Hi, I know next to nothing about abstract algebra (had one intro-class years ago), so sorry if this is a dumb question. I was browsing through "A book on abstract algebra" by Pinter and had a thought. In the chapter called "ruler and compass" (chapter 30) he talks about how abstract algebra can prove the impossibility of certain constructions using ruler and compass: doubling the cube, trisecting any angle, and squaring the circle. I was wondering if abstract algebra can also suggest new instruments which include those prohibited constructions as possibilities? Like can it describe new tools which are able to perform those constructions which haven't been invented yet? Or is it really saying that there can never be ANY tool, or combinations of ANY tools, which can perform those feats? Thanks for any insight on this. edit: Or maybe a better way to ask this would be to ask if it can suggest new "operations" which mimic the behavior of real-life tools.