Desargues' Configuration

Author

Eric W. Weisstein
December 19, 2023

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Combinatorics/DesarguesConfiguration.nb.

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

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

Source

Figure

v={{0,0},{1,0},{2,(3/4)},{(1/2),(3/2)},{(671/816),(5/4)},{(7/5),(11/8)},{(68/97),(204/97)},{(331/136),0},{(799/234),(565/312)},{(113/37),(339/296)}};
lines={{1,2,8},{8,10,9},{9,6,5},{7,6,3},{7,4,1},{1,3,10},{9,3,2},{7,5,2},{8,5,4},{10,4,6}};

ConfigurationDiagram[v,lines,.3]

cg=ConfigurationGraph[v,lines,VertexLabels->Automatic]

cgname=RecognizeGraph[cg]

GraphData[cgname,"Graph","Configuration"]

RecognizeGraph@LeviGraph[lines]

GraphData["DesarguesGraph","Configurations"]

RecognizeGraph/@LeviGraph/@%

Construction

From Ed Pegg, Jr.
Mathematical Games, Episode 12: Configurations
Dec. 2023

linesets={
{{1,5,9},{1,3,7},{4,8,10},{2,5,10},{3,4,6},{3,8,9},{2,4,7},{5,7,8},{1,2,6},{6,9,10}},
{{1,5,6},{2,4,6},{3,4,7},{5,7,8},{1,3,9},{2,8,9},{3,5,10},{4,8,10},{6,9,10},{1,2,7,11},{3,8,11},{4,5,11}},
{{1,3,6},{2,4,6},{1,2,7},{3,4,7},{5,6,8},{2,8,9},{3,5,9},{1,5,10},{7,9,10},{4,8,10,12},{1,4,11},{2,3,11},{2,5,12},{7,11,12}},
{{3,4,5},{1,4,6},{2,5,6},{1,2,8},{3,7,8,11,12},{1,3,9},{6,7,9},{2,4,10},{5,7,10},{8,9,10,13},{1,3,11},{2,4,11},{1,7,12},{2,4,12},{3,6,12},{2,3,13},{5,7,13},{6,11,13}},
{{1,2,6},{3,4,6},{1,3,7},{2,5,7},{5,6,8},{1,8,9},{4,5,9},{2,4,10},{3,8,10},{7,9,10},{1,5,11},{2,3,11},{4,8,11}},
{{2,3,6},{4,5,6},{2,4,7},{3,5,7},{1,6,8},{1,3,9},{4,8,9},{1,2,10},{5,9,10},{7,8,10},{1,6,11},{2,4,11},{1,3,12},{2,5,12},{3,5,13},{4,8,13},{4,9,13},{11,12,13}},
{{1,3,5},{1,2,6},{3,4,6},{1,4,8},{6,7,8},{2,7,9},{4,5,9},{2,3,10},{5,7,10},{8,9,10},{1,4,11},{2,3,11},{2,3,12},{4,5,12},{1,4,13},{2,5,13},{7,12,13},{2,4,14},{5,11,14},{6,7,14}},
{{2,3,5},{1,5,6},{2,4,6},{1,2,8},{4,7,8},{1,3,9},{6,7,9},{3,4,10},{5,7,10},{8,9,10},{1,2,11},{3,4,11},{1,3,12},{4,5,12},{7,11,12},{1,4,13},{2,3,13},{6,7,13}},
{{1,3,5},{1,2,6},{3,4,6},{1,4,8},{5,7,8},{2,7,9},{4,5,9},{2,3,10},{6,7,10},{8,9,10},{1,4,11},{2,3,11},{1,7,12},{2,3,12},{4,5,12},{2,4,13},{5,7,13},{6,11,13}}
};

points={{0.866`,-0.032`},{-2.115`,1.134`},{1.9`,1.373`},{-0.843`,-0.147`},{-1.173`,-1.241`},{-0.014183647417451612`,0.31227847463560826`},{0.08992729344663442`,-1.0865281941078129`},{1.3528545868932689`,-0.9320563882156256`},{1.6812387678260028`,0.4513858118203225`},{-1.7331390925491106`,0.17124028110842537`}};

ConfigurationGraph[points,linesets[[1]],VertexLabels->Automatic]