Plot3D[Log[-((x + I*y)! - (x - I*y)!)^-2], {x, -5, 15}, {y, -5, 15}, PlotPoints -> 69] produces http://gosper.org/groovy.png , whose ridges indicate where z! is real. It's three families of "hyperbolas" plus the big ridge for the real axis, which is also the axis of symmetry. What are the asymptotes of the "hyperbola" just above the real axis in the first quadrant, heading off northwest in the second quadrant? Spoiler (Please ogle the png first): StringReverse[".)x-(gol ekil ylhguor gniworg trap yranigami eht ,etotpmysa tnerappa on htiw ,latnoziroh drawot kcab stcelfni *yllaudarg* cra tsewhtron ehT .noitseuq kcirt a saw siht tuB .x gol/1 ekil egrevnoc ot smees trap yranigami ehT ).foorp a sah nailuJ( .cra drawtsae eht fo etotpmysa eht si sixa laer eht taht raelc ytterp s'tI"] Experimentally, the ridges bunch gradually with increasing imaginary part, perhaps reminiscent of the bunching of the nontrivial roots of Ϛ. --rwg