h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Hi Henry,

I liked your entry regarding tesseracts...but can you tell me what happens in the case of a sphere?

A friend of mine asked me a playful question:

"what is the true shape of a snooker ball?", and I thought that your entry about tesseracts and your knowledge of maths might be helpful.

Cheers!

I only really provided the ascii picture of the tesseract, though I do know something about them. The tesseract is the 4 dimensional analogue of a cube, the hypersphere is the 4 dimensional analogue of a sphere. I dont know of a better way to describe it than as all the points at some distance from some fixed centre point, in 4 dimensional space. Thats the same as the definitions for a circle or a sphere, except with '4' instead of '2' or '3'.

Or you can try imagining what a hypersphere passing through our space would look like: a point, expanding out to a sphere of increasing size, then shrinking again back to a point and disappearing. Same sortof idea as how a 3-d sphere looks like passing through a 2-d plane, except everything 'up a dimension'.

The true shape of a snooker ball? You mean as in a solid sphere? I dont think its anything more fancy than that, just points such that their distance from the centre is less than the radius.

Thanks Henry,

I'm obliged for your swift answer. I'll let you know if the question about the snooker ball has any interesting answer!

Dear HenryS,

Further to your reply I am right in saying that the dim's of time and space are different things?

For instance the fourth dimension is often refered to as time. Does this mean that the fourth spatial dimension is the same as time?

If time and the fourth spatial dimension are the same, perhaps they appear different only because we are looking at different aspects of them. Eg. The elevational view on a cone is very different to the plan view of the same cone.

If this interests you and you have a moment to answer I would appreciate your thoughts.

Space/time dimensions: from a mathematical point of view, when we talk about '3 dimensional space', '4 dimensional space', 'n dimensional space' whatever, there is no link to where we are actually living intended. A cube, or tesseract, or whatever mathematical object, is a thing in some mathematical space, which may resemble things in our real physical space, but theres no need for it to. What I mean is that mathematicians deal with (say) 4 dimensional space as an imaginary construct, with no link to reality, so its meaningless to ask 'where' the 4th dimension is, or if it is space or time. It's all invented and exists only in our minds (though aspects of what we imagine may relate to things we observe in the outside world, they are certainly not the same things).

Mathematicians routinely work with dimensions higher than 3, even spaces which have effectively infinitely many dimensions or no way to define how many dimensions at all. We have no need (and its sometimes impossible) to think of them as part of our real space, so don't even try