What is unique feature of 3 dimensional space?

Question

For example in 1D space (and only in 1D) phase
transitions do not exist

Answer 1

That's not entirely true. You could have a collection of points on a 1D line that were tightly bound together (a solid), then suddenly the points start repelling each other and fly apart (a gas).

One unique feature of 3D space is that gravitational, electrical, and magnetic field strength goes as the inverse square of the distance. If space were 2D, the field strength would go as the inverse of the distance (as is clearly evident in problems involving infinitely long cylinders of electric charge).

Answer 2

In 3-space, the order of spatial transformations makes a difference, whereas in lower dimensions, it does not. Consider a 90-degree rotation in a plane or a 2-unit jump along a line in the "positive" direction.

If the first rotation in the plane is followed by a 45-degree or other amount of rotation back to the starting orientation, the ending orientation is the same as the order of transformations were reversed -- the 45-degree tilt back followed by the 90-degree rotation.

Same on the line -- if the 2 units forward is followed by 3 units back, you end up at the same location as if you went back 3 units first and then 2 units forward.

Now consider 3-space, with x, y, and z orthogonal axes. Say x and y lay in a plane with y pointing up and z points toward you. Place your fist on the vertex with your thumb pointed up. Now rotate in the x-y place 90 degrees clockwise. Then swing 90 degrees using the y axis for rotation. Your thumb points left. Now do the transformations in reverse order -- swing counterclockwise around the y-axis first (thumb points left), and then rotate clockwise 90 degrees in the x-y placne. Your thumb points up. This means that the order of transformations in 3-space is important, but is not in 2- or 1-space. This is the non-commutativity of operations and it is why the order of multiplication of matrices is important.

Answer 3

3D... when you specify any thing in 3D you always refer to the world coordinates i.e. the free real space around you. While the other dimensions such as 1D and 2D just exist on paper and you don't have any existence of them in the real world. This is what it is unique for.

Answer 4

Let V = volume (e.g., of a liquid)
In 3-D, V = V(x1,x2,x3), and density = mass/V
In 2-D, V = V(x1,x2), and density = mass/V
So, vh = specific volume of liquid, & vg = specific volume of vapor, and we have from the first law of thermo that delta h = T(sg -sh) at constant pressure, where sg and sh are specific entropies of vapor and liquid respectively. Hence, a phase change. I.e., if my memory serves me correctly.

Any set of basis vectors that describe position wrt a given coordinate system has a count of 3.

Answer 5

3 dimensional space has depth ie 3D movies