Function Repository Resource:

LatticePointsArrangement

Source Notebook

Get lattice points in a variety of arrangements

Contributed by: Sander Huisman

ResourceFunction["LatticePointsArrangement"][arrangement,n]

generates the first n lattice points based on arrangement.

ResourceFunction["LatticePointsArrangement"][]

returns all the possible arrangements.

Details

The following arrangements can be chosen as arrangement:

Examples

Basic Examples (1)

Create 20 points in a counter clock wise spiral starting in the eastern direction:

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Scope (2)

Create some points based on zigzagging diagonally in the first two quadrants:

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Visualize the points:

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LatticePointsArrangement without any arguments returns all the possible arrangements:

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Currently 160 arrangements are known:

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Applications (1)

Create the Ulam prime spiral:

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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:

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$ShowRoute[output_Association] := Module[{colors}, colors = (ColorData["Rainbow"] /@ Subdivide[Length[output["Coordinates"]] - 1.0]); Graphics[{Line[output["Coordinates"], VertexColors -> colors], Disk[Last@output["Coordinates"], 0.2]}] ];$

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$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[] ] ];$

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$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]]|> ];$