Hesse Configuration

Author

Eric W. Weisstein
December 13, 2023

This notebook downloaded from https://mathworld.wolfram.com/notebooks/Configurations/HesseConfiguration.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/HesseConfiguration.html.

©2023 Wolfram Research, Inc. except for portions noted otherwise

Source

https://en.wikipedia.org/wiki/Hesse_configuration

Import["https://upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Hesse_configuration.svg/480px-Hesse_configuration.svg.png"]

Length[lines={
{1,2,3},{4,5,6},{7,8,9},
{1,4,7},{2,5,8},{3,6,9},
{1,5,9},{2,6,7},{3,4,8},
{3,5,7},{1,8,6},{9,2,4}}]

e@@@lines

g=Graph[Range[9],UndirectedEdge@@@e@@@lines,VertexCoordinates->Flatten[Table[{i,j},{j,3},{i,3}],1],VertexLabels->Automatic]

RecognizeGraph@g

Graph[EdgeList[g]]

levi=LeviGraph[lines]

RecognizeGraph@%

Through[{VertexCount,EdgeCount}[levi]]

GroupOrder[GraphAutomorphismGroup[levi]]

Möbius–Kantor configuration

Removing any one point and its four incident lines from the Hesse configuration produces another configuration of type (Subscript[8, 3]) (Subscript[8, 3]), the Möbius–Kantor configuration.

LeviGraph/@Table[Select[lines,FreeQ[#,i]&],{i,9}]

RecognizeGraph/@%

Pappus

Removing from the Hesse configuration the three lines belonging to a single triple produces a configuration of type (Subscript[9, 3]) (Subscript[9, 3]), the Pappus configuration

LeviGraph/@Join@@@Subsets[Partition[lines,3],{3}]

RecognizeGraph/@%

GraphData

GraphDataString

<<MathWorld`Graphs`

file=OpenWrite["~/hesse.m"]

WriteString[file,GraphDataString[
CanonicalGraph@levi,
"StandardName"->"HesseGraph",
"Name"->"Hesse graph",
"Embeddings"->None,
"Classes"->{"Incidence"},
Information->"HesseGraph",
"Configurations"->{lines},
"Skip"->{"Arboricity","ConnectedDominatingSets","ConnectedInducedSubgraphPolynomial","ConnectedInducedSubgraphs","ConstructionNumber","CyclicEdgeConnectivity","GracefulLabeling","LinklesslyEmbeddable","MaximalIrredundantSets","MinimumDistinguishingLabelings","MinimalDominatingSets","MinimumCyclicEdgeCut","MinimumCyclicEdgeCuts","MinimumWeightFractionalEdgeColoring","Minors","ProjectivePlanar","QuadraticEmbeddingConstant","Skewness","VertexCuts"}
]]

Close[file]