A couple of qs re canon: why power 2? why p:? Isn't the I. sufficient? Cheers,
Mike Sent from my iPad > On 18 Jul 2023, at 22:14, Raul Miller <rauldmil...@gmail.com> wrote: > > 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 ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm