Double helix in large large cardinals and iteration of elementary embeddings

We consider iterations of general elementary embeddings and, using this notion, point out helices of consistency-wise implications between large large cardinals.Up to now, large cardinal properties have been considered as properties which cannot be accessed by any weaker properties and it has been known that, with respect to this relation, they form a proper hierarchy. The helices we point out significantly change this situation: the same sequence of large cardinal properties occurs repeatedly, changing only the parameters.As results of our investigation of this helical structure, we have (i) new characterizations of extendible cardinals and of Vopěnkan cardinals in terms of elementary embeddings from the universe V, (ii) a new large large cardinal property which looks like Shelahness properly between Vopěnkaness and almost hugeness, and (iii) a more direct relation between Vopěnkaness and Woodinness than already known.We also consider limits of the helices and relations with the axioms I1–I3.