One thing I was considering was categorizing shapes by their symmetry (#@~.@rotall"3) and testing fewer possibilities for shapes with higher symmetry. (because it's less effort to test the symmetry of one shape than to canonicalize all potential downstream shapes). But I hadn't developed that thought.
-- Raul On Tue, Jul 18, 2023 at 2:12 PM 'Michael Day' via Programming <programm...@jsoftware.com> wrote: > > My not particularly small laptop manages to do that task in 19sec! > > Running jpm for extendall ^: (i.7) - taking ca 2.5 sec on this m/c - > seems to reveal > 0.08 sec for ~70500 reps in rotx, > 0.26 - ~280000 in roty, > 0.33, ~320000 in rotz > > I haven't looked deeply into the rots (!) - perhaps you could do something > otherthan use powers up to 3 . > > The representation is nice & concise. Wikipedia mentions dual graphs, > with cubes as nodes and face-face adjacency as links, but afaics, this > loses the 3rd dimension, even the 2nd in some cases; eg the V-tricube > is identical to the straight-3 polyomino in this rep. Arthur O'Dwyer > @ quuxplusone.github.io, subject polycube snakes & ouroboroi > uses srd .. as straight, right, down etc for the sub-set of singly > connected > (I think) pc's. > > All for now - got to go out, > > Mike > > On 15/07/2023 22:03, Raul Miller wrote: > > https://en.wikipedia.org/wiki/Polycube > > > > Today, I've been playing with polycubes (which are polyhedrons formed > > from adjacent cubes). > > > > This requires both a representation -- a rank 3 bit array works > > nicely, and a mechanism for working with cubic symmetry. For that, I > > went with: > > > > rotx=: |."2@(2 1&|:) NB. rotate in the yz plane about the x axis > > roty=: |.@(1 0 2&|:) NB. rotate in the xz plane about the y axis > > rotz=: |.@(2 1 0&|:) NB. rotate in the xy plane about the z axis > > > > rotall=: {{ > > 'X Y Z'=. y > > rotx^:X roty^:Y rotz^:Z x > > }}"3 1&(4 4 4#:(i.20),24+i.4) > > > > Here, rotall gives all 24 rotations of cubic symmetry for a given > > collection of 3d cubes represented by a bit array. > > > > I decided to use a brute force mechanism to generate polycubes. In > > other words, pad the array representation and add a single cube in > > each of the locations adjacent to an existing cube: > > > > extend=: {{ > > Y=. (-2+$y){.(1+$y){.y > > new=. ($#:I.@,)Y < (_1&|. +. 1&|. +. 1&|."2 +. _1&|."2 +. 1&|."1 +. > > _1&|."1) Y > > ~.new {{canon 1 (,:x)} y}}"1 3 Y > > }} > > > > Here, Y is the y argument with an extra 0 of padding in all > > directions. And 'new' is the list of indices of empty locations > > adjacent to cube locations. > > > > I also need a verb (canon) to reduce these generated polycubes to > > canonical form. A lot of this is removing unnecessary rows, columns > > and planes of zeros. The rest of it is picking arbitrarily (but > > consistently) from the 24 possible symmetric rotations: > > > > offset=: {{ 1 i.~ +./+./"2 x |: *y }} > > trim=: {{ (0 offset y)}."3 (1 offset y)}."2 (2 offset y)}."1 y }} > > rtrim=: trim&.|.&.:(|."2)&.:(|."1) > > > > canon=: {{ rtrim {. (/: +/@p:@I.@,"3) rotall trim y }}^:2 > > > > Finally, I want to work with all polycubes of a specific order, and to > > get larger batch I can use; > > > > extendall=: {{ ~.;<@extend"3 y }} > > > > And, testing, this works: > > > > #@> extendall&.>^:(i.8)<1 1 1 1$1 > > 1 1 2 8 29 166 1023 6922 > > > > But, this is slow -- since I'm brute forcing all possibilities then > > discarding duplicates, I need almost seven seconds on the little > > laptop I'm currently using, to generate the above sequence. > > > > (Also, for visualization of individual polycubes, > > https://code.jsoftware.com/wiki/Scripts/Plot_3D works passibly well.) > > > > Does anyone here have the geometric insight to improve on this approach? > > > > Thanks, > > > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
