Mathematical importance of the golden ratio I know the golden ratio is the limit of the ratios of consecutive Fibonacci numbers and that it appears when studying many related combinatorial objects (such as the sequences of zeros and ones with no consecutive ones and the Fibonacci substitution $1 \mapsto 10$, $0 \mapsto 1$). I have also read  which states that it is, in some sense, the hardest irrational to approximate by rationals. I'm also familiar with its geometrical definition.  Are there any other interesting properties or applications of this number within mathematics, maybe outside of combinatorics? I'm not interested in its applications within physics, aesthetics or any other discipline.  Edit: This is not a duplicate, since I'm explicitly asking for a different answer than the one given for the other question.
Uses of the 'Golden Ratio' I have heard much about the numerous appearances of the ratio found in nature: 1.6180339887.  Are there any actual mathematical uses that have been found of this number? What are its advantages? Just curiosity, really.
Does infinity exist in physical systems? Sometimes people fail at asking a question by being too broad, unclear like . So I'll take a stab at what I believe to be the same question, but more concise and clearly stated:     Does infinity exist in either terms of structure, parameters, or   measurement in any physical systems?   Can infinity have any real connection with reality?  Or is infinity purely a mathematical concept just used by physicists as a convenient way to describe the very large, an approximation?  I have heard that if you model a physical system (), and you run into infinity as a solution, then you have either made an error in your calculations or your model is wrong.