N-dimensions: part 2

There is an interesting situation with a cube when adding dimensions. First, we start with a rectangle in 2 dimensions that becomes a cube in 3 dimensions. Although we can stop there it is possible to add more to understand if the intuition we had applies now.

Les’s recap some trigonometry first:

We can measure the angle of the diagonal of a square of 1 by 1 using the cosine between one of the sides and the hypotenuse.

When measuring the diagonal of a 1-by-1 rectangle we get:

Cos(α) = 1/H = 1/(√2)

Now let’s add one more dimension now we have a cube:

In this case, we can measure the angle of the diagonal of the cube in the same manner.

The result in this case is Cos(α) = 1/√3

The thing is...

At this point, we can add the number of dimensions we want and still get the length of the diagonal using the cosine with n≥2

Cos(α) = 1/√n

If we take the limit of Cos(α) when n->∞ we get:

Cos(α) = 1/√n -> 0!

That means the diagonal in n dimensions is going to be perpendicular to the axis at some point if we keep increasing the number of dimensions!

What do you think this means? Or how this could affect the way you model a problem in n dimensions?