sphere

[ydoaPs]

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?

[DQW]

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.

[ydoaPs]

so, is it IN 1 dimension but IS 0-dimensional? that still leaves the question of how you know which points

[DQW]

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.

[ydoaPs]

what is c?

[DQW]

c: center; r: radius

[ydoaPs]

i'm stupid, i was thinking in two dimensions....i got it now...i think

[YT2095]

wouldn`t a 2D sphere look like a Sine wave? (Taking Time into Account). and a Circle if you ignore Time.

[ydoaPs]

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.

[ydoaPs]

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?

[DQW]

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].

[DQW]

PS : Notice that the equation of the 0-sphere in cartesian co-ordinates can be written :

 

[math] (x-c)^2=r^2 [/math]

[YT2095]

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.

[ydoaPs]

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

[DQW]

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.

[ydoaPs]

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.

[DQW]

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 ?

[ydoaPs]

it is the x-y plane as i said in my last post.

[ydoaPs]

[DQW]

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.

[ydoaPs]

it didn't work

[ydoaPs]

each equation only makes half a sphere. the spheres meet at the x-y plane. maybe that has something to do with it

[ydoaPs]

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?

[DQW]

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.

[DQW]

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 ?