Wolfram Function Repository
Instant-use add-on functions for the Wolfram Language
Function Repository Resource:
Lattice
Gives lattice points in a variety of arrangements
Contributed by:
Sander Huisman
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ResourceFunction["LatticePointsArrangement"][arrangement,n] generates the first n lattice points based on arrangement. |
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ResourceFunction["LatticePointsArrangement"][] returns all the possible arrangements. |
Details
The following arrangements can be chosen as arrangement:
| "CCWSpiralEast" |
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| "CCWSpiralNorth" |
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| "CCWSpiralWest" |
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| "CCWSpiralSouth" |
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| "CWSpiralEast" |
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| "CWSpiralNorth" |
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| "CWSpiralWest" |
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| "CWSpiralSouth" |
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| "CCWDiamondEast" |
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| "CCWDiamondNorth" |
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| "CCWDiamondWest" |
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| "CCWDiamondSouth" |
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| "CWDiamondEast" |
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| "CWDiamondNorth" |
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| "CWDiamondWest" |
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| "CWDiamondSouth" |
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| "ZigZagEastQ1" |
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| "ZigZagNorthQ1" |
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| "ZigZagWestQ2" |
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| "ZigZagNorthQ2" |
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| "ZigZagWestQ3" |
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| "ZigZagSouthQ3" |
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| "ZigZagEastQ4" |
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| "ZigZagSouthQ4" |
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| "ZigZagEastQ12" |
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| "ZigZagWestQ12" |
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| "ZigZagNorthQ23" |
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| "ZigZagSouthQ23" |
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| "ZigZagWestQ34" |
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| "ZigZagEastQ34" |
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| "ZigZagNorthQ14" |
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| "ZigZagSouthQ14" |
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| "ZigZagNorthEastQ12" |
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| "ZigZagNorthWestQ12" |
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| "ZigZagWestNorthQ23" |
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| "ZigZagWestSouthQ23" |
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| "ZigZagSouthEastQ34" |
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| "ZigZagSouthWestQ34" |
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| "ZigZagWestSouthQ14" |
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| "ZigZagWestNorthQ14" |
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| "ZigZagEastQ123" |
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| "ZigZagSouthQ123" |
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| "ZigZagNorthQ234" |
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| "ZigZagEastQ234" |
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| "ZigZagEastQ134" |
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| "ZigZagNorthQ134" |
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| "ZigZagEastQ124" |
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| "ZigZagSouthQ124" |
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| "ZigZagNorthWestQ123" |
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| "ZigZagWestNorthQ123" |
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| "ZigZagWestSouthQ234" |
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| "ZigZagSouthWestQ234" |
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| "ZigZagSouthEastQ134" |
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| "ZigZagEastSouthQ134" |
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| "ZigZagNorthEastQ124" |
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| "ZigZagEastNorthQ124" |
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| "ZigZagNorthEastQ123" |
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| "ZigZagWestSouthQ123" |
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| "ZigZagWestNorthQ234" |
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| "ZigZagSouthEastQ234" |
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| "ZigZagSouthWestQ134" |
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| "ZigZagEastNorthQ134" |
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| "ZigZagNorthWestQ124" |
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| "ZigZagEastSouthQ124" |
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| "DiagonalZigZagEastQ1" |
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| "DiagonalZigZagNorthQ1" |
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| "DiagonalZigZagNorthQ2" |
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| "DiagonalZigZagWestQ2" |
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| "DiagonalZigZagSouthQ3" |
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| "DiagonalZigZagWestQ3" |
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| "DiagonalZigZagEastQ4" |
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| "DiagonalZigZagSouthQ4" |
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| "DiagonalZigZagEastQ12" |
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| "DiagonalZigZagSouthQ23" |
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| "DiagonalZigZagNorthQ23" |
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| "DiagonalZigZagEastQ34" |
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| "DiagonalZigZagWestQ34" |
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| "DiagonalZigZagNorthQ14" |
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| "DiagonalZigZagWestQ12" |
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| "DiagonalZigZagSouthQ14" |
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| "DiagonalZigZagEastQ123" |
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| "DiagonalZigZagSouthQ123" |
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| "DiagonalZigZagNorthQ234" |
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| "DiagonalZigZagEastQ234" |
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| "DiagonalZigZagWestQ134" |
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| "DiagonalZigZagNorthQ134" |
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| "DiagonalZigZagWestQ124" |
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| "DiagonalZigZagSouthQ124" |
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| "CCWDiagonalQ1" |
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| "CWDiagonalQ1" |
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| "CCWDiagonalQ2" |
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| "CWDiagonalQ2" |
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| "CCWDiagonalQ3" |
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| "CWDiagonalQ3" |
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| "CWDiagonalQ4" |
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| "CCWDiagonalQ4" |
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| "CCWDiagonalQ12" |
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| "CWDiagonalQ23" |
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| "CCWDiagonalQ23" |
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| "CWDiagonalQ34" |
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| "CCWDiagonalQ34" |
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| "CWDiagonalQ14" |
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| "CWDiagonalQ12" |
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| "CCWDiagonalQ14" |
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| "CCWDiagonalQ123" |
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| "CWDiagonalQ123" |
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| "CCWDiagonalQ234" |
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| "CWDiagonalQ234" |
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| "CCWDiagonalQ134" |
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| "CWDiagonalQ134" |
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| "CCWDiagonalQ124" |
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| "CWDiagonalQ124" |
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| "CWCornerQ1" |
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| "CCWCornerQ3" |
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| "CWCornerQ2" |
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| "CCWCornerQ4" |
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| "CWCornerQ3" |
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| "CCWCornerQ1" |
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| "CWCornerQ4" |
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| "CCWCornerQ2" |
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| "CWCornerQ12" |
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| "CCWCornerQ12" |
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| "CCWCornerQ23" |
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| "CWCornerQ23" |
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| "CCWCornerQ34" |
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| "CWCornerQ34" |
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| "CCWCornerQ14" |
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| "CWCornerQ14" |
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| "CWCornerQ123" |
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| "CCWCornerQ123" |
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| "CWCornerQ234" |
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| "CCWCornerQ234" |
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| "CWCornerQ134" |
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| "CCWCornerQ134" |
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| "CWCornerQ124" |
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| "CCWCornerQ124" |
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Examples
Basic Examples (1)
Create 20 points in a counterclockwise spiral starting in the eastern direction:
| In[1]:= |
![ResourceFunction["LatticePointsArrangement"]["CCWSpiralEast", 20]](https://www.wolframcloud.com/obj/resourcesystem/images/e42/e42fe59d-9cd4-4f48-a4e0-e23057debd8e/1-0-1/0599238130174053.png){kind=small}
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| Out[1]= |
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Scope (2)
Create some points based on zigzagging diagonally in the first two quadrants:
| In[2]:= |
![pts = ResourceFunction["LatticePointsArrangement"][
"DiagonalZigZagEastQ12", 100];](https://www.wolframcloud.com/obj/resourcesystem/images/e42/e42fe59d-9cd4-4f48-a4e0-e23057debd8e/1-0-1/6d7d5b987ace128d.png){kind=small}
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Visualize the points:
| In[3]:= |
![ListPlot[pts, Joined -> True, PlotMarkers -> Automatic]](https://www.wolframcloud.com/obj/resourcesystem/images/e42/e42fe59d-9cd4-4f48-a4e0-e23057debd8e/1-0-1/1b9b4ae1244c1a1c.png){kind=small}
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| Out[3]= |
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LatticePointsArrangement without any arguments returns all the possible arrangements:
| In[4]:= |
![arr = ResourceFunction["LatticePointsArrangement"][]](https://www.wolframcloud.com/obj/resourcesystem/images/e42/e42fe59d-9cd4-4f48-a4e0-e23057debd8e/1-0-1/36dd93afe4b959f9.png){kind=small}
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| Out[4]= |
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Currently 136 arrangements are known:
| In[5]:= |
![Length[arr]](https://www.wolframcloud.com/obj/resourcesystem/images/e42/e42fe59d-9cd4-4f48-a4e0-e23057debd8e/1-0-1/1819aba1d2758309.png){kind=small}
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| Out[5]= |
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Applications (1)
Create the Ulam prime spiral:
| In[6]:= |
![pts = ResourceFunction["LatticePointsArrangement"]["CCWSpiralEast", 100000];
Graphics[{PointSize[Small], Point[pts[[Prime[Range[1, PrimePi[Length[pts]]]]]]]}]](https://www.wolframcloud.com/obj/resourcesystem/images/e42/e42fe59d-9cd4-4f48-a4e0-e23057debd8e/1-0-1/4bebca3a08eca428.png){kind=small}
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| Out[7]= |
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Neat Examples (1)
Recreate the OEIS A316667 sequence of a horse moving on a spirally numbered board and moving to the lowest available unvisited square at each step:
| In[8]:= |
![ClearAll[ShowRoute, MakeMove, FindSequence]
horsejump = Select[Tuples[Range[-2, 2], 2], Norm[#] == Sqrt[5] &];
ShowRoute[output_Association] := Module[{colors},
colors = (ColorData["Rainbow"] /@ Subdivide[Length[output["Coordinates"]] - 1.0]);
Graphics[{Line[output["Coordinates"], VertexColors -> colors], Disk[Last@output["Coordinates"], 0.2]}]
]
MakeMove[spiral_Association, visited_List] := Module[{poss, hj},
poss = Table[Last[Last[visited]] + hj, {hj, horsejump}];
poss = DeleteMissing[{spiral[#], #} & /@ poss, 1, \[Infinity]];
poss = Select[poss, FreeQ[visited[[All, 2]], Last[#]] &];
If[Length[poss] > 0,
First[TakeSmallestBy[poss, First, 1]]
,
Missing[]
]
]
FindSequence[start_ : {0, 0}, grid_] := Module[{positions, j, next},
positions = {{grid[start], start}};
PrintTemporary[Dynamic[j]];
Do[
next = MakeMove[grid, positions];
If[next =!= Missing[],
AppendTo[positions, next]
,
Break[];
]
,
{j, \[Infinity]}
];
<|"Coordinates" -> positions[[All, 2]], "Indices" -> positions[[All, 1]]|>
]
grid = ResourceFunction["LatticePointsArrangement"]["CCWSpiralEast", 10000];
grid = Association[MapIndexed[#1 -> #2[[1]] &, grid]];
ShowRoute[FindSequence[{0, 0}, grid]]](https://www.wolframcloud.com/obj/resourcesystem/images/e42/e42fe59d-9cd4-4f48-a4e0-e23057debd8e/1-0-1/7e4df7b0f71d8f28.png)
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| Out[15]= |
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License Information
This work is licensed under a Creative Commons Attribution 4.0 International License