Then when you multiply by the [1,1] vector you end up with a vector of successive terms of the Fibonnaci sequence. If you do more and more multiplications before finally multiplying by [1,1] you will get closer and closer to the golden ratio.

If you instead of [1,1] multiply the resulting matrix by something else such as [2,15]:

Note that the resulting vector's x and y value are still in the golden ratio.

Check out "Exact real computer arithmetic with continued fractions" by JE Vuillemin for a reason for this. Roughly, you could be said to be calculating convergents of the continued fraction expansion for phi [1; 1 1 1 ...]. The paper goes into some enormous detail about different ways of calculating but in this case you can think of generalized continued fraction as a possibly endless string of 2x2 matrices waiting to be multiplied. The two column vectors in your left-most matrix hold the convergents.

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