Quote:
Originally Posted by mhss1992
I seriously don't understand how those dimensions beyond the third work. I mean, it's impossible to imagine something like an axis that's perpendicular to the 3 original dimensions, and everything that's said about it is always shown in 3d or 2d. So, what does it really mean?
|
EDIT: Darn, you figured it out before I posted this, but it still might help you with higher dimensional concepts.
It is only hard to imagine how the higher dimensions work because we are not in them.
Let us look at how we form new dimensions: a 0-dimensional point is connected to another 0-dimensional point at right angles to create a line in the 1st dimension. A 1-dimensional line is connected to another 1-dimensional line at right angles to form a square in the 2nd dimension. A 2-dimensional square is connected to another 2-dimensional square at right angles to form a cube in the 3rd dimension. A 3-dimensional cube is connected to another 3-dimensional cube at right angles to form a hypercube in the 4th dimension (and so on).
We formed the 4th dimension in the same way we formed the others, so why do we have such a hard time imagining how a 4-dimensional hypercube works? As I said, it is because we are not in that dimension. A person in the 2nd dimension thinks of the 3rd dimension in the same way a person in the 3rd dimension thinks of the 4th dimension.
The reason the higher dimensions are always represented in the lower dimensions is (again) because we are not in those dimensions. We do not have computer monitors that have 4-dimensional displays, so we have to use projections of 4-dimensional objects instead. We can project any object in any dimension into any other dimension.
In the case of leonid's signature, we have projected a 4-dimensional hypercube into 3-dimensional space, and then taken that 3-dimensional representation (which I called a shadow) and projected it into the 2nd dimension (and then animated it rotating
). The reason I used the word shadow is because objects in a certain dimension always cast shadows in the dimension directly below them (e.g. our 3-dimensional bodies cast 2-dimensional shadows). So 4-dimensional objects cast 3-dimensional shadows, 3-dimensional objects cast 2-dimensional shadows, 2-dimensional objects cast 1-dimensional shadows, etc.
We could have directly gone from the 4th dimension to the 2nd dimension if we wanted to do so as shown at the far right of this picture:
http://upload.wikimedia.org/wikipedi.../Dimoffree.svg
It is important to note that I am only talking about spatial dimensions. In a space-time view of the dimensions, the 4th dimension is commonly referred to as 'time.' Let me explain: the 0th dimension has no direction, the 1st dimension has forwards and backwards; the 2nd dimension has forwards, backwards, left, and right; the 3rd dimension has forwards, backwards, left, right, up, and down; and the 4th dimension has forwards, backwards, left, right, up, down,
forwards in time, and backwards in time.
If you are interested, here is a good video on spatial dimensions
http://www.youtube.com/watch?v=uDaKz...eature=related and here is a good series on space-time dimensions
http://www.youtube.com/user/10thdim
Well, that was fun. I am going to go eat some spaghetti now.