What's the evidence for the existence of higher dimensions?
By Engineer Saviour  Blaze Labs
In physics, the inverse square law relation is quite common. This relation is valid for the gravitational attraction between matter, for the electrical forces between charges and for magnetic forces between moving charges. A force that varies with the square of the distance means that the force will increase with the square of the distance if we reduce the distance, and it will decrease with the square of the distance if we increase the distance.
Electromagnetic energy decreases as if it were dispersed over the area of an expanding sphere, 4π R^{2} where radius R is the distance the energy has travelled. The amount of energy received at a point on that 3D sphere diminishes as 1/R^{2}. This clearly shows the origin of the inversesquare law.
Here is a table showing the volume and surface area of hyperspheres of different dimensions:
Dimension (n)

Shape

Volume

Surface Area

2

circle

π r^{2}

2πr

3

sphere

(4/3)π r^{3}

4πr^{2}

4

4sphere

(1/2)π^{2} r^{4}

2π^{2} r^{3}

5

5sphere

(8/15)π^{2} r^{5}

(8/3)π^{2} r^{4}

6

6sphere

(1/6)π^{3} r^{6}

π^{3} r^{5}

7

7sphere

(16/105)π^{3} r^{7}

(16/15)π^{3} r^{6}

As a result, a force that varies with the square of the distance can be considered as a conventional 1dimensional force vector (xaxis) that is scattered into 2 additional dimensions (y, z) due to the 3dimensional nature of space. The square power of the distance indicates the projection of such a force over a 3D spherical surface area. But what happens if the force is also acting in higher order dimensions? What if the force is originating force is being projected on a higher dimensional surface area? Are there forces which vary to other powers than the inverse square law?
The Casimir force related by the above equation is known to vary as the inverse d^{4}, which is two orders of dimensions higher than the more common forces, and coincides with a force projected over the surface area of a 5D hypersphere (see table above). Such force that varies with the fourth power of the distance can be thus considered as a force vector that is scattered in a 5dimensional space. Therefore, it is evident that the field that originates the Casimir force is a 5dimensional field, that it is in fact a hyperspace field that produces the corresponding effects in our restricted 3D vision of our universe.
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