Joerg, we must have different definitions. To my sight, snowflakes seem quite capable of tiling space over a wide range of elevations from plains to alpine. These tilings are non-compact, but opaque and reflective nonetheless. Let's not get distracted from the physical question of whether or not a broken-symmetry "rotoflake" could have fractal spiral arms. It's a possibility, but perhaps not the most likely. Another possibility is redemption of Gosper's "flowsnake". Obviously it is not a natural snow flake, but have we been too dense to consider that it could describe the shape of an *unnatural* snowflake? Here are a few more alternatives, made from a generalization of the Holladay-Ulam snowflake (see Mma code at the end): https://0x0.st/zkpa.png The pinwheels on the bottom right seem to be the most likely outcome of an experiment with a rotating hexagonal seed. Assume T = -15C, i.e. thin snow crystal. Rotation will produce a +/- pressure differential on the left/right halves of each prism facet. Pressure directly affects saturation, so the differential will alternatively encourage and suppress dendrite growth. If you cut, say, the left half off every axial dendrite, essentially you get a pinwheel. Could the experiment be done with Plate on a Pedestal?
http://www.snowcrystals.com/designer/designer.html
I don't think it would be infeasible to motorize rotation of the sapphire substrate, but then alignment becomes an issue. Alignment would also be an issue with rotating e-needles. The other issue would be additional heat from electronics. If an adequate solution could be found, I would say go for it! For one, it would lead to new and more unique works of art. More practically, natural snowflakes are not made sitting still, so experiments should develop motion control to probe the affect of rotational motion on formation. Happy 2020, --Brad ==========Mma Code========== MoveStar = {Cos[Pi/3 #], Sin[Pi/3 #]} & /@ Range[6]; ONRep[inds_] := ON[or_, x_] :> ( ON[or + MoveStar[[Mod[x + #, 6] /. {0 -> 6}]], Mod[x + #, 6] /. {0 -> 6}] & /@ inds) DelDup[ONs_] := Flatten[Select[ Cases[ONs, ON[#, _]] & /@ Union[ONs[[All, 1]]], Length[#] == 1 &]] Iterate[data_, inds1_, inds2_] := With[ {it1 = DelDup[Flatten[data /. ONRep[inds1]]]}, Union@Join[DelDup[Flatten[Join[data, it1] /. ONRep[inds2]]], it1, data]] DepictRepCol = ON[x_, y_] :> { Blend[{Darker@Blue, Darker@Cyan}, y/6], EdgeForm[Thick], Polygon[x + 1/Sqrt[3] {Sin[Pi/3 #], Cos[Pi/3 #]} & /@ Range[6]]}; AXIOM = MapIndexed[ON[#1, #2[[1]]] &, MoveStar[[1 ;; 6]]]; (* Example: *) Graphics[Nest[ Iterate[#, {-1, 0, 1}, {-2, -1, 1, 2}] &, AXIOM, 15] /. DepictRepCol] On Tue, Dec 31, 2019 at 6:57 AM Joerg Arndt <arndt@jjj.de> wrote:
Note that this flake tiles the plane. This does not seem to be the case with real snowflakes.