Dimensions beyond 4

Factual Questions
21

Generally speaking (within science at least), your definition of a dimension fits. However, as I understand it, string theory and such require 10 time-space dimensions, which temperature does not qualify as.

As for the definition of a point with x,y,z,t,T… well, a point doesn’t have a T. T is average kinetic energy. That point has KE, but it doesn’t have T.

That aside, though, you can’t define two separate objects with the same x,y,z,t and different T’s–for example, if one ball is at 1,1,1,1,1, the second ball CAN’T also be at 1,1,1,1, and simply have a different T (3 balls in a box each have a different x,y,z at given time t; to count as a fifth time-space dimension, they’d need to be able to have the same x,y,z,t, but a different coordinate w that uniquely defines them).

Note I also realize that x,y,z aren’t hard-set as the 3d space dimensions (a simple example being spherical coordinates, which can also be used to define the same point)… perhaps if I sit and ponder how x,y,z and r, theta, tau (or whatever it is you use for spherical coordinates, it’s been a long time) can be the same thing for long enough I can grok a fifth coordinate in a “spherical” system :slight_smile:

22

It’s extremely useful at proving that the number of points on a line is equal to the number of points in a plane. (Whereas that number is much larger than, say, the number of integers.)

It doesn’t prove that there’s really only one dimension, but I was just answering the letter of the challenge: prove that a 3-dimensional point can be encoded in a single coordinate. Such a one-to-one correspondence does exist, and the proof of it was a historical moment in set theory.

23

But that isn’t what I said. I said you need three coordinates to specify a point in space. You cannot point to or visit the location encoded in your single number directly; you need to first extract the three coordinates from it using some sort of algorithm.

24

Umm. . . By using the algorithm posted by Ben, one coordinate does specify a point in a 3-D space. And in order to go from any coordinate system to a point in a space, you need some sort of algorithm. The origin is an arbitrary point; the axes are arbitrary lines through the origin; the scaling of the axes, etc.

What makes the space 1-D, 3-D or whatever-D is the topology. If two points are “close” in the space, are the coordinates for the point “close”? The number of dimensions of the space is the minimum number of coordinates needed to specify a point subject to “close” points having “close” coordinates.

25

andy_fl wrote :

Thanks andy, that is also what I recall as a description for dimensions. If we can assume it is accurate, I may have a neat way to answer jharmon’s original question.
Imagine taking a piece of relatively taut rope or string and making waveforms with it.
You can do this by moving it really fast and bringing it back to it’s original position. the “hump” or waveform you create will propagate down the string to the other end. (where it will be absorbed or reflected)
Here is a crude ascii drawing of such a thing with three “snapshots” at different times.

,.·´¯`·., t = 1 sec

,.·´¯`·., t = 2 sec

,.·´¯`·., t = 3 sec

okay simple so far.
now do it again except make the waveform only 1/2 the height, and ALSO have someone on the other end create a 1/2 sized waveform at the same time.

Below are three “snapshots” of what that would look like.

,.·-·.,.·-·., t = 1 sec

,.·´¯`·., t = 2 sec

,.·-·.,.·-·., t = 3 sec
So what?
Well - Take a specific look at t=2 seconds for both diagrams.
They are the same. If you froze time at t=2, just looking at the pictures of the waveform you could not tell them apart. So a simple description of where the waveform is in space Using the 3 dimensions (x,y,z) is insufficient to describe the waveform fully.

You need to describe the momentum stored at the time of the snapshot. Since momentum is a vector quantity, it also needs a direction, and that direction is in 3 dimensional space, so to accurately describe momentum, you need 3 bits of data, or 3 momentum dimensions.
Does this help?
Or is my decade old education on this terribly out of date?

BTW, the other 3 dimensions [IIRC] are 3 of angular momentum, which means not only do you need 3 dimensions to describe how fast and in what direction an object is moving, but you need 3 to fully describe how it could also be spinning.

giving you a total of :
3 spatial dimensions,
3 linear momentum dimensions,
3 angular momentum dimensions, and
1 time dimension.

for a total of 10.

26

I don’t know if that IS right, but it SOUNDS right. I was starting to wonder if perhaps you could look at things such as spin as a dimension (ie, help figure out this answer using the 4 quantum numbers).

27

A recent prior thread.

28

This doesn’t jive with current mathematics. If one adds two of these quozrog coordinates using traditional mathematics, they will rarely get the correct sum.

for example
x1 = .11
y1 = .08
z1 = .05

x2 = .11
y2 = .04
z2 = .06

add them and you get:

x3 = .22
y3 = .12
z3 = .11
if you change that to the q dimension you get (q3 = .211221)
however if you just look at the “q” dimension numbers and use normal mathematics you get:
q1 = .100185
q2 = .100146

q3 = .200331

So for this three-in-one dimension theory you’d have to throw away any and all traditional mathematics from addition on up. Does Cantor talk about this at all in his proof?

I also have a big logic problem with this statement. If said line lies in said plane, then they can not have same number of points. For all the points in the line are in the plane, but there are plenty of other points in the plane not in the line. Sure, there are an infinite number in each, BUT all infinities are not equal. Mathematics even has a way (L’Hopital’s rule) to compare infinities.

29

Sqwert,

Calculus does not deal with actual infinities. The infinites that appear in calculus are potential infinities. Cantor’s transfinite numbers are actual infinities. A cardinal number can be finite or infinite. Two sets have the same cardinality if you can find a one-to-one correspondence between them. It turns out that a set is infinite if and only if there is a proper subset of it with the same cardinality.

You are correct that not all infinities are equal. Cantor also showed that you cannot find a one-to-one correspondence between the integers and the real numbers, and that the cardinality of a set is always less that the cardinality of its power set (the set of all the subsets of the given set).

But the cardinality of the line is the cardinality of the plane, as Ben’s post showed. The fact that a plane has the same cardinality as a line imbedded in it tells us that there must be an infinite number of points in the plane and in the line.

The point (Ha!) of showing the correspondence between the line and a plane is only to show that the two sets have the same size. It is not intended to demonstrate a useful way to refer to the points in a plane using one coordinate. Which is a good thing because, as you pointed out, if fails to do so.

30

Also, Sqwert, the “q” numbers add and subtract just fine, you just have to use different rules for them. Instead of carrying the excess from the kth place to the k+1th place, you carry it to the k+nth place in the number for an n-dimensional space.

Which brings another point to mind. It seems that the “q” numbers don’t quite store all the information that the others do, because in order to convert back to n different coordinates, or do any meaningful manipulations, one has to know what n is. This information is contained in a vector of coordinates, but is not contained in the “q” numbers.

31

It would be more accurate to say reality has no dimensions - variables representing position in space only exist when human interpretation specifies them. Unless that is you posit an organizing intelligence which specifies the positions of entities independantly of human interpretation.

32

No, because the number of dimensions specfies how each point relates to each other point.

33

The answer is pretty obvious if you just review the basics and go from there. I set up a website just for this at: http://www.beyondfourdimensions.com/. You’re only 20 minutes from enlightenment, me bucko!

34

[moderating]
Welcome to the Straight Dope, JanH. We ask that on our board you participate in discussions rather than simply pointing people off to your website. I will leave the link, as it is relevant, but please don’t spam your site further.

We encourage our guests and new members to visit the “About This Message Board” section of the site to get a feeling for our culture.

Thank you.
[/moderating]

35

You’re being Humpty Dumpty. You use words to mean whatever you want, without any rigor or precision. You admit you can’t even define “exist” but you go on to make it an assumption. You mix dimensions of space and time to mean something different than what physicists mean by space-time. You talk about infinities but don’t understand them and use them to prove your statements by sheer hand-waving. You don’t understand that dimensions, as stated earlier in this thread, are composed of the number of elements needed to uniquely describe a point in them and so turn them from a mathematical concept into a metaphor. Then you go on to use “subjectively” and “objectively” as necessary components without any attempt to define them. Of course you can get to the conclusion you desired. You started with it as an assumption and then filled in as many words as you felt like doing. But you could have just gone there directly with as much meaning. Nothing in between was rigorous, logical, or accurate. You could prove that the fifth dimension is Hogwarts with equal ease.

Sorry. We get a lot of people here who link to their websites and wait for us to be impressed. That always fails.

36

Doubly so here. I read the web site and my only response was: It’s no Time Cube.

37

[NOT moderating]
My first reaction was, who creates a website with a couple of paragraphs per page and no navigation system except for linear reading? That whole website is equivalent to one short blog post.

My second reaction was exactly the same as Exapno’s.

38

Don’t try to understand 5 dimensions, or 10 dimensions, or 23 dimensions, or whatever. Try to understand n dimensions, for an arbitrary value of n, and then let n be whatever you like. It really is a lot easier that way.

39

I endorse this post.

40

Help me understand this. Would this mean something like…where you are (your X, Y, and Z coordinates), when you are there, what color you are while you’re there, what mood you are in, how old you are at the time, and so on?