why do you say they are 'higher'?
consider...
one dimension ....length...a line.
sideways to that (at a right angle) is width.
Sideways to both of them is height...
they're all at right angles to each other.
none of them are 'higher' or 'lower'.

Now imagine a dimension at right angles to all of them.
then more dimensions at right angles to them too.....ad infinitum.