How To Imagine the Tenth Dimension
video.google.com
video.google.com
You can't "visualize" the fourth dimension (or I haven't met anyone who could... and the kind of math/physics where this comes up was a common topic for me for a while).
However, you can understand it pretty easily. Its really just another coordinate. You can do algebra with it. If I say someone is at (3, 2, 5, 6) and moves (1, 2, 3, 4) they are now at (4, 4, 8, 10)...
Want a 5th dimension? Add another comma/number.
People who "visualize" the higher dimensions actually visualize 2 or 3 dimensions and know how to generalize (from practice/experience/theorems).
Sure, on paper you can have as many dimensions as you like, just add coordinates, but that won't help us come up with the grand unified theory, or find other new laws of our universe, which might have pretty useful practical applications.
Unfortunately we are limited in seeing beyond our primary senses.
Actually, I wonder, when people usually ask this question, are they just using the word dimension as a buzz word or do they know what it means?
BTW the video is ok until 4:30. Then it quickly degrades into nonsense, not just more complicated than needed.
This is the problem. The 4th dimension is not time. We can represent time as the 4th dimension in a system if we want to, but you can have systems where the 4th dimension is not time.
A good example is when people create a matrix representing links in webpages. If you limit your universe to online 100 webpages and their interconnections, you can get a vector for each webpage. The vector is a point in 100 dimension space. The 4th dimension is not "time" but "if the 4th website on the list links to me."
Also, visualizing an n dimensional vector is not very interesting. A vector indicates a point, and a point looks the same in any number of dimensions.
"there is really little point in trying to graphically represent anything beyond spacetime. Possibly 3D space with a time vector is useful (i.e. an object representing 3D space moving along a vector) because it is still helpful to explain the concepts. Beyond that just stick with the numbers or you risk confusing yourself on what is being talked about"
"Time is the fourth dimension" is the single most damaging sentence for your understanding of time and space.
To have an n-dimensional spacetime where time is not the nth dimension makes no sense.
The video didn't expand on the person living in a 2D space example ... if such of thing would be possible, he would be able to explain the presence of a third dimension using time (going so far as to consider time as being the third dimension).
NO! Absolutely not! 2D spacetime does not look like 3D space at all!
In 3D space, I can take a 2D angle bracket, flip it in the third dimension, and get its mirror image. This will seem like magic to 2D creatures. In 2D spacetime, I can do no such thing. No matter how long I wait (i.e. move in the third time dimension), the angle bracket won't transform into its mirror image.
In 4D space, I can take a 3D right-hand glove, flip it in the fourth dimension, and get a left-hand glove. This will seem like magic to 3D creatures. In 3D spacetime I can do no such thing. No matter how long I wait, a right-hand glove will never transform into a left-hand glove.
Do you understand?
Do you see how adding time to n-dimensional space does absolutely nothing to explain n+1 space?
But it's early in the morning and I haven't had breakfast. Talk to you when my brain joins me.
To perceive time as a dimension is useful because it helps you realize you actually perceive four dimensions the same way a flatlander perceives three.
If you want to do space-time calculations, yes.
because it helps you realize you actually perceive four dimensions the same way a flatlander perceives three
Absolutely not. We perceive time as time and space as space. The concept of spacetime does not help us understand or visualize higher dimensional spaces at all. It only confuses the issue, which is exactly the original complaint against the video this whole discussion is about.
Edit: One more example: Can you visualize the difference between a 3D cube moving in time, and a 4D cube moving in the 4th dimension? If you project the 4D cube down to 3D, it will look like a double cube that twists itself inside and out, but the 3D cube moving in time will just look like a 3D cube. Spacetime does not help you understanding 4D space.
Obligatory link: http://news.ycombinator.com/item?id=1385543