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 Thoughts on hyper-operations of rational but non-integer orders? Gottfried Ultimate Fellow Posts: 764 Threads: 118 Joined: Aug 2007 09/02/2019, 03:40 PM (This post was last modified: 09/10/2019, 02:33 AM by Gottfried.) 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 Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Messages In This Thread Thoughts on hyper-operations of rational but non-integer orders? - by VSO - 09/01/2019, 04:34 AM RE: Thoughts on hyper-operations of rational but non-integer orders? - by Gottfried - 09/02/2019, 03:40 PM RE: Thoughts on hyper-operations of rational but non-integer orders? - by tommy1729 - 09/09/2019, 10:38 PM

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