Georges Configuration

Author

Eric W. Weisstein
August 2, 2026

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

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

©2026 Wolfram Research, Inc. except for portions noted otherwise

Configuration data

The Georges graph is the Levi graph of the Georges configuration. The following code recovers the two incidence classes from graph-distance parity and then numbers the 25 configuration points from 1 through 25.

g=GraphData["GeorgesGraph"]

{VertexCount[g],EdgeCount[g],Union[VertexDegree[g]],BipartiteGraphQ[g]}

vertices=VertexList[g];distances=(GraphDistance[g,First[vertices],#1]&)/@vertices;pointVertices=Pick[vertices,EvenQ/@distances];lineVertices=Complement[vertices,pointVertices];pointIndex=AssociationThread[pointVertices->Range[Length[pointVertices]]];lines=(Sort[Lookup[pointIndex,AdjacencyList[g,#1]]]&)/@lineVertices;

Each row below gives the three points incident with one configuration line.

lines

ConfigurationNotation[lines]

LeviGraph[lines]//RecognizeGraph

Geometric realization

Kocay gave a rational coordinatization by assigning integer homogeneous coordinates and closing the construction with a rational point on a cubic. The exact coordinates below follow that construction, with parameter choices selected so that there are no unintended incidences. Point numbers and incidence triples use Grunbaum's numbering.

configurationLines={{1,2,4},{1,10,12},{4,9,10},{5,8,9},{2,3,5},{3,4,6},{5,6,7},{7,8,13},{6,8,11},{3,9,11},{2,10,11},{12,13,14},{7,14,22},{1,14,17},{15,21,22},{15,16,20},{16,17,19},{13,17,18},{16,18,24},{19,20,21},{21,23,24},{12,22,23},{15,23,25},{20,24,25},{18,19,25}};

homogeneousPoints={{2646361189,-2646361189,-10847216446},{2175941630,-29109973079,-119319380906},{1,1,1},{1,0,0},{12509661,8257124,-4004587},{0,1,1},{4169887,-27550,-4114787},{0,1,-2},{1,0,1},{275645,0,-15115199},{0,1,0},{1338149,-1392089,-2748218},{10,-1,-8},{1,-1,-2},{11,-9,-5},{4,-3,-1},{1,-1,0},{6,3,-8},{2,1,-3},{76,-67,-49},{30,-27,-19},{43441078,-86245227,-130540355},{1,1,0},{108,-51,-53},{8,4,-1}};

ConfigurationDiagram[pointCoordinates,configurationLines]

ConfigurationDiagram[pointCoordinates,configurationLines,VertexLabels->None]

The corresponding exact homogeneous line coordinates are listed in the same order as the incidence triples above.

lineCoordinates=Function[coordinates,With[{reduced=(coordinates/(GCD@@Abs[coordinates]))},Sign[FirstCase[reduced,Except[0]]] reduced]]/@(Cross@@homogeneousPoints[[#1[[{1,2}]]]]&)/@configurationLines;lineCoordinates

A projective transformation selects an affine chart in which all 25 points are finite. Each configuration line is drawn between the two most widely separated incident points.

projectiveTransform={{-20,15,-19},{-14,-13,-28},{-23,-13,-11}};pointCoordinates=N[(((Most[#1])/(Last[#1]))&)/@(projectiveTransform.#1&)/@homogeneousPoints];lineSegments=Function[line,First[MaximalBy[Subsets[pointCoordinates[[line]],{2}],Function[pair,EuclideanDistance@@pair]]]]/@configurationLines;

Verification

There are 25 distinct points and 25 distinct lines. The exact incidence calculation finds precisely the 75 prescribed point-line incidences and no others.

actualIncidences=(Flatten[Position[homogeneousPoints.#1,0]]&)/@lineCoordinates;{Length[homogeneousPoints],Length[configurationLines],Total[Length/@actualIncidences],actualIncidences===configurationLines,And@@(#1\[Cross]#2≠{0,0,0}&)@@@Subsets[homogeneousPoints,{2}],And@@(#1\[Cross]#2≠{0,0,0}&)@@@Subsets[lineCoordinates,{2}]}

Reference

Grunbaum, B. Configurations of Points and Lines. Providence, RI: American Mathematical Society, 2009, pp. 311-316.

Kocay, W. "A Note on the Georges Configuration and a Paper of Grunbaum." Bulletin of the Institute of Combinatorics and its Applications 58, 103-111, 2010.