= Daniel Asimov <asimov@msri.org> wrote: Pretending that spheres, balls, and Euclidean spaces can have real dimensions: ="Olivier GERARD" <olivier.gerard@gmail.com> Very delicate hypothesis. You would have to prove that there is a continuous family of geometrical objects, to whom volume and surface make sense, etc. and according precisely to these formulae. ="Bill Gosper" <billgosper@gmail.com> This exposes a bug in the terminology "unit ball", which really ought to mean "unit diameter ball". MLeBrun> I don't know balls, but, prompted by this mysterious maximum, a number of years ago I asked this list if it was always possible to construct a fractal of any given real dimension (ie we're not just restricted to values of the "classical" type of log-mumble expressions), and y'all said "yes"--and I think moreover even gave an explicit method. Starting with such a fractal as the "space", can you then hand-wave up a suitably-constructed "unit subset" that degenerates to the Euclidean ball at integer dimensions? The content formulae over this continua of constructions will of course likewise degenerate to the expression under discussion here at integer dimensions, but it might have additional twiddle factors that vanish at those values--hence different maxima as it interpolates in-between. So another tack, symbolic instead of geometric: the volume formula can be constructed directly by iterated integration. I think the usual fractional iterated integration operator also just changes factorial to gamma. If that generalization is unique, it would seem that the generalized fractional dimensional geometric construction must also be. If not, the converse. <MLB

From the McCarthy festschrift, http://gosper.org/fst.pdf p "145" = 17 and thereabouts derives the triadic dragon (and its Fourier series) with 0<=dimension<=1 adjustable via the parameter w. See the trajectories plot.

Caution: there are some typos in that collection of valuations at the bottom of p145. Fix: gosper.org/fstfix.pdf I'm not sure why we always use the triadic dragon to illustrate continuously variable dimension. The ordinary Heighway Dragon ought to work as well. --rwg