It's not clear that Carmack's purpose in using the Fibonacci series was to take successive ratios to approximate "the golden ratio" (if it was the golden ratio

It's not clear that Carmack's purpose in using the Fibonacci series was to take successive ratios to approximate "the golden ratio" (if it was the golden ratio he cared about, he could just as well have directly calculated (1 + sqrt(5))/2). He quite possibly could have been interested in the Fibonacci sequence for some other reason, or no reason at all.