I think I would have found the Futurama episode “Mobius Dick” funnier if I could actually grasp the concept.
Wikipedia is way too convoluted for me to derive a proper comprehension on this. I know in layman’s terms, the fourth dimension is “time”, but I don’t really “get” how we can visualize this.
I especially don’t get how the Klein bottle really looks like in the fourth dimension. I know it’s being conceptualized in 3D, but I don’t really understand it.
I’m curious about this in another sense – is it still realistic to consider time a 4th dimension considering that it’s been found to be inextricably linked to space, as in “space-time”? Is it more correct to say there are 3 dimensions of “space-time”, than 3 of space and one of time?
Going into the other 10 or so theoretical dimensions is going to be way over my head…
Did you see the recent thread What is a “Dimension”?
It’s better to say that time is a fourth dimension, not the fourth dimension.
Get a piece of paper. Draw a circle. Draw two dots, one inside the circle, one outside the circle.
Now, how can you get from one dot to the other, without crossing the circle? If you’re restricted to two dimensions (i.e. you have to stay on the paper), it’s impossible, but if you can move into the third dimension (above the paper) it’s trivially easy. You could get a piece of wire or something, put one end of the wire at each dot, and have it curve up above the circle.
Now, if you looked at this from above (and didn’t have any depth perception), it would look like the wire was intersecting the circle, but it doesn’t really, because it moves up into the third dimension to go “around” it.
Likewise, when you look at a 3-D model of a Klein bottle, it looks like the surface intersects itself, but it really doesn’t, because it moves into a fourth dimension (a fourth spatial dimension—nothing to do with time) to go “around” it.
Because of this, you could say there are 4 (not 3) dimensions of “space-time.”
Here’s the way it was explained to me. I don’t know if it’s right or not, but it makes sense.
Think of a cube. That cube has three dimensions: length, width, and height. If we took away it’s height, the cube would cease to exist and it would just be a plane. If we took away it’s width, the plane would cease to exist and it would just be a line.
But what if we took away it’s time? Time ends and what are we left with? Nothing. The line only exists because it has a variable called time. Time could be 30 seconds, in which case the line or the plane or the cube would exist for 30 seconds and then be gone. Time could be 100 years, in which case the object would be gone after 100 years.
Thus, to exist, an object requires time, and one, two, or three spatial dimensions.
Here’s an explanation that doesn’t make my brain explode.
Come over and visit me. I live on Lake Street (east-west) at the corner of Elm Ave (north-south), on the 5th floor (up-down). Those are three spacial dimensions. Here’s the 4th dimension: I’ll only be there at 6:00. At other times I will be in other locations.
Time is still qualitatively different from the spatial dimensions, though, so physicists will often refer to “3+1 dimensional space” or the like.
To illustrate: Let’s say you have a stick, and you have coordinates for the point at one end of the stick, and coordinates for the point at the other end of the stick. You could ask a question like “how tall is the stick?”, meaning “what is the difference between the z coordinates of the ends”. Likewise, you could ask how wide the stick is, or how deep it is. But all of these numbers can change, if you rotate the stick around a different way. If you really want to say something about the stick itself, that doesn’t depend on the frame of reference you’re measuring it in, you could take something like delta X^2 + delta Y^2 + delta Z^2, which would give you the same value no matter how you rotated the stick.
Well, in relativity, if you measure the distance between two events, or the time between them, you get different answers if you rotate your reference frame, too, except here, the “rotations” are reference frames moving at different velocities. But like with the stick, there’s still a calculation you can make that will always give the same answer, no matter what reference frame you use. Except now, that calculation is delta X^2 + delta Y^2 + delta Z^2 - delta T^2. The time part gets a minus sign, unlike the space parts which all get a plus sign.
The problem I have with this argument is that time shouldn’t have any effect on the cube’s cubicity. It’s a hexahedron with equal quadrilateral spaces, 12 edges, and eight vertices, regardless of its duration.
True. But imagine a particle moving around inside the cube, in constant motion. What are the coordinates of the particle?
Well, actually the cube appearing and disappearing in 30 seconds vs 100 years would be more like a box in 4-D space time, with the length of the box equal to the conversion between space and time. The reason it’s a box is that it’s other dimensions don’t change as time changes, in the same way that moving up and down a cube the width and depth don’t change. This 4-D box has 8 cubic sides (The cube at time 0; the cube at time 1; and each of the cubes six original sides extended from time 0-1), 16 corners (the original 8 corners at time =0 and time=1). It also has 24 2-dimensional faces, and 32 1-dimenesional edges which again are combinations of the original faces and edges at time =0 or 1, and the original edges and points extended from T=0 to T=1.
As an alternative you can consider a pyramid. It has a square base, that decreases in dimension as you go towards the time. I could consider a 4-D space-time pyramid, as the following. A cube which suddenly appears and then slowly shrinks to nothingness. In the same way a 3-d pyramid had a height cross section that was a square decreasing in size, my 4-D pyramid has a time cross section which is a cube that is slowly decreasing in size.
Beating the dead horse of “how it was explained to me”:
Draw a line. With this line, you can measure length, which is one dimension.
Cross that line with another line at a right angle. You can now measure length and width, which are two dimensions.
Cross those crossed lines with another line at a right angle again. You can now measure length, width, and height, which are three dimensions.
How long did it take you to draw those lines? That’s time, which is a fourth dimension.
Scientifically, it’s probably claptrap, but it helped it all make sense to me as a kid.
Our brains are wired to deal with visualizing 3 spatial dimensions and one time dimension. To geometrically visualize more than that we need to either focus on just 3 or less spatial ones at a time, work by way of 3 or less spatial dimension analogies, or deal with maths that handle n-dimensional geometries without actually picturing it.
Cross those crossed lines with another line at a right angle again. You can now measure four dimensions.
Cross those crossed lines with another line at a right angle again. You can now measure five dimensions.
Cross those… what do you mean you can’t? Oh, that’s right, because we live in three-dimensional space. But the concept is the same; nothing magically switches in the definition after three. If you want to think of 17-dimensional space, it’s just like 3-dimensional space, only with more dimensions.
The slogan “Time is the fourth dimension” has perhaps confused many people at a more fundamental level than it has helped.
When we say something is N-dimensional, we mean that, in some sense or another (the exact relevant sense can depend on the context in which we say such a thing), it has the same structure as lists of N many numbers. For example, it is often useful to refer to an event by giving four numbers specifying where and then when it occurs (three for where and one for when); in this sense, there are 3 + 1 dimensions involved in specifying where and when things occur, in our particular world of 3d space and 1d time.
But you can use lists of numbers for lots of things other than specifying where and when things occur. When you calculate the final grades for a class of 60 students, you get a 60-dimensional result, which has nothing to do with time (or physical space, for that matter).
Incidentally, relativists don’t usually, in practice, refer to time being the fourth dimension. We refer to it as the zeroth dimension. We could refer to it as the fourth dimension, and it’d work about as well, but this is the convention that caught on.
If you’re not really interested in the detailed mathematics, and really just want to have an intuition on what 4D concepts are “like,” I recommend looking instead at 2D. If you think about what 3D would seem like to a 2D person (or whatnot), you can somewhat extrapolate what 4D would be like to us lowly 3D folks.
The book Flatland is essentially about this. And a dose of political satire. It’s also highly entertaining and short and, I believe, public domain.
Also, actually 4D wouldn’t really help with the Futurama ep in question – they were mostly riffing on scifi cliches of trippy “alternate dimensions.”
Flatland is the only thing I’ve ever read that made me not embarrassed as a wannabe science geek for not understanding 4D space. I still don’t completely get it, but that book certainly helped. And yes, it is public domain.
Here’s something to think about:
You (a three dimensional object ) grow and move about a worldline in four dimensions.
A 2D tv image changes over the course of a broadcast.
A 1 D line on an oscilloscope sweeps out a curve when analyzing an event.
Even a zero dimensional quantum singularity has a exists throughout a past, present and future, instead of only existing for an instant.
Time is the measure of change.
Time is NOT the fourth dimension.
Time is the FIRST dimension!
(does that blow your mind?):rolleyes:
Time and 3-D space are combined into 4-D spacetime in relativity, I think that’s the best way of stating it.
I did see that, thank you. But it didn’t really capture what I wanted to know specifically about “the” fourth dimension. I did watch the video linked though… still a little confused.
Thanks for this. It doesn’t allow me to visualize a fourth dimension, but does explain why my 3D brain can’t see it.
And Chronos, thanks for trying, but that a little too technical and had a little too much math in it. Kudos to you for actually understanding this stuff, though!
An object in a true 4th spacial dimension, if on par with the 3 we know and love so well, would hold a geometry that extends beyond, into a dimension we have no access to. However, just as a 3d object casts a 2D shadow, a 4D object would cast a 3D “shadow”.
Even more, if pushed through the volume of 3D space, we would be witness to a confounding, morphing cross-section of the 4D object.
Google hypercubes, and tesseracts for countless animated mathematical models.
I prefer to keep time, or duration as a separate conceit between spacial and temporal dimensions.
It would only cast a “shadow” if photons traveled freely in all of the spatial dimensions and came from an angle outside of the usual three spatial ones.
As far as the 4D object moving through the three extended dimension, well the other spatial ones are considered to be likely very small, so that would apply only to very very small objects … yes, it would predict that at that very small level objects would appear to change form as they moved, maybe changing “flavors” or something like that …