N-Cluster

Author

Eric W. Weisstein
December 15, 2025

This notebook downloaded from https://mathworld.wolfram.com/notebooks/ComputationalGeometry/N-Cluster.nb.

For more information, see Eric's MathWorld entry https://mathworld.wolfram.com/N-Cluster.html.

©2025 Wolfram Research, Inc. except for portions noted otherwise

Initialization

MakeBoxes[Coords[x_,y_],TraditionalForm]:=RowBox[{"(",RowBox[{MakeBoxes[x,TraditionalForm],",",MakeBoxes[y,TraditionalForm]}],")"}]

6-Configuration

config={{0,0},{132,-720},{546,-272},{960,-720},{1155,540},{546,1120}};

Plot

Show[Graphics[{
{PointSize[.05],Red,Point/@config},
Text[Coords@@#,If[#=={0,0},{-1,1}150,#+{350,Sign[Last[#]]100}],FormatType->TraditionalForm]&/@config}],AspectRatio->Automatic,Axes->True,Ticks->{{500,1000,1500},Automatic},PlotRange->All]

Cluster Configuration

dist=Norm/@(Subtract@@@Subsets[config,{2}])

And@@(IntegerQ/@dist)

And@@(CollinearQ/@Subsets[Point/@config,{3}])

And@@(ConcyclicQ/@Subsets[config,{4}])

7-Configuration

config={{0,0},{49595290,0},{26127018,932064},{32142553,411864},{17615968,238464},{7344908,411864},{19079044,-54168}};

Plot

With[{s=5 10^4,x=1 10^6},
Show[Graphics[{
{PointSize[.03],Red,Point/@config},
Text[Coords@@#,If[Last[#]==0,{x+#[[1]],s},#+{x,s Sign[Last[#]]}],{-1,0},FormatType->TraditionalForm]&/@config}],Axes->True,PlotRange->All,AspectRatio->1,Method->{"AxesInFront"->False},ImageSize->300]
]

Cluster Configuration

𝒸={1,Sqrt[2002]}/2227;

dist=Norm/@(Subtract@@@Subsets[𝒸 #&/@config,{2}])

And@@(IntegerQ/@dist)

And@@(CollinearQ/@Subsets[Point/@(𝒸 #&/@config),{3}])

And@@(ConcyclicQ/@Subsets[𝒸 #&/@config,{4}])