# Explain how a hypersphere is possible

1. Apr 25, 2006

### Quantum1332

Can someone explain how a hypersphere is possible. Because obviously it would not add up to 360 degrees. Could it just be a 3d sphere rotating in a 4d (or more dimensions) hyperspace. Can someone shed some insight.

2. Apr 25, 2006

### hossi

what should add up to 360 degrees? This seems to me a weird definition of a hypersphere. A sphere is defined to be all the points that have the same distance to some central point. You can do that in an arbitrary number of dimensions. Best,

B.

3. Apr 25, 2006

### Quantum1332

i was talking about the degrees. Hyper means higher dimensions, a hypersphere is a sphere that exists in higher dimensions, as in a 4d sphere.

4. Apr 25, 2006

### dav2008

What exactly is your question then?

Any point on a circle is defined by a radius at an angle from the x-axis. Since the radius is constant by definition for a circle, you only need one angle to define any point on a circle. This is where you are getting 360 degrees from. (or 2pi radians)

In a sphere, once again the radius is constant but now you need two angles to define a point on the sphere: one angle measured from the x-axis and another from the z-axis. You can't really say there are a certain amount of degrees or radians in a sphere; you can say there are 4pi steradians.

It follows that a hypersphere for 4 dimensions would be defined by all points a distance R from its center, and any point could be defined by three angles. Are you asking if there is something like a radian or steradian in 4-d?

Last edited: Apr 25, 2006
5. Apr 26, 2006

### hossi

Thanks for pointing that out! Talking about degrees... Maybe you should try to figure out what you actually mean by hypersphere. It has some angles and lives in higher dimensions won't do. As far as I know, a sphere in d dimensions is the d-1 dimensional subset which contains all points that have the same distance to some central point. What you can do then, is to compute e.g. the surface or the volume (for fixed radius), both of which will depend on d, have some powers of \pi in it, and (if I recall that correctly) some \Gamma functions. The surface is not in general 2 \pi.

Best,

B.

6. Apr 26, 2006

### Quantum1332

What I don't understand is how you can have a sphere in a 4d hyperspace. So what i want to know is, if it would basically be a sphere that is suspeded withing a 4d hyperspace.

7. Apr 26, 2006

Staff Emeritus

The object is called a 3-sphere. Just as a spherical surface in 3-space is two dimensional (latitude and longitude), so a spherical volume in 4-space is three dimensional. It's the solution of $$(x-a)^2 + (y-b)^2 + (z-c)^2 + (w-d)^2 = R^2$$. You can't really visualize it; each little piece of it looks like three dimensional space.

8. Apr 26, 2006

### josh1

This thread should be moved to the Tensor Analysis & Differential Geometry forum

9. Apr 26, 2006

### Quantum1332

Other dimensions are very hard to imagined. I have another question, why can't we see these other dimensions.

10. Apr 26, 2006

### marcus

Quantum1332 I will offer to make a deal with you.
Start your threads in a more appropriate forum and I will give you some hints about how to imagine a sphere in one higher dimension than usual.

Your question about imagining higher dim spheres could just as well have been asked before 1915. It is a classical type question. That is not bad! We need all that classical understanding----you should be asking. but it does not belong here.

So if you open it in, say, "Special and General Relativity"
forum it will not be so off - topic and you SHOULD get a fuller response.

11. Apr 27, 2006

### Quantum1332

I am by no means a physicist, but i know a good bit about this topic. Could the reason as to why we can't see this other dimension be because light passes beneath it. You are right maybe it should be moved, but you dont have to be so angry about it.

12. Apr 27, 2006

### marcus

not angry at all Q1332. guess my tone of voice didnt come thru clearly.

I am just suggesting that if you want a good response you should probably abandon this thread and start a thread in the appropriate forum where people like to talk about hyperspheres and stuff.

you MIGHT get a reasonable response here but more likely in, say Relativity forum, or in one of the maths.

13. Apr 27, 2006

### Gokul43201

Staff Emeritus
Is your problem specifically with a hypersphere, or with any object embedded in a 4d space ?

When you are using math, you need to be very specific about your description. What does not add to 360 degrees ? And the answer is not "I'm talking about the degrees". The degrees add to 360 degrees ? That doesn't help, does it ? You need to tell us what "it" is.

I think that will work.

If you have a 2-sphere in 4d space$(x_1,x_2,x_3,x_4)$, the surface of which is defined by $x_1^2 + x_2^2 + x_3^2 = a^2$ and rotate this about either $x_1,~x_2~or~x_3$, I imagine you'll get a 3-sphere with the same center and radius.