11 dimensions-myth or reality?

4 dimensional space in "Interstellar" (2014) movie

 I have seen recently a lot of posts on social media platforms mentioning the 11 dimensions. In most cases, these posts were misinforming and were bringing this topic in a mystical zone. But what really are the higher dimensions?

According to the theory of relativity, we live in a four dimensional world (3 dimensions of space and 1 of time).

There are plenty of theories which contain the higher dimensions. The most famous theories of this kind are the String Theory (requiers 10 dimensions- 9 of space and 1 of time) and the M-Theory ( a version of String Theory, requiers 11 dimensions- 10 of space and 1 of time).

What do dimensions mean?
The simplest way to think about dimensions is through coordinates. A point on a line (1D) needs just one coordinate (lenght), a point on a surface (2D) needs two (lenght and width), and a point in space (3D) needs three (lenght,width and height). A higher dimensional space will add new coordinates.

Take an example: We have a horizontal 1-dimensional line. We construct another line perpendicular (at 90 degrees) on the one given. Now we have a square, which is bidimensional. We repeat this step, adding this time the height. Now, we have a cube (tridimensional). We observe that each face of the cube is a square. Therefore, when we build a tesseract (4 dimensional) each face needs to be a square. Don't forget to built the extra dimension. Impossible to think about such a shape, isn't it? Although, we can imagine a 3D analogue of it.

A square (2 dimensional), cube (3 dimensional) and a 3D model of a tesseract (4 dimensional)

  
Where are the higher dimensions?
In general, the extra dimensions are theorized to be hidden, or in scientific terms, compactified, at extremely small scales, making them imperceptible to us.

Even though higher dimensions are a very useful tool in some math and physics problems, their existence can't be proven or disproven with the experimental tools we have now. What do you think? Are the higher dimensions just a mathematical artifice or they represent our reality?