Bokowski-Schewe Configuration

Author

Eric W. Weisstein
July 31, 2026

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

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

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

Sources

Sources: J. Bokowski and L. Schewe, There Are No Realizable 15_4- and 16_4-Configurations, Revue Roumaine de Mathematiques Pures et Appliquees 50 (2005), 483-493; 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; attached source notebook.

Configuration data

The first 15 points are exact affine coordinates from the attached source notebook. Points 16 through 18 are the three corresponding ideal points, represented by homogeneous direction vectors.

homogeneousPoints={{0,,1},{(1/4) (-21-9 (Sqrt[5])),,1},{(1/4) (21+9 (Sqrt[5])),,1},{0,-2 (Sqrt[3]),1},{-3,(Sqrt[3]),1},{3,(Sqrt[3]),1},{0,(Sqrt[3]),1},{(3/2),-((Sqrt[3])/2),1},{-(3/2),-((Sqrt[3])/2),1},{(1/4) (3+3 (Sqrt[5])),,1},{(1/4) (9+3 (Sqrt[5])),,1},{(1/2) (-6-3 (Sqrt[5])),-((Sqrt[3])/2),1},{0,,1},{(1/2) (21+9 (Sqrt[5])),,1},{(1/2) (-21-9 (Sqrt[5])),,1},{3,0,0},{(3/2),-((3 (Sqrt[3]))/2),0},{(3/2),((3 (Sqrt[3]))/2),0}}

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

Illustration

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

displayPoints={{0,-1.0000000000000000000000000000000000000000181125189159118466`29.221848749616356},{-0.5300115471747448394380559867058800833960979782876977001055`29.25900279073188,0.3060023094349489678876111973411760166792218126465867213104`29.25900279073188},{0.2572842744474721121653287139786073561233662148532557384906`29.374946735644876,0.14854314511050557756693425720427852877532783322555910953`29.374946735644876},{0,-0.1708203932499369089227521006193828706321873191353493084658`29.454388466948387},{-0.1334795290170525955395753273598256663723350933788904927167`29.439898176180584,0.0770644420092997793581017222753413449405771340648061155024`29.439898176180584},{0.1053542560595193822795682266745877938128343035792427661231`29.456112946190718,0.0608263080962362767194015322053926799825283705289967776895`29.456112946190718},{0,0.0679892667105595216196041575687458575620082012952736760046`29.494311857224314},{0.0615327051373106308624731252537048428092678146788258857149`29.482648907517717,-0.0355259238749921609189871159347638698427320600712627629047`29.482648907517717},{-0.070167953070000628743830841105270299210703204079483543222`29.477290012778344,-0.0405114865934498905803608106445627500135317144877377961813`29.477290012778344},{0.075594503891805692591080560092454518044942650975602933138`29.42234430108758,0.1848810992216388614817838879815090963910128092698421519758`29.42234430108758},{0.1737590457051265749823842732799874521463196167176993816256`29.377346408779104,-0.2243219634229495372535578337644690115455009945167659223442`29.377346408779104},{-0.3845568838263210716370998204209228849835109087512266911931`29.365431151138942,-0.0524127614512203931398804405079652517758107488313128950327`29.365431151138942},{0,0.4999999999999999999999999999999999999999999999999999999998`29.346787486224652},{0.6339745962155613532362768292470638165286154927559753715517`29.221848749616356,-0.3660254037844386467637231707529361834713911368860743783648`29.221848749616356},{2.36602540378443864676372317075293618347`28.847829346321834,1.36602540378443864676372317075293618347`28.847829346321834},{1.`29.397940008672037,0},{-1.3660254037844386467637231707529361834711854098087089524093`29.02392060537752,2.36602540378443864676372317075293618347`29.02392060537752},{0.3660254037844386467637231707529361834714026269051903140278`29.397940008672037,0.6339745962155613532362768292470638165285973730948096859722`29.397940008672037}}

segments={{{-0.5300115471747448394380559867058800833960979782876977001055`29.25900279073188,0.3060023094349489678876111973411760166792218126465867213104`29.25900279073188},{0.6339745962155613532362768292470638165286154927559753715517`29.221848749616356,-0.3660254037844386467637231707529361834713911368860743783648`29.221848749616356}},{{2.36602540378443864676372317075293618347`28.847829346321834,1.36602540378443864676372317075293618347`28.847829346321834},{-0.3845568838263210716370998204209228849835109087512266911931`29.365431151138942,-0.0524127614512203931398804405079652517758107488313128950327`29.365431151138942}},{{-0.070167953070000628743830841105270299210703204079483543222`29.477290012778344,-0.0405114865934498905803608106445627500135317144877377961813`29.477290012778344},{2.36602540378443864676372317075293618347`28.847829346321834,1.36602540378443864676372317075293618347`28.847829346321834}},{{0,0.4999999999999999999999999999999999999999999999999999999998`29.346787486224652},{0,-1.0000000000000000000000000000000000000000181125189159118466`29.221848749616356}},{{-0.3845568838263210716370998204209228849835109087512266911931`29.365431151138942,-0.0524127614512203931398804405079652517758107488313128950327`29.365431151138942},{0.6339745962155613532362768292470638165286154927559753715517`29.221848749616356,-0.3660254037844386467637231707529361834713911368860743783648`29.221848749616356}},{{0,0.4999999999999999999999999999999999999999999999999999999998`29.346787486224652},{0.1737590457051265749823842732799874521463196167176993816256`29.377346408779104,-0.2243219634229495372535578337644690115455009945167659223442`29.377346408779104}},{{-0.1334795290170525955395753273598256663723350933788904927167`29.439898176180584,0.0770644420092997793581017222753413449405771340648061155024`29.439898176180584},{1.`29.397940008672037,0}},{{-0.3845568838263210716370998204209228849835109087512266911931`29.365431151138942,-0.0524127614512203931398804405079652517758107488313128950327`29.365431151138942},{1.`29.397940008672037,0}},{{0,-1.0000000000000000000000000000000000000000181125189159118466`29.221848749616356},{2.36602540378443864676372317075293618347`28.847829346321834,1.36602540378443864676372317075293618347`28.847829346321834}},{{-0.5300115471747448394380559867058800833960979782876977001055`29.25900279073188,0.3060023094349489678876111973411760166792218126465867213104`29.25900279073188},{1.`29.397940008672037,0}},{{0,-0.1708203932499369089227521006193828706321873191353493084658`29.454388466948387},{-1.3660254037844386467637231707529361834711854098087089524093`29.02392060537752,2.36602540378443864676372317075293618347`29.02392060537752}},{{0.1737590457051265749823842732799874521463196167176993816256`29.377346408779104,-0.2243219634229495372535578337644690115455009945167659223442`29.377346408779104},{-1.3660254037844386467637231707529361834711854098087089524093`29.02392060537752,2.36602540378443864676372317075293618347`29.02392060537752}},{{0.6339745962155613532362768292470638165286154927559753715517`29.221848749616356,-0.3660254037844386467637231707529361834713911368860743783648`29.221848749616356},{-1.3660254037844386467637231707529361834711854098087089524093`29.02392060537752,2.36602540378443864676372317075293618347`29.02392060537752}},{{0,-1.0000000000000000000000000000000000000000181125189159118466`29.221848749616356},{-1.3660254037844386467637231707529361834711854098087089524093`29.02392060537752,2.36602540378443864676372317075293618347`29.02392060537752}},{{-0.070167953070000628743830841105270299210703204079483543222`29.477290012778344,-0.0405114865934498905803608106445627500135317144877377961813`29.477290012778344},{0.3660254037844386467637231707529361834714026269051903140278`29.397940008672037,0.6339745962155613532362768292470638165285973730948096859722`29.397940008672037}},{{0,-0.1708203932499369089227521006193828706321873191353493084658`29.454388466948387},{0.3660254037844386467637231707529361834714026269051903140278`29.397940008672037,0.6339745962155613532362768292470638165285973730948096859722`29.397940008672037}},{{-0.5300115471747448394380559867058800833960979782876977001055`29.25900279073188,0.3060023094349489678876111973411760166792218126465867213104`29.25900279073188},{2.36602540378443864676372317075293618347`28.847829346321834,1.36602540378443864676372317075293618347`28.847829346321834}},{{0,-1.0000000000000000000000000000000000000000181125189159118466`29.221848749616356},{0.3660254037844386467637231707529361834714026269051903140278`29.397940008672037,0.6339745962155613532362768292470638165285973730948096859722`29.397940008672037}}}

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]]}