ydoaPs Posted July 17, 2005 Share Posted July 17, 2005 i have no idea what anything above a 2-sphere would look like. a 2-sphere is like the surface of a ball. a 1-sphere is a circle. a 0-sphere is two points. why is a 0-sphere two points? i would have thought that a 0-sphere would be nothing since it is the set of all points a certain distance from the center in zero dimensions. how do you know which two points the sphere is? Link to comment Share on other sites More sharing options...
DQW Posted July 17, 2005 Share Posted July 17, 2005 i would have thought that a 0-sphere would be nothing since it is the set of all points a certain distance from the center in zero dimensions[/u']. That's your error (the underlined). Look at the other n-spheres again and you'll see what's wrong. Link to comment Share on other sites More sharing options...
ydoaPs Posted July 17, 2005 Author Share Posted July 17, 2005 so, is it IN 1 dimension but IS 0-dimensional? that still leaves the question of how you know which points Link to comment Share on other sites More sharing options...
DQW Posted July 17, 2005 Share Posted July 17, 2005 how do you know which two points the sphere is?The 0-sphere, I believe is the locus described below : {x : |x-c|=r; x,r,c in |R } This describes the pair of points c-r and c+r, where c,r are in |R. PS : forgive the sloppy formatting - I'm too sleepy now. Link to comment Share on other sites More sharing options...
ydoaPs Posted July 17, 2005 Author Share Posted July 17, 2005 what is c? Link to comment Share on other sites More sharing options...
DQW Posted July 17, 2005 Share Posted July 17, 2005 c: center; r: radius Link to comment Share on other sites More sharing options...
ydoaPs Posted July 17, 2005 Author Share Posted July 17, 2005 i'm stupid, i was thinking in two dimensions....i got it now...i think Link to comment Share on other sites More sharing options...
YT2095 Posted July 17, 2005 Share Posted July 17, 2005 wouldn`t a 2D sphere look like a Sine wave? (Taking Time into Account). and a Circle if you ignore Time. Link to comment Share on other sites More sharing options...
ydoaPs Posted July 17, 2005 Author Share Posted July 17, 2005 both sine waves and circles are one dimensional. they only have length. a two-sphere is like the surface of a ball, but it has no thickness. Link to comment Share on other sites More sharing options...
ydoaPs Posted July 17, 2005 Author Share Posted July 17, 2005 so, if c, r, and x are real numbers, then the equation of a 0-sphere is |x-c|=r. the equation of a 1-sphere is [imath](x-h)^2+(y-k)^2=r^2[/imath] where (h, k) is the center and r is the radius. what is the equation for a 2-sphere? is there a general equation for a n-sphere? Link to comment Share on other sites More sharing options...
DQW Posted July 17, 2005 Share Posted July 17, 2005 so, if c, r, and x are real numbers, then the equation of a 0-sphere is |x-c|=r. the equation of a 1-sphere is where (h, k) is the center and r is the radius. what is the equation for a 2-sphere? is there a general equation for a n-sphere?The equation in cartesian co-ordinates is the natural extension : [math]\sum_{i=1}^{n+1} (x_i-c_i)^2=r^2[/math] In general, I believe the locus given by |x-c|=r holds for all dimensions, where now, x and c are elements of [imath] \mathbb {R}^{n+1} [/imath] and r is in [imath] \mathbb {R}[/imath]. Link to comment Share on other sites More sharing options...
DQW Posted July 17, 2005 Share Posted July 17, 2005 PS : Notice that the equation of the 0-sphere in cartesian co-ordinates can be written : [math] (x-c)^2=r^2 [/math] Link to comment Share on other sites More sharing options...
YT2095 Posted July 17, 2005 Share Posted July 17, 2005 both sine waves and circles are one dimensional. they only have length. a two-sphere is like the surface of a ball, but it has no thickness. incorrect, it has Amplitude also! Peak and trough from the base-line (zero). in fact a 3D sine wave would be like cutting a ball in half across its diameter and rotating ONE of the sections 180 degrees around, making one of them a "Dome" and the other a "Cup", and the point of contact between each would be almost Zero. Link to comment Share on other sites More sharing options...
ydoaPs Posted July 17, 2005 Author Share Posted July 17, 2005 i was speaking of the curve itself. the aplitude is not a part of the curve it is a property of the curve that is used to determine its shape. edit: your "3-d sine wave" is only two dimensional. it has no thickness Link to comment Share on other sites More sharing options...
DQW Posted July 17, 2005 Share Posted July 17, 2005 YT : pogo is right. Sine waves and circles are only 1d spaces. You can specify any location on a specific sine wave or circle using just one number. To describe a sine wave or a circle, however, you must have it live in a 2 (or more) dimensional space. Link to comment Share on other sites More sharing options...
ydoaPs Posted July 17, 2005 Author Share Posted July 17, 2005 i tried to graph a 2-sphere, but it didn't work. equations had to be in terms of z, so i had to use: [imath]z=\sqrt{-x^2-y^2+9}[/imath] and [imath]z=-\sqrt{-x^2-y^2+9}[/imath]. i thought that would give me a 2-sphere at the origin with a radius of 3, but it didn't. i got: . what did i do wrong? the split is along the x y plane. if it matters, this is my first try at using graphcalc. Link to comment Share on other sites More sharing options...
DQW Posted July 17, 2005 Share Posted July 17, 2005 Hmm...can't see too clearly. Do you have a bigger version of that ? It looks like you have most of the 2-sphere except for a "ring" near the x-y plane. Or is that the y-z plane ? Link to comment Share on other sites More sharing options...
ydoaPs Posted July 17, 2005 Author Share Posted July 17, 2005 it is the x-y plane as i said in my last post. Link to comment Share on other sites More sharing options...
ydoaPs Posted July 17, 2005 Author Share Posted July 17, 2005 Link to comment Share on other sites More sharing options...
DQW Posted July 17, 2005 Share Posted July 17, 2005 I have no idea how graphcalc works or how it calculates square roots. I'm not sure if there's a "division by zero" hazard it tries to avoid. Have you tried moving the center away from the origin (or the x-y plane) ? Can't make a strong case for why that should work, but it's not too much more effort, so may be worth a shot, until someone comes up with a real fix. Link to comment Share on other sites More sharing options...
ydoaPs Posted July 17, 2005 Author Share Posted July 17, 2005 it didn't work Link to comment Share on other sites More sharing options...
ydoaPs Posted July 17, 2005 Author Share Posted July 17, 2005 each equation only makes half a sphere. the spheres meet at the x-y plane. maybe that has something to do with it Link to comment Share on other sites More sharing options...
ydoaPs Posted July 17, 2005 Author Share Posted July 17, 2005 if the equation of an n-sphere is [math]\sum_{i=1}^{n+1} (x_i-c_i)^2=r^2[/math], what is the volume of said n-sphere? Link to comment Share on other sites More sharing options...
DQW Posted July 17, 2005 Share Posted July 17, 2005 It looks like it chooses not to plot when z gets very close to 0. I can't see why this happens, but then, I know precious little about numerical algorithms. In fact, the only one I'm familiar with is Newton-Raphson, and I don't see that having any problems, but I have absolutely no idea what real graphing softwares use now. Link to comment Share on other sites More sharing options...
DQW Posted July 17, 2005 Share Posted July 17, 2005 if the equation of an n-sphere is [math]\sum_{i=1}^{n+1} (x_i-c_i)^2=r^2[/math]' date=' what is the volume of said n-sphere?[/quote']Being a surface, it has no volume (or content). Do you mean the volume enclosed ? Link to comment Share on other sites More sharing options...
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