So this was kind of a revelation for me. Maybe someone else will find it interesting or perhaps even useful. I'm actually kind of excited by it.
The basic open chords you learn as a beginner (C, A, G, E, and D major) seem pretty different. They look different and you just have to learn them by rote memorization. Later, when you learn about chord construction and triads, and what notes are where on the fretboard, you start to see how those chords are constructed, but at least to me they still seemed like 5 very different patterns.
I recently realized however that they are all kind of the same pattern. You can derive each by taking any one and shifting the individual open and fretted positions down a string. So as an example, start with a basic open E chord. You play the open E string, the 2nd fret of the A and D strings, the 1st fret of the G string, and the open B and high E strings. That's the basic E shape. So now shift everything down a string. The open E becomes open A. Instead of fingering the 2nd fret on the A string, finger the 2nd fret on the D string. Instead of fingering the 2nd fret on the D string, finger the 2nd fret on the G string. One wrinkle, a really important one that this whole thing hinges on, is that when you move from the 3rd string to the 2nd string, you have to move the fret position one fret closer to the bridge in order to compensate for the unique tuning of the B string (vis-a-vis the G string). So make the 1st fret on the G string (from the E chord) the 2nd fret on the B string. Finally, the open note on the B string becomes an open note on the high E string. Oh, and because you have 2 E strings, whatever is on the high E string (string 1) gets copied to the low (6th) E string. So in short, you shift everything down a string and compensate for the unique tuning of the B string. When you do that, you see that the E shape chord you stared with has been transformed into an A shape chord. Repeat that procedure with the A shape chordand it gets transformed into a D shape chord, and doing it yet again gives you a G shape chord (albeit the one where you play the 3rd fret of both the B and high E strings, which to my ear is the prettiest). Doing it one final time gives you the C shape chord.
Something else I learned in doing this is that for the basic chords you learn, there's basically only 2 different note sequences (assuming you consider all 6 strings and cycle back from the high to low E strings) - E, A, and D chords use the sequence root, 5th, root, 3rd, 5th, root. C and G chords can use that sequence also, but the forms most people learn first are root, 3rd, 5th, root, 3rd, root.
You can derive the different minor chord shapes in the same way. One surprise I had when I did that is that a minor chord emerged that I was unfamiliar with - Cm [331013]. The full chord would be very difficult (maybe impossible) to play but the top or bottom 4 strings would probably be more manageable (I haven't tried yet) and are still a complete Cm chord. I'm guessing it's not taught early on though because the Cm chord is only diatonic to the Bb, Ab, and Eb major scales which are not very common guitar keys.
