I lol'd at this maths joke.

[Cheomesh]

Anything that has really good grip, rubber on concrete in dry weather, steel on steel if it seizes up. Do a bit of trig and you will find that the slip angle for a coefficient of friction of 1 is 45 degrees, so anything that can stay put on a 45 degree angle has a coefficient of friction equal to or greater than 1.

You a physics major...?

 

[Qhartb]

I think the "fourth dimension" that the Klein Bottle exists in is either an imaginary dimension (as time is generally considered to be the fourth dimension), or an actual fourth dimension that I have absolutely no understanding of.

I'm not sure what you mean by an "imaginary dimension." Imagine a 3-d unit sphere centered at the origin. Done? The 3 dimensions that sphere existed in were all imaginary, none of them being the actual height, width, or depth we know live in. In that sense, yes, the 4th dimension is also imaginary.

It really gets confusing trying to visualize 4d geometry using time, the most convenient 4th dimension in the physical world. It works in some simple cases, but once you try rotating along axes that aren't parallel to the time dimension, it's more of a hassle than it's worth. Even simple things like taking distance become troublesome. For instance, which is the longer distance, Paris now to London next week, or New York now to the Sun in a minute? (It depends on an arbitrary conversion factor between time and space.)

Higher physical dimensions, such as those theorized in string theory, are irrelevant to higher dimensional geometry. In such cases, it's the math that suggests higher dimensions rather than higher dimensions suggesting the math.

It's probably easiest just to take a set of points at face value. An example 3-d sphere is the set of points p such that f(p)=0 where f(p)=p1^2 + p2^2 + p3^2 - 1. Similarly, a 4-d sphere is the set of point p such that f(p)=0 where f(p)=p1^2 + p2^2 + p3^2 + p4^2 - 1. And similarly in any number of dimensions. The notion of higher dimensional spaces isn't really that abstract; fairly simple economic models routinely use a new dimension to represent each product in a market.

If you think shapes in higher dimensional Euclidean space are hard to think about, just wait until someone gives you an unusual distance metric (or just assures you that such a metric exists).

 

[Dan]

You a physics major...?

engineering

 

[Cheomesh]

I'm not sure what you mean by an "imaginary dimension." Imagine a 3-d unit sphere centered at the origin. Done? The 3 dimensions that sphere existed in were all imaginary, none of them being the actual height, width, or depth we know live in. In that sense, yes, the 4th dimension is also imaginary.

It really gets confusing trying to visualize 4d geometry using time, the most convenient 4th dimension in the physical world. It works in some simple cases, but once you try rotating along axes that aren't parallel to the time dimension, it's more of a hassle than it's worth. Even simple things like taking distance become troublesome. For instance, which is the longer distance, Paris now to London next week, or New York now to the Sun in a minute? (It depends on an arbitrary conversion factor between time and space.)

Higher physical dimensions, such as those theorized in string theory, are irrelevant to higher dimensional geometry. In such cases, it's the math that suggests higher dimensions rather than higher dimensions suggesting the math.

It's probably easiest just to take a set of points at face value. An example 3-d sphere is the set of points p such that f(p)=0 where f(p)=p1^2 + p2^2 + p3^2 - 1. Similarly, a 4-d sphere is the set of point p such that f(p)=0 where f(p)=p1^2 + p2^2 + p3^2 + p4^2 - 1. And similarly in any number of dimensions. The notion of higher dimensional spaces isn't really that abstract; fairly simple economic models routinely use a new dimension to represent each product in a market.

If you think shapes in higher dimensional Euclidean space are hard to think about, just wait until someone gives you an unusual distance metric (or just assures you that such a metric exists).

So what do you do for a living?

 

[Qhartb]

So what do you do for a living?

I is programmer. Me tooked more maths in college than most programmers, though. IMO, math is actually a lot more fun when it's about proving things rather than number crunching.

 

[Solaris]

It most certainly is my friend.

I for one find non-elucidian geometry rather fun.

 

[Cheomesh]

I is programmer. Me tooked more maths in college than most programmers, though. IMO, math is actually a lot more fun when it's about proving things rather than number crunching.

Heh, I just dropped my CS major for a Computer Programming major at my college. I *think* this is a good choice.

 

[Wanted Bob]

Klein Bottle confuses me.

Funny gif though.