All known natural laws ORIGINATE from math (from symmetries). See any good physics textbook, or click here to learn how: http://www.emmynoether.com/index.1.htm
Thus our respect of symmetries (= beauty) and our evolutionary inherited genetic drive to serach for beauty, perfection, etc (=symmetry) - what you call "intuitive feeling of beauty".
Let's now go over another incorrect claim you wrote: There is maths that matches nature-patterns, and maths that doesn't. Some maths you get a number of solutions to an equation; and some are absurd and discarded while others fit nature.
Nothing can be away from truth. Math actually gives you ALL correct solutions corresponding to ANY and ALL possible phenomena. But we, humans, often interested only in SOME of them. For example, throw a rock in the air and its motion will be described by the quadratic equation: r(t) = ro + vot + gt2/2 (in vector form). Suppose, we want to find out when a thrown by our hand rock falls on the ground.
As any quadratic equation, it has TWO solutions - one (t1) is usually positive and one (t2) is negative. It is INCORRECT to throw away negative solution as many professors do. Negative time simply means that if the same rock would continue to move BEFORE we thrown it, then it would cross ground level t2 BEFORE it arrived to our hand. So, math is actually MORE smart than we are.
Another example: tunneling via potential barrier in quantum mechanics. Mathematical solution of Shroedinger equation (for the amplidude of wave representing particle penetrating via barrier) consists of TWO terms: Aexp(kx) + Bexp(-kx) and many teachers usually discard first term as "non-physical", explaining that amlitude can not increase to infinity at x = oo (infinitely far under barrier), but can decrease to zero - thus the second term is the "correct one". But they forget that if the particle moves FROM THE RIGHT (not in the direction +x, but in the direction -x, thus penetrating the barrier FROM OUTSIDE), then its amplitude should be describet by the FIRST TERM Aexp(kx), because it increases in x and dexreases in -x direction.
So, agaim math covers MORE cases than we are interested in. Math is MORE general than our limited thinking. And it is ALWAYS correct. We, humans are often incorrect or just do not know math.