   <i>(Site not responding. Last check: 2007-11-06)</i></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> The <b>hypersphere</b> with <a href="/topics/Radius" title="Radius" class=fl>radius</a> r starts from point 0 (real element of the initial data set), moves to the center of mass of all captured points, point 1 (FORmal ELement). </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> From this point one more data element is found within distance r, the mass center is re-calculated and shifts to point 2. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> All points captured during a single voyage of the <b>hypersphere</b> trough the expression space are collected in a single provisional cluster.</td></tr> <tr><td></td><td colspan=2><font color=gray>obesitygene.pbrc.edu /pti/forel.htm</font>   (344 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://www.cise.ufl.edu/~snsk/Research/NIPS05/html/node8.html">Solution to the Intersection of Spheres problem</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Once again, based on the geometry of the problem, it is trivial to compute the new radii and centers of the corresponding <b>hyperspheres</b>. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> The end result of the recursion is a low dimensional linear subspace in which lies the Null Space represented as a single <b>hypersphere</b> (parameterized as a <a href="/topics/Radius" title="Radius" class=fl>radius</a> and a center). </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> If all the constraints are met, we simply pick a point on the final <b>hypersphere</b> that improves the generalization capacity of the classifier.</td></tr> <tr><td></td><td colspan=2><font color=gray>www.cise.ufl.edu /~snsk/Research/NIPS05/html/node8.html</font>   (285 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://www.mazepath.com/uncleal/comprom.htm">THE MATHEMATICAL IMPOSSIBILITY OF COMPROMISE</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> N variables then create an N-dimensional unit <b>hypersphere</b> with each line being one of its diameters (the pure variable without interaction with others) and every line's center being the <b>hypersphere's</b> center. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Exact maths for volumes and surface areas of <b>hyperspheres</b> are presented at the end. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> As N (the number of variables and <a href="/topics/Dimension" title="Dimension" class=fl>dimensions</a>) increases, <b>hypersphere</b> volume and surface area maximize, then almost symmetrically reflect and rapidly approach zero.</td></tr> <tr><td></td><td colspan=2><font color=gray>www.mazepath.com /uncleal/comprom.htm</font>   (739 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><u>Untitled Document</u>   <i>(Site not responding. Last check: 2007-11-06)</i></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> The familiar breakdown of a <a href="/topics/Sphere" title="Sphere" class=fl>sphere</a> in 3-space into two hemispheres can also be applied to the <b>hypersphere</b> in 4-space: while in 3-space we use two disks joined along their boundaries, the analog in 4-space is to join two solid <a href="/topics/Sphere" title="Sphere" class=fl>spheres</a> by gluing them along their bounding <a href="/topics/Sphere" title="Sphere" class=fl>spheres</a>. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> It turns out that the <b>hypersphere</b> can be broken down into two linked solid tori glued together along their surfaces. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> To return to the <b>hypersphere</b>, we "smooth out" the polyhedral version to obtain the desired decomposition of the <a href="/topics/3_sphere" title="3_sphere" class=fl>3-sphere</a> in 4-space.</td></tr> <tr><td></td><td colspan=2><font color=gray>www.socsci.uci.edu /imbs/CONFERENCES/4-D/ABSTRACTS.htm</font>   (1979 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://www.hypersphere.com/hs/abouths.html">About the Hypersphere</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> As a model of the universe, the <b>HYPERSPHERE</b> shows how things emerge in time and are enfolded back into fabric of the universe. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> The <b>HYPERSPHERE</b> also shows how all things in the universe are interconnected, even when they appear to be separate from one another. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Since the <b>HYPERSPHERE</b> is a model of the universe, we can say that seven-ness is a basic feature of the natural world.</td></tr> <tr><td></td><td colspan=2><font color=gray>www.hypersphere.com /hs/abouths.html</font>   (539 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://www.hypersphere.com.au/services.htm">HyperSphere:: Services</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> These affordable service packages are designed to give you the time, knowledge and resources to stop putting marketing in the ‘too hard basket’ – at a fraction of the cost of hiring an experienced marketing person on staff. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>HyperSphere</b> design <a href="/topics/Will" title="Will" class=fl>will</a> ensure your market always receives a consistent visual message from your business. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>HyperSphere</b> helps ensure that your website design, content, choice of technology and online strategy are aligned with your marketing objectives.</td></tr> <tr><td></td><td colspan=2><font color=gray>www.hypersphere.com.au /services.htm</font>   (499 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://matcmadison.edu/alehnen/sphere/hypers.htm">The Sphere Game in n Dimensions</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> In particular, as the <a href="/topics/Dimension" title="Dimension" class=fl>dimension</a> gets large the probability distribution approaches a normal distribution with a mean of the equatorial value and a standard deviation which goes to zero like the reciprocal of the square root of the <a href="/topics/Dimension" title="Dimension" class=fl>dimension</a>. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> In this analysis the length L is generated by randomly selecting an element of “area” on the surface of the <b>hypersphere</b>. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> When the <b>hypersphere</b> stops moving, the <a href="/topics/Euclidean-space" title="Euclidean space" class=fl>Euclidean</a> distance between the marked point and the point of tangency of the <b>hypersphere</b> to the hyperplane of the floor of the box is measured.</td></tr> <tr><td></td><td colspan=2><font color=gray>matcmadison.edu /alehnen/sphere/hypers.htm</font>   (2045 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://www.kdc.com.au/strategy.htm">HyperSphere:: Strategy Development</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> A <b>HyperSphere</b> marketing communications strategy <a href="/topics/Will" title="Will" class=fl>will</a> provide you with a practical, actionable plan full of ideas and information to help guide your business from where it is now to where you want it to be. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> This is applicable where the client supplies ample, up-to-date market research data and a business/marketing plan to be used as a basis for devising the marketing communications plan. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>HyperSphere’s</b> role in this case is to analyse and synthesise the information, tie it together, providing external perspective and professional insights, and formulate a marketing communications strategy and action plan designed to achieve the agreed objectives.</td></tr> <tr><td></td><td colspan=2><font color=gray>www.kdc.com.au /strategy.htm</font>   (675 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><u>Abstract/Artistic</u>   <i>(Site not responding. Last check: 2007-11-06)</i></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> This <b>hypersphere</b> is constructed of 20 Jatoba disks surrounding a central Birch <a href="/topics/Sphere" title="Sphere" class=fl>sphere</a>. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> This is a <a href="/topics/Sphere" title="Sphere" class=fl>sphere</a> constructed of 12 disks of Pau Ferro projected outward from a central 12-sided polyhedron (dodecahedron) made of Olivewood. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> The <a href="/topics/Sphere" title="Sphere" class=fl>sphere</a> is constructed of 12 disks projected outward from a 12 sided polyhedron (dodecahedron).</td></tr> <tr><td></td><td colspan=2><font color=gray>www.waynehall.ca /Gallery-artistic.htm</font>   (248 words)</td></tr> </table> </td> </tr> </table><script language="JavaScript"> <!-- // This function displays the ad results. // It must be defined above the script that calls show_ads.js // to guarantee that it is defined when show_ads.js makes the call-back. function google_ad_request_done(google_ads) { // Proceed only if we have ads to display! 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