A 2D circle is defined as a shape where every point on it''s perimeter is equidistant from the center of the circle in 2 dimensions. A 3D sphere is the same thing, only in all 3 dimensions. So a 4D sphere (hypersphere, I guess?) would be...?

Would be a 4D sphere : x^2 + y^2 + z^2 + t^2 = R^2

Fruny, could you show me a graphical presentation of that?

A sphere which exists in both space and time, which is very convenient if you have a look into relativity, because as you are most likely aware space and time are actually one thing, which Einstein referred to as space-time.

R2 in the equation above would only give the magnitude of one ray (vector).

Here is a theory on 4D for you.
If you are trying to think of an 3d object and how time is relevant to it, think of the object being like earth. Earth, I believe would be an example of a 4D sphere. The reason why I have chosen a planet is because change is very notable over time, and if you took earth at the same place and rotation in space (lets just say) in 1500AD and 2001AD, the face of the earth would have undergone massive changes, which affected the object. This massive change however is not a necessary feature, (I am only using it as a way of introducing the concept of time in relation to the sphere), and hence if I reference it at different points in time the sphere will have different properties, possibility even to a point where it is only an instance of its former state.

Well, that is my mixed up definition of a 4D sphere it is probably incorrect, but if it is good stimulus for your own thoughts, well that is enough.

I'm not sure exactly how hyperspheres work, but I'm pretty sure to get a consistent one in real spacetime you have to multiply the t-values by i. So...

x² + y² + z² - t² = r²

Can someone correct me on that? I'm sure it must be more complicated. It's something to do with real time being all lumpy and imaginary time being homogeneous, or something

Edited by - Dracoliche on November 18, 2001 5:47:16 AM

well here x,y,z,t is just a notation convention. It has nothing to do with the metric tensor ( ds^2 = dx^2 + dy^2 + dz^2 - c^2 dt^2 )

As for a graphical representation of the 4D sphere, check google ...

Or you can see a unit quaternion as a ''4D'' sphere : the vector (i,j,k) part as the radius vector, and the scalar as rotation around that vector ... et voilà, a 4D sphere.

Anon said:
Fruny, could you show me a graphical presentation of that?

I think there''s a theorem that says its impossible to visualise anything in 4D with a 2D representation. Even in the 3D world we only see things in 2 dimensions, but our brain converts 3D to 2D (having two eyes helps a bit as well).
If you make one of the dimensions a time dimension you could show something changing in time, but there you are still making 3 into 2 for the brain to interpret.

Representations of higher dimensional objects are general referenced as "shadows" cast on a lower dimension. So, just as an object in 3d space casts a 2d shadow, an object in 4d space casts a 3d shadow. Without having done the work, but just as a conjecture, I think it would be interesting to take a 4d object, cast it''s shadow on to 3d and then cast the shadow of that shadow onto 2d. Of course, since I''m posting as anonymous, I''m kind of a shadow here myself so what I think only counts in 2d, eh? :-)

Hello guys!

Very interesting discussion about math theory you have here . Well, anyone asked for a graphical presentation of a 4D object. Well, it''s possible (looking somewhat weird, but it exists!).
For better understanding let''s analyze the coordinate systems we are used to (cartesian coordinates!):

2D (R²)... every single dot in this set of numbers can be accessed by 2 axes (x and y value).

3D (R³)... every single dot in this set of numbers can be accessed by 3 axes (x, y and z value).

4D ... so what you guess, eh? exactly, we need 4 axes. btw, you can even draw a 17D-polygon (for example).
I can''t figure out what this should help, but in theory it''s possible.

Funny to think it all over, eh?

Indeterminatus

--si tacuisses, philosophus mansisses--

Hello guys!

Very interesting discussion about math theory you have here . Well, anyone asked for a graphical presentation of a 4D object. Well, it''s possible (looking somewhat weird, but it exists!).
For better understanding let''s analyze the coordinate systems we are used to (cartesian coordinates!):

2D (R²)... every single dot in this set of numbers can be accessed by 2 axes (x and y value).

3D (R³)... every single dot in this set of numbers can be accessed by 3 axes (x, y and z value).

4D ... so what you guess, eh? exactly, we need 4 axes. btw, you can even draw a 17D-polygon (for example).
I can''t figure out what this should help, but in theory it''s possible.

Funny to think it all over, eh?

Indeterminatus

--si tacuisses, philosophus mansisses--