How to visualise four and higher dimensions. As we discuss in mathematics any n-dimensional space?

Answer 1

Yes, we can do math in any n-dimensional space. Remember that the plane can be seen as the collection of all pairs (x,y) and 3-dimensions as all triples (x,y,z). Given that, it is easy enough to play with any number of variables algebraically. For example:

circle in 2D: x^2+y^2=r^2
sphere in 3D: x^2+y^2+z^2=r^2
hyper-sphere in 4D: x^2+y^2+z^2+w^2=r^2.
etc.

line in 2D: y=ax+b
plane in 3D:z=ax+by+c
hyper-plane in 4D: w=ax+by+cz+d.
etc.

of course, after a bit it becomes easier to write subscripts instead of x,y,z, etc, so we use x_1, x_2, x_3, ...x_n for the variables in n-dimensions.

It is very common for physicists studying a system of particles to use 6n dimensions where n is the number of particles: 3 for the spatial position and 3 more for the velocity for each particle. This leads to a very large number of dimensions very quickly.

Visualization is a bit more difficult and takes some practice. A decent place to start are the books 'Flatland' and 'Sphereland'.

Answer 2

personally I find 5 and up impossible

but 4 dimensions you can try

it's pretty simple and it may have happened to you already. Take a wireframe model of a cube, I mean a drawing. The norm is to "decide" that the bottom left part is at the front, and the upper right part is at the back.

But it could just be the opposite, right?

So what you can do is this. Look at the image in a classical way (i.e. lower left is supposed to be at the front). When you get tired of doing this, after a minute, force yourself to look at it the other way, i.e. bottom left is back, and upper right is at the front.

After a while it may switch "back and fourth" almost of itself.

Well, in 4D space, you could do this with a real 3D wireframe cube, if you "rotated" it, not around an axis but around a plane, a plane going through the middle of the 3D cube.

A 3D being would think it saw a wireframe model of a cube kind of swimming through itself before taking an inverted appearance. But what would have really happened would have been a rotation in 4D space.

Hope this example helps

Answer 3

It's not really easy to visualize fourth dimensions and higher, but here is something to try:

Something in the fourth dimension can be "visualized" by using 2 sets of planes to represent the four variables x, y, z, and say t

You could choose xy plane and zt plane, or xz and yt plane, etc
to get an idea of what various "cross sections" or projections look like. Such a technique could be generalized to n dimensions.

Another way of visualizing a 4th dimension is to simply observe the world. It is full of 3 dimensional objects changing in the fourth dimension of time.

Answer 4

Thanks for the question.

Most answers are extremely thought provoking.
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I have aricle created on this subject by me for you.

Here are my thoughts.
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The Fourth Dimension

By
Minootoo

In projective geometry, the experts deal with "Point figure and Line figure"(Projective Geometry by O’Hara and Ward is the name of the book). This book will be in reference section at Bombay University. It will be also at Ruparel college in Matungga(near western railway station). Parsi book society in Dadar Parsi coloney should have one (I had donated it) .

If you study only point figure, you will find that there is no limit, you can have "n" point figure, "n" being an integer, and it can be very large (but mathematical few).

Hint: Cube or a rectangle is 8 point figure in space. Any corner can be an origin, and the tree lines,represent 3 dimensions.

In classical Math you have Three Dimensions. In space geometry you can locate any point in space using this method.

In Engineering the three dimension together with +ve and -ve direction represent 6 degree of freedom.

Lot of time people loosely calls it six dimensional freedoms or six dimensions, only.

This means you can have “Any No. of dimension in projective geometry” or work in classic geometry with three dimensions.

PS

There may be an advantage in creating the 4th dimension to simplify the calculations, because by carefully picking and choosing a particular additional dimension (when applicable), calculations can all be done using linear formulas. It can be great trick to simplify the work load.

By the way complex problems can only be solved using principles of projective geometry without getting confused.

Problems involving multiple relative (relativity theory) motions are lot easy to solve using projective geometry.

The idea of having time as 4th dimension suggested by one of the answerer also appeals to me, it can reduce the calculation time in most cases significantly.

Answer 5

a flat picture is a 2-d representation of 3-d space, so maybe thinking about how each inch can be flattened and then add a third dimension...

Answer 6

I doesn't mean that whatever we have in math should have a real model in the world.
things that weare talking about in math are imaginary.
so maybe the 4D doesn't.