you should read carefully what I wrote I did not say that get all algebraic numbers with the HuzitaHatori axioms for origami See for example Theorem  in One Two and MultiFold Origami Axioms by Alperin and LangIn particular this is not original research  I asked you to discuss before making changes I think you should do it next time Note that it is not me who started this war  For the rest I gave the answers already  I will wait couple of days and undo you changes I am  sure think this type of construction worth to be mentioned here