  We introduce the flower calculus, a deep inference proof system for
intuitionistic first-order logic inspired by Peirce's existential graphs. It
works as a rewriting system over inductive objects called ''flowers'', that
enjoy both a graphical interpretation as topological diagrams, and a textual
presentation as nested sequents akin to coherent formulas. Importantly, the
calculus dispenses completely with the traditional notion of symbolic
connective, operating solely on nested flowers containing atomic predicates. We
prove both the soundness of the full calculus and the completeness of an
analytic fragment with respect to Kripke semantics. This provides to our
knowledge the first analyticity result for a proof system based on existential
graphs, adapting semantic cut-elimination techniques to a deep inference
setting. Furthermore, the kernel of rules targetted by completeness is fully
invertible, a desirable property for both automated and interactive proof
search.
