Wolfram Cloud Page

Basic Examples

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

In[]:=

[◼]	LatticePointsArrangement

["CCWSpiralEast",20]

Out[]=

{{0,0},{1,0},{1,1},{0,1},{-1,1},{-1,0},{-1,-1},{0,-1},{1,-1},{2,-1},{2,0},{2,1},{2,2},{1,2},{0,2},{-1,2},{-2,2},{-2,1},{-2,0},{-2,-1}}

Scope

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

In[]:=

pts=

[◼]	LatticePointsArrangement

["DiagonalZigZagEastQ12",100];

Visualize the points:

In[]:=

ListPlot[pts,JoinedTrue]

Out[]=

LatticePointsArrangement

without any arguments returns all the possible arrangements:

In[]:=

arr=

[◼]	LatticePointsArrangement

[]

Out[]=

{CCWSpiralEast,CCWSpiralNorth,CCWSpiralWest,CCWSpiralSouth,CWSpiralEast,CWSpiralNorth,CWSpiralWest,CWSpiralSouth,CCWDiamondEast,CCWDiamondNorth,CCWDiamondWest,CCWDiamondSouth,CWDiamondEast,CWDiamondNorth,CWDiamondWest,CWDiamondSouth,ZigZagEastQ1,ZigZagNorthQ1,ZigZagWestQ2,ZigZagNorthQ2,ZigZagWestQ3,ZigZagSouthQ3,ZigZagEastQ4,ZigZagSouthQ4,ZigZagEastQ12,ZigZagWestQ12,ZigZagNorthQ23,ZigZagSouthQ23,ZigZagWestQ34,ZigZagEastQ34,ZigZagNorthQ14,ZigZagSouthQ14,ZigZagNorthEastQ12,ZigZagNorthWestQ12,ZigZagWestNorthQ23,ZigZagWestSouthQ23,ZigZagSouthEastQ34,ZigZagSouthWestQ34,ZigZagWestSouthQ14,ZigZagWestNorthQ14,ZigZagEastQ123,ZigZagSouthQ123,ZigZagNorthQ234,ZigZagEastQ234,ZigZagEastQ134,ZigZagNorthQ134,ZigZagEastQ124,ZigZagSouthQ124,ZigZagNorthWestQ123,ZigZagWestNorthQ123,ZigZagWestSouthQ234,ZigZagSouthWestQ234,ZigZagSouthEastQ134,ZigZagEastSouthQ134,ZigZagNorthEastQ124,ZigZagEastNorthQ124,ZigZagNorthEastQ123,ZigZagWestSouthQ123,ZigZagWestNorthQ234,ZigZagSouthEastQ234,ZigZagSouthWestQ134,ZigZagEastNorthQ134,ZigZagNorthWestQ124,ZigZagEastSouthQ124,DiagonalZigZagEastQ1,DiagonalZigZagNorthQ1,DiagonalZigZagNorthQ2,DiagonalZigZagWestQ2,DiagonalZigZagSouthQ3,DiagonalZigZagWestQ3,DiagonalZigZagEastQ4,DiagonalZigZagSouthQ4,DiagonalZigZagEastQ12,DiagonalZigZagSouthQ23,DiagonalZigZagNorthQ23,DiagonalZigZagEastQ34,DiagonalZigZagWestQ34,DiagonalZigZagNorthQ14,DiagonalZigZagWestQ12,DiagonalZigZagSouthQ14,DiagonalZigZagEastQ123,DiagonalZigZagSouthQ123,DiagonalZigZagNorthQ234,DiagonalZigZagEastQ234,DiagonalZigZagWestQ134,DiagonalZigZagNorthQ134,DiagonalZigZagWestQ124,DiagonalZigZagSouthQ124,CCWDiagonalQ1,CWDiagonalQ1,CCWDiagonalQ2,CWDiagonalQ2,CCWDiagonalQ3,CWDiagonalQ3,CWDiagonalQ4,CCWDiagonal4,CCWDiagonalQ12,CWDiagonalQ23,CCWDiagonalQ23,CWDiagonalQ34,CCWDiagonalQ34,CWDiagonalQ14,CWDiagonalQ12,CCWDiagonalQ14,CCWDiagonalQ123,CWDiagonalQ123,CCWDiagonalQ234,CWDiagonalQ234,CCWDiagonalQ134,CWDiagonalQ134,CCWDiagonalQ124,CWDiagonalQ124,CWCornerQ1,CCWCornerQ3,CWCornerQ2,CCWCornerQ4,CWCornerQ3,CCWCornerQ1,CWCornerQ4,CCWCornerQ2,CWCornerQ12,CCWCornerQ12,CCWCornerQ23,CWCornerQ23,CCWCornerQ34,CWCornerQ34,CCWCornerQ14,CWCornerQ14,CWCornerQ123,CCWCornerQ123,CWCornerQ234,CCWCornerQ234,CWCornerQ134,CCWCornerQ134,CWCornerQ124,CCWCornerQ124}

Currently 136 arrangements are known:

In[]:=

Length[arr]

Out[]=

136

Applications

Create the Ulam prime spiral:

In[]:=

pts=

[◼]	LatticePointsArrangement

["CCWSpiralEast",100000];Graphics[{PointSize[Small],Point[pts[[Prime[Range[1,PrimePi[Length[pts]]]]]]]}]

Out[]=

Neat Examples

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[]:=

ClearAll[ShowRoute,MakeMove,FindSequence]horsejump=Select[Tuples[Range[-2,2],2],Norm[#]

&];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,∞];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,∞}];<|"Coordinates"->positions[[All,2]],"Indices"->positions[[All,1]]|>]grid=

[◼]	LatticePointsArrangement

["CCWSpiralEast",10000];grid=Association[MapIndexed[#1->#2[[1]]&,grid]];ShowRoute[FindSequence[{0,0},grid]]

Out[]=