Properties of 6 Dimensional Spheres

[starkind]

basically, yes. the ball inside a sphere has one more dimension than the sphere itself.

Remember that the surface of a three dimensional ball is two dimensional, and in general a surface has one less dimension than the volume under it.

if this is still confusing, think of the ocean on the ball of the earth. If you sail on the ocean, you only need two coordinates to determine your position. The Earth is a three dimensional ball with a two dimensional surface.

If you need to tell someone where you are above or below the ocean, you need to add another coordinate for altitude or depth. So as always, the ball is one dimension less than the surface of the ball.

As I said before, I am not sure what the rules are for embedding.

 

[benk99nenm312]

The answer for the inside of your sphere must still be just 3-space. Unless you are trying to argue that having put a spherical boundary around a region of the regular world, you have somehow manufactured new degrees of freedom within. And even if you take this approach, they would surely not be extra flat-space dimensions.

A futher complication in your story is that you want to add time as a dimension. You can't just do this in a simple way, treating time as one more spatial direction.

Of course, I myself would argue that geometry with spatiotemporal scale is a very interesting subject to discuss. But that is a complicated topic and first we need to sort out our intuitions about purely spatial spaces.

Why would it still be 3-space? It is not just a boundary. I am establishing the sphere to have an inside, making it a ball, which should raise the value of the dimension by 1, correct?

Just as a side note, I find it interesting that if I ask one question to two people, I get two different answers. Starkind says that it would raise the value of the dimension inside, you say that the inside would remain the same as the space beyond the surface. Which answer is correct?

 

[starkind]

I noticed that too, but it could just be that we have not established basic definitions. As I keep saying, words are slippery, and of limited use in logic. however, you can find my supporting text if you go to wikipedia and search for n-sphere. I think the link was above in another post, but I am being called away. Excuse me for not providing it in this post.

 

[apeiron]

This is getting a little nuts, so not sure that it is worth continuing.

However both responses say the inside of your ball is 3-space.

 

[benk99nenm312]

This is getting a little nuts, so not sure that it is worth continuing.

However both responses say the inside of your ball is 3-space.

Are you saying 3-space as meaning 3 dimensions total (adding the spatial dimensions that it is embedded in), or are you talking merely about the sphere/ball and nothing else?

 

[cristo]

Discussion in the BtSM forum should be about professionally researched theories. This thread is way too nuts to remain, thus I am locking it.