* Dan Asimov <asimov@msri.org> [Dec 27. 2015 07:48]:
Aha.
By the way, just in case I haven't mentioned this:
For some time I wondered about the possibility of an analogue of the map-of-France in higher dimensions. (Not the space-filling curve per se, but the Franceoid shape that it fills.)
For example, can you start by taking a truncated octahedron and surrounding it with 14 copies of itself, then iterate, so that in the limit we have a bounded shape which when surrounded by 14 copies of itself results in the identical shape only larger?
Apparently not, so an analogue of the map-of-France in 3D seems not to exist, at least not with this particular strategy.
But with Joe Gerver (Rutgers Camden), a couple of years ago we were able to find such things in 4 and 8 dimensions (based on the quaternions and octonions, respectively).
Since there are no real division algebras except in dimensions 1, 2, 4, 8, is it possible that such shapes don't exist in any other dimensions?
For dim=3, see, http://jjj.de/tmp-cubic/ (all found March 2015, and posted here). There are STL files so you can whirl these objects around (DO THIS). These are self-similar (not only self-affine, which is rather trivial to obtain)! It appears these critters exist only for third powers for the number of similar subsets that the object can be decomposed into. these are built from either cubes, truncated octahedra, or rhombic dodecahedra. No proper success with curves giving these as "tiles" as in my draft (except for the well known 3D-Peano which gives a cube). I can build self-affine objects that give some of the "islands" ("tiles" for me) that we are talking about. Best regards, jj
—Dan
On Dec 26, 2015, at 3:34 PM, rwg <rwg@sdf.org> wrote:
On 2015-12-26 12:01, Dan Asimov wrote:
Yes, but unless the mirroring is periodic with respect to the levels of recursion, can the end result be self-similar? —Dan
No. Interesting point. Only countably many self-similar ones. But more than 900K !-) --rwg
On Dec 26, 2015, at 10:54 AM, Bill Gosper <billgosper@gmail.com> wrote: When building a frac-tile, you have, at every level of recursion, the option of mirror imaging. Hence, e.g., uncountably many slight variations of "France".
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