9
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I am trying to produce these lozenge tilings as a way of encoding plane partitions. I need to produce something like:

enter image description here

but am using demonstration code like this:

coversQ[parent_, child_] := 
 And[Length[parent] >= Length[child], 
  Min[Take[parent, Length@child] - child] >= 0]

planepartitionQ[par_] := 
 MatchQ[par, {{___Integer} ..}] && 
  If[Length[par] > 1, 
   And @@ MapThread[coversQ, {Drop[par, -1], Rest[par]}], True]

PlanePartitions[n_Integer] := Module[{l1, l2, l3, l4, z, w},
  l1 = z @@@ IntegerPartitions[n];
  l2 = l1 /. k_Integer /; (k > 1) :> w @@ IntegerPartitions[k];
  l3 = l2 /. z[x_w, y : (1 ...)] :> Thread[z[x, y], w] /.
      
      z[x__w] :> Outer[z, x] /. 
     z[x__w, y : (1 ...)] :> 
      Outer[z, x, Sequence @@ ({y} /. 1 -> w[1])] /. w -> Sequence;
  l4 = l3 /. 
     z[x___List, y : (1 ..)] :> z[x, Sequence @@ Transpose[{{y}}]] /. 
    z -> List; Cases[Union[l4], _?planepartitionQ]
  ]
PlanePartitionDiagram[l_List] := Module[{i, j, k},
  Graphics3D[
   Table[Cuboid[{j, -i, k}],
    {i, Length[l]},
    {j, Length[l[[i]]]},
    {k, l[[i, j]]}
    ]
   ]
  ]
Show[PlanePartitionDiagram[{{3, 3, 2, 1}, {0, 3, 2, 1}, {0, 3, 2, 1},
    {0, 0, 0, 1}}]]

producing the slightly less pleasing:

enter image description here

Is there a way to produce figures like this lozenge tiling in Mathematica?

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1
  • 3
    $\begingroup$ Try ViewPoint -> {Infinity, -Infinity, Infinity} as an option to get the projected geometry undistorted. $\endgroup$
    – Roman
    Aug 13 '20 at 7:05
9
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You can add the options ViewProjection, ViewPoint, and ViewVertical to make it appear as if it isometric:

coversQ[parent_,child_]:=And[Length[parent]>=Length[child],Min[Take[parent,Length@child]-child]>=0]

planepartitionQ[par_]:=MatchQ[par,{{___Integer}..}]&&If[Length[par]>1,And@@MapThread[coversQ,{Drop[par,-1],Rest[par]}],True]

PlanePartitions[n_Integer]:=Module[{l1,l2,l3,l4,z,w},l1=z@@@IntegerPartitions[n];
l2=l1/.k_Integer/;(k>1):>w@@IntegerPartitions[k];
l3=l2/.z[x_w,y:(1...)]:>Thread[z[x,y],w]/.z[x__w]:>Outer[z,x]/.z[x__w,y:(1...)]:>Outer[z,x,Sequence@@({y}/.1->w[1])]/.w->Sequence;
l4=l3/.z[x___List,y:(1..)]:>z[x,Sequence@@Transpose[{{y}}]]/.z->List;Cases[Union[l4],_?planepartitionQ]]
PlanePartitionDiagram[l_List]:=Module[{i,j,k},
Graphics3D[{EdgeForm[{Black,Thickness[0.01]}],
Table[
Cuboid[{j,-i,k}]
,
{i,Length[l]},
{j,Length[l[[i]]]},
{k,l[[i,j]]}
]},
Boxed->False,
ViewProjection->"Orthographic",
ViewPoint->{1,1,1},
Lighting -> {{"Directional", 
   Yellow, {{0, 0, 1}, {0, 0, 0}}}, {"Directional", 
   Blue, {{0, 1, 0}, {0, 0, 0}}}, {"Directional", 
   Red, {{1, 0, 0}, {0, 0, 0}}}}
]
]
PlanePartitionDiagram[{{3,3,2,1},{0,3,2,1},{0,3,2,1},{0,0,0,1}}]

(should work with 11.2 and up).

enter image description here

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7
  • 1
    $\begingroup$ Is it possible to remove the frame, and colour the walls with cells of the respective colour? $\endgroup$
    – apkg
    Aug 13 '20 at 9:10
  • 1
    $\begingroup$ Boxed -> False option removes the box around it. Perhaps the coloring can be achieved by placing lights strategically in certain, see the documentation of Lighting reference.wolfram.com/language/ref/Lighting.html Otherwise one has to decompose the cubes into polygons with different colors. $\endgroup$
    – SHuisman
    Aug 13 '20 at 9:13
  • 1
    $\begingroup$ Ok, the boxed option is in the Graphics3D part? $\endgroup$
    – apkg
    Aug 13 '20 at 9:14
  • 1
    $\begingroup$ Yes, or the Show. $\endgroup$
    – SHuisman
    Aug 13 '20 at 9:15
  • 1
    $\begingroup$ I've modified my answer to reflect the right colors… $\endgroup$
    – SHuisman
    Aug 13 '20 at 10:01
3
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Here is different way to achieve the coloring:

PlanePartitionDiagram[l_List, col_, {offsetx_, offsety_, offsetz_}] :=
  Module[{i, j, k},
  Graphics3D[
   Prepend[Glow[col]]@Table[
     Cuboid[{j + offsetx, -i + offsety, k + offsetz}],
     {i, Length[l]},
     {j, Length[l[[i]]]},
     {k, l[[i, j]]}
     ]
   ]
  ]

Show[
 PlanePartitionDiagram[{{3, 3, 2, 1}, {0, 3, 2, 1}, {0, 3, 2, 1}, {0, 0, 0, 1}}, Red, {10^-2, 0, 0}],
 PlanePartitionDiagram[{{3, 3, 2, 1}, {0, 3, 2, 1}, {0, 3, 2, 1}, {0, 0, 0, 1}}, Blue, {0, 10^-2, 0}],
 PlanePartitionDiagram[{{3, 3, 2, 1}, {0, 3, 2, 1}, {0, 3, 2, 1}, {0, 0, 0, 1}}, Yellow, {0, 0, 10^-2}],
 Lighting -> None,
 Boxed -> False,
 ViewProjection -> "Orthographic"
 ]

enter image description here

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