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,
>>> 
>> 
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