One of the interesting things about both major and minor triads is their geometry, particularly their resemblance to the Pythagorean triangle with sides in the ratio 3-4-5.
Consider that an octave is divided into 12 semitones. For the sake of example, let us start from the note C (this is an arbitrary choice - these ideas are just as true starting from Gb, or A of course).
If we divide the octave using the numbers 3-4-5 (that is, taking intervals of 3, 4 and 5 semitones, or minor 3rd, major 3rd and perfect 4th) we get the notes C-Eb-G and then C one octave up. This is the minor triad with the octave added (simply, a pure minor chord).
If we take 4-3-5, we get C-E-G-C, the major.
The combinations 5-4-3 and 5-3-4 yield the notes C-F-A and C-F-Ab respectively - which are the second inversions of the major and minor chords on the related key of F (the fourth or subdominant).
Continuing the idea of relating geometry to types of chord, we may note that a 4-4-4 triangle (equilateral triangle) corresponds to an augmented chord/triad (C-E-G#). A diminished 6th chord (C-Eb-F#-A), being a series of minor thirds, relates to the shape 3-3-3-3, or a square.
Note that the sides of squares and equilateral triangles cannot be distinguished from each other by length. Analogously, the dim 6 and aug chords don't really belong to any key - you can invert (rotate) them and they sound the same, in terms of intervals (not, of course in absolute pitch).
Music is indeed mathematics, at least in an analytical sense. In a practical sense, it can be something much more, of course. Perhaps that is why it functions so well as a bridge between the logical and super-logical mind.