Wolfram Language

Core Geometry

Mesh from Arrays

Generating color tetrominoes and building chessboards or arbitrary geometric shapes from patterns is easy in Version 11.

In[1]:=
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arrays = {\!\(\* TagBox[ RowBox[{"(", "", GridBox[{ {"1", "0", "0", "0"}, {"1", "0", "0", "0"}, {"1", "0", "0", "0"}, {"1", "0", "0", "0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\), \!\(\* TagBox[ RowBox[{"(", "", GridBox[{ {"1", "1", "0", "0"}, {"0", "1", "0", "0"}, {"0", "1", "0", "0"}, {"0", "0", "0", "0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\), \!\(\* TagBox[ RowBox[{"(", "", GridBox[{ {"1", "1", "0", "0"}, {"1", "0", "0", "0"}, {"1", "0", "0", "0"}, {"0", "0", "0", "0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\), \!\(\* TagBox[ RowBox[{"(", "", GridBox[{ {"0", "1", "0", "0"}, {"1", "1", "0", "0"}, {"0", "1", "0", "0"}, {"0", "0", "0", "0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\), \!\(\* TagBox[ RowBox[{"(", "", GridBox[{ {"1", "0", "0", "0"}, {"1", "1", "0", "0"}, {"0", "1", "0", "0"}, {"0", "0", "0", "0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\), \!\(\* TagBox[ RowBox[{"(", "", GridBox[{ {"1", "1", "0", "0"}, {"1", "1", "0", "0"}, {"0", "0", "0", "0"}, {"0", "0", "0", "0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\), \!\(\* TagBox[ RowBox[{"(", "", GridBox[{ {"1", "1", "0", "0"}, {"0", "1", "1", "0"}, {"0", "0", "0", "0"}, {"0", "0", "0", "0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\)};
In[2]:=
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col = ColorData[97, "ColorList"];
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Table[ArrayMesh[arrays[[i]], MeshCellStyle -> {2 -> col[[i]]}], {i, 7}]
Out[3]=

Build a 3D chessboard.

In[4]:=
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m = Join @@ ConstantArray[{{{0}, {1}, {0}, {1}, {0}, {1}, {0}, {1}}, {{1}, \ {0}, {1}, {0}, {1}, {0}, {1}, {0}}}, 4];
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p = First /@ Position[Flatten[m], 1];
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style = {{1, All} -> {Thick, Black}, {3, All} -> White, {3, #} & /@ p -> Black};
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r = ArrayMesh[ConstantArray[1, {8, 8, 1}], MeshCellStyle -> style]
Out[7]=

Construct a Seidel mesh, a region with tunnels going in every direction without crossing.

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In[8]:=
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seidelArray[{r_, s_, t_}] := Table[If[Mod[i, 4] == 2 && Mod[j, 4] == 2 || Mod[i, 4] == 0 && Mod[k, 4] == 0 || Mod[j, 4] == 0 && Mod[k, 4] == 2, 0, 1], {i, 3 + r 4}, {j, 3 + s 4}, {k, 3 + t 4}] transparentMesh[mr_?MeshRegionQ] := HighlightMesh[ BoundaryMesh[mr], {Style[1, None], Style[2, Opacity[0.5]]}]
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transparentMesh[ArrayMesh[seidelArray[{2, 2, 2}]]]
Out[9]=

Implement Conway's Game of Life.

In[10]:=
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gameOfLife = {224, {2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}}; board = RandomInteger[1, {40, 40}];
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sim = NestList[Last[CellularAutomaton[gameOfLife, #, {{0, 1}}]] &, board, 70];
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ListAnimate[ArrayMesh /@ sim]
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Play Animation
Stop Animation

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