Thoughts on hyper-operations of rational but non-integer orders? - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: Thoughts on hyper-operations of rational but non-integer orders? (/showthread.php?tid=1234) Thoughts on hyper-operations of rational but non-integer orders? - VSO - 09/01/2019 I can't seem to find the right angle to approach this concept intuitively. Does anyone have any ideas of how to consider hyper-operations in a way that isn't recursive, such as to accept non-integers? RE: Thoughts on hyper-operations of rational but non-integer orders? - Gottfried - 09/02/2019 I am not sure I get your problem correctly.            Take the function $f: b^z$ as to be iterated, with, say $b=sqrt(2)$ .              Assume one plane on a math-paper and look for easiness only the lines and their crossings of the coordinate-system of integer complex numbers $z_0$ .               Now take another paper, position it 10 cm above and for every point of the crossings (and ideally also of the lines) mark the values of $z_1=b^{z_0}$. Then repeat it with a third plane, again 10 cm above, marking $z_2=b ^{b^{z_0}}$ . After that, try to connect the related points of the zero'th, the first and the second plane by a weak string, say a spaghetti or so. Surely except of the fixpoints in $z_0$ it shall be difficult to make a meaningful and smooth curve - and in principle it seems arbitrary, except at the fixpoints, where we simple stitch a straight stick through the iterates of the $z_0$ at the fixpoint.            Of course the spaghetti on the second level is then no more arbitrary but must be - point for point - be computed by one iteration. But the spaghatti in the first level follow that vertically orientated curve, where a fictive/imaginative plane of paper is at fractional heights and the fractional iterates would be the marks on the coordinate-papers at the "fractional (iteration) height".                                                      I'd liked to construct some physical example, showing alternative paths upwards between the fixed basic planes, with matrial curves made by an 3-D-printer, but I've not yet started to initialize the required data. But I think, that mind-model alone makes it possibly already sufficiently intuitive for you. A somewhat better illustration is in my answer at MSE, see https://math.stackexchange.com/a/451755/1714 RE: Thoughts on hyper-operations of rational but non-integer orders? - tommy1729 - 09/09/2019 I think the OP refers to concepts like , what i called " semi- super " operators. Like the semisuper operator of the semisuper operator of f(x) is the super of f(x). This is extremely difficult. Do not confuse with the functional half-iterate of the superfunction. Regards  Tommy1729