  The purpose of this note is to report, in narrative rather than rigorous
style, about the nice geometry of $6$-division points on the Fermat cubic $F$
and various conics naturally attached to them. Most facts presented here were
derived by symbolic algebra programs and the idea of the note is to propose a
research direction for searching for conceptual proofs of facts stated here and
their generalisations. Extensions in several directions seem possible (taking
curves of higher degree and contact to $F$, studying higher degree curves
passing through higher order division points on $F$, studying curves passing
through intersection points of already constructed curves, taking the duals
etc.) and we hope some younger colleagues might find pleasure in following
proposed paths as well as finding their own.
