I was recently working on a math problem, which involved some horrifically messy calculations, but which resulted in a surprisingly clean, beautiful, and useful result, and I was wondering what a good expression would be to describe such a thing. My first thought was "diamond in the rough", but that doesn't really convey that it started out as something really ugly, and in any case, the expression reminds me of a song I don't like. Is there a better expression for this? The best I can think of is "ugly duckling", but that expression focuses mainly on the earlier (ugly) phase, rather than the fact that beauty came seemingly out of nowhere.
The butterfly emerged from its chrysalis See YouTube, Monarch Butterfly Emerging from Chrysalis
chrysalis, from Merriam-Webster:
the hardened outer protective layer of a pupa
and a pupa is, according to Merriam-Webster:
an intermediate usually quiescent stage of a metamorphic insect (such as a bee, moth, or beetle) that occurs between the larva and the imago, is usually enclosed in a cocoon or protective covering, and undergoes internal changes by which larval structures are replaced by those typical of the imago
and an imago is, from Merriam-Webster:
an insect in its final, adult, sexually mature, and typically winged state
In this case, the imago is the butterfly.