Bokowski-Pilaud Configuration

Author

Eric W. Weisstein
July 31, 2026

This notebook downloaded from https://mathworld.wolfram.com/notebooks/Configurations/Bokowski-PilaudConfiguration.nb.

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

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

Sources

Source: J. Bokowski and V. Pilaud, Enumerating Topological (n_k)-Configurations, Computational Geometry 47 (2014), 175-186, https://doi.org/10.1016/j.comgeo.2012.10.002; https://arxiv.org/abs/1210.0306.

Configuration data

The incidence table is exact. The homogeneous point coordinates were digitized from the paper's vector figure and are approximate; points 17 and 18 are its two ideal points. The paper proves an exact realization over Q[cuberoot(108+12 sqrt(93))].

homogeneousPoints={{99.758`,-425.351`,1},{357.797`,-348.992`,1},{478.019`,-313.351`,1},{654.199`,-261.18`,1},{253.098`,-425.621`,1},{406.547`,-348.992`,1},{357.797`,-537.801`,1},{478.039`,-425.582`,1},{357.797`,-184.531`,1},{477.98`,-72.371`,1},{429.25`,-261.09`,1},{582.66`,-184.531`,1},{181.578`,-348.992`,1},{357.797`,-296.75`,1},{477.758`,-261.18`,1},{736.328`,-184.531`,1},{1,0,0},{-0.7311061496077306`,-0.6822637305366297`,0}}

incidence={{13,14,15,16},{9,12,16,17},{4,8,7,16},{3,6,5,16},{4,15,11,17},{4,12,10,18},{3,11,9,18},{3,8,10,15},{14,9,7,2},{14,6,8,18},{13,5,7,18},{13,2,6,17},{14,11,12,1},{13,9,10,1},{8,5,1,17},{4,3,2,1},{2,5,12,15},{6,7,10,11}}

Illustration

The displayed affine chart is chosen only for illustration; the incidence data above define the configuration.

displayPoints={{0.19292836452897585`,-0.822613452362361`},{0.4298476298714171`,-0.4192695412317196`},{0.4881063273883904`,-0.3199634445356346`},{0.5476109160736019`,-0.2186261658304329`},{0.37763970336174363`,-0.6350559395353921`},{0.4613922333909487`,-0.3960727746498534`},{0.5179651673466404`,-0.778548129146389`},{0.5340140898958367`,-0.4754147347937093`},{0.37437164963730485`,-0.19307924571536794`},{0.41204115670309327`,-0.062387193087073856`},{0.4426350629872015`,-0.2692314236350109`},{0.49353299907948545`,-0.15630408446287122`},{0.27672739354000997`,-0.5318686543871789`},{0.4105236285406957`,-0.34048045894585877`},{0.46921776741527466`,-0.2565112389400522`},{0.551863386216044`,-0.13830236324278425`},{1.`29.397940008672037,0},{0.5882715057933372`,0.5489713256088838`}}

segments={{{0.27672739354000997`,-0.5318686543871789`},{0.551863386216044`,-0.13830236324278425`}},{{0.37437164963730485`,-0.19307924571536794`},{1.`29.397940008672037,0}},{{0.5179651673466404`,-0.778548129146389`},{0.551863386216044`,-0.13830236324278425`}},{{0.37763970336174363`,-0.6350559395353921`},{0.551863386216044`,-0.13830236324278425`}},{{0.4426350629872015`,-0.2692314236350109`},{1.`29.397940008672037,0}},{{0.5476109160736019`,-0.2186261658304329`},{0.5882715057933372`,0.5489713256088838`}},{{0.4881063273883904`,-0.3199634445356346`},{0.5882715057933372`,0.5489713256088838`}},{{0.5340140898958367`,-0.4754147347937093`},{0.41204115670309327`,-0.062387193087073856`}},{{0.37437164963730485`,-0.19307924571536794`},{0.5179651673466404`,-0.778548129146389`}},{{0.5340140898958367`,-0.4754147347937093`},{0.5882715057933372`,0.5489713256088838`}},{{0.5179651673466404`,-0.778548129146389`},{0.5882715057933372`,0.5489713256088838`}},{{0.27672739354000997`,-0.5318686543871789`},{1.`29.397940008672037,0}},{{0.49353299907948545`,-0.15630408446287122`},{0.19292836452897585`,-0.822613452362361`}},{{0.41204115670309327`,-0.062387193087073856`},{0.19292836452897585`,-0.822613452362361`}},{{0.19292836452897585`,-0.822613452362361`},{1.`29.397940008672037,0}},{{0.5476109160736019`,-0.2186261658304329`},{0.19292836452897585`,-0.822613452362361`}},{{0.37763970336174363`,-0.6350559395353921`},{0.49353299907948545`,-0.15630408446287122`}},{{0.5179651673466404`,-0.778548129146389`},{0.41204115670309327`,-0.062387193087073856`}}}

configurationGraphic=Graphics[{StandardGray,Line/@segments,RGBColor[0.16,0.48,0.72],PointSize[0.016],Point[displayPoints]}]

Checks

The result gives the point count, line count, line sizes, point degrees, and maximum normalized 3 by 3 determinant residual.

{Length[homogeneousPoints],Length[incidence],Union[Length/@incidence],Union[Values[Counts[Flatten[incidence]]]],Max[Flatten[Function[line,(Quiet[Abs[Det[#1]]]&)/@Subsets[(Normalize/@N[homogeneousPoints,50])[[line]],{3}]]/@incidence]]}