Grünbaum (Subscript[20, 4]) Configuration

Author

Eric W. Weisstein
July 31, 2026

This notebook downloaded from https://mathworld.wolfram.com/notebooks/Configurations/Gruenbaum20-4Configuration.nb.

For more information, see Eric's MathWorld entry https://mathworld.wolfram.com/Gruenbaum20-4Configuration.html.

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

Sources

Sources: B. Grunbaum, Musings on an Example of Danzer's, European Journal of Combinatorics 29 (2008), 1910-1918, https://doi.org/10.1016/j.ejc.2008.01.004

Configuration data

Exact affine coordinates from the attached source notebook are stored as homogeneous projective coordinates {x,y,1}.

homogeneousPoints={{0,1,1},{,(1/4) (-1+(Sqrt[5])),1},{,(1/4) (-1-(Sqrt[5])),1},{,(1/4) (-1-(Sqrt[5])),1},{,(1/4) (-1+(Sqrt[5])),1},{,,1},{,,1},{,,1},{,,1},{,,1},{,(1/4) (-1+(Sqrt[5])),1},{,(1/4) (-1+(Sqrt[5])),1},{,,1},{,,1},{,,1},{0,,1},{,,1},{,,1},{,,1},{,,1}}

incidence={{1,3,13,14},{1,4,10,9},{1,6,12,17},{1,7,11,20},{2,4,15,6},{2,5,12,11},{2,8,14,18},{2,9,13,16},{3,5,7,8},{3,10,6,19},{3,11,15,17},{4,13,7,18},{4,12,8,20},{5,15,9,19},{5,14,10,16},{6,11,18,16},{7,12,19,16},{8,13,19,17},{9,14,20,17},{10,15,20,18}}

Illustration

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

displayPoints={{0,1.`30.},{0.9510565162951535721164393333793821434056986341257502224475`30.,0.3090169943749474241022934171828190588601545899028814310677`30.},{0.5877852522924731291687059546390727685976524376431459910721`30.,-0.8090169943749474241022934171828190588601545899028814310677`30.},{-0.5877852522924731291687059546390727685976524376431459910721`30.,-0.8090169943749474241022934171828190588601545899028814310677`30.},{-0.9510565162951535721164393333793821434056986341257502224475`30.,0.3090169943749474241022934171828190588601545899028814310677`30.},{0.1267183905942635837068362809603162391784062379064743102951`30.,-0.2898997114029781425578756881127874826255656742599984543246`30.},{-0.1267183905942635837068362809603162391784062379064743102951`30.,-0.2898997114029781425578756881127874826255656742599984543246`30.},{-0.23655287340841685924089709777999313562963995857612992108`30.,-0.2101002885970218574421243118872125173744343257400015456754`30.},{-0.3148691457953567400818197815283479592951189565633237611983`30.,0.0309324136211959008431580543544143741884788568011921385423`30.},{-0.2729161064971163890868861731107248094862895015374730163342`30.,0.1600505920038566750545485284627665669513665532959264303898`30.},{-0.0678814435159129245391272084020537468261602861014556523811`30.,0.3090169943749474241022934171828190588601545899028814310677`30.},{0.0678814435159129245391272084020537468261602861014556523811`30.,0.3090169943749474241022934171828190588601545899028814310677`30.},{0.2729161064971163890868861731107248094862895015374730163342`30.,0.1600505920038566750545485284627665669513665532959264303898`30.},{0.3148691457953567400818197815283479592951189565633237611983`30.,0.0309324136211959008431580543544143741884788568011921385423`30.},{0.23655287340841685924089709777999313562963995857612992108`30.,-0.2101002885970218574421243118872125173744343257400015456754`30.},{0,0.1000993931863303647751515668488919008461355448881502305713`30.},{0.0952001801670501890136981478889941662088954947663617531683`30.,0.0309324136211959008431580543544143741884788568011921385423`30.},{0.0588369470783506591677090725582624923522459518050186579142`30.,-0.080982110214361083230733837778860324611546629245267253828`30.},{-0.0588369470783506591677090725582624923522459518050186579142`30.,-0.080982110214361083230733837778860324611546629245267253828`30.},{-0.0952001801670501890136981478889941662088954947663617531683`30.,0.0309324136211959008431580543544143741884788568011921385423`30.}}

segments={{{0,1.`30.},{0.5877852522924731291687059546390727685976524376431459910721`30.,-0.8090169943749474241022934171828190588601545899028814310677`30.}},{{0,1.`30.},{-0.5877852522924731291687059546390727685976524376431459910721`30.,-0.8090169943749474241022934171828190588601545899028814310677`30.}},{{0,1.`30.},{0.1267183905942635837068362809603162391784062379064743102951`30.,-0.2898997114029781425578756881127874826255656742599984543246`30.}},{{0,1.`30.},{-0.1267183905942635837068362809603162391784062379064743102951`30.,-0.2898997114029781425578756881127874826255656742599984543246`30.}},{{0.9510565162951535721164393333793821434056986341257502224475`30.,0.3090169943749474241022934171828190588601545899028814310677`30.},{-0.5877852522924731291687059546390727685976524376431459910721`30.,-0.8090169943749474241022934171828190588601545899028814310677`30.}},{{0.9510565162951535721164393333793821434056986341257502224475`30.,0.3090169943749474241022934171828190588601545899028814310677`30.},{-0.9510565162951535721164393333793821434056986341257502224475`30.,0.3090169943749474241022934171828190588601545899028814310677`30.}},{{0.9510565162951535721164393333793821434056986341257502224475`30.,0.3090169943749474241022934171828190588601545899028814310677`30.},{-0.23655287340841685924089709777999313562963995857612992108`30.,-0.2101002885970218574421243118872125173744343257400015456754`30.}},{{0.9510565162951535721164393333793821434056986341257502224475`30.,0.3090169943749474241022934171828190588601545899028814310677`30.},{-0.3148691457953567400818197815283479592951189565633237611983`30.,0.0309324136211959008431580543544143741884788568011921385423`30.}},{{0.5877852522924731291687059546390727685976524376431459910721`30.,-0.8090169943749474241022934171828190588601545899028814310677`30.},{-0.9510565162951535721164393333793821434056986341257502224475`30.,0.3090169943749474241022934171828190588601545899028814310677`30.}},{{0.5877852522924731291687059546390727685976524376431459910721`30.,-0.8090169943749474241022934171828190588601545899028814310677`30.},{-0.2729161064971163890868861731107248094862895015374730163342`30.,0.1600505920038566750545485284627665669513665532959264303898`30.}},{{0.5877852522924731291687059546390727685976524376431459910721`30.,-0.8090169943749474241022934171828190588601545899028814310677`30.},{-0.0678814435159129245391272084020537468261602861014556523811`30.,0.3090169943749474241022934171828190588601545899028814310677`30.}},{{-0.5877852522924731291687059546390727685976524376431459910721`30.,-0.8090169943749474241022934171828190588601545899028814310677`30.},{0.2729161064971163890868861731107248094862895015374730163342`30.,0.1600505920038566750545485284627665669513665532959264303898`30.}},{{-0.5877852522924731291687059546390727685976524376431459910721`30.,-0.8090169943749474241022934171828190588601545899028814310677`30.},{0.0678814435159129245391272084020537468261602861014556523811`30.,0.3090169943749474241022934171828190588601545899028814310677`30.}},{{-0.9510565162951535721164393333793821434056986341257502224475`30.,0.3090169943749474241022934171828190588601545899028814310677`30.},{0.23655287340841685924089709777999313562963995857612992108`30.,-0.2101002885970218574421243118872125173744343257400015456754`30.}},{{-0.9510565162951535721164393333793821434056986341257502224475`30.,0.3090169943749474241022934171828190588601545899028814310677`30.},{0.3148691457953567400818197815283479592951189565633237611983`30.,0.0309324136211959008431580543544143741884788568011921385423`30.}},{{0.1267183905942635837068362809603162391784062379064743102951`30.,-0.2898997114029781425578756881127874826255656742599984543246`30.},{-0.0678814435159129245391272084020537468261602861014556523811`30.,0.3090169943749474241022934171828190588601545899028814310677`30.}},{{-0.1267183905942635837068362809603162391784062379064743102951`30.,-0.2898997114029781425578756881127874826255656742599984543246`30.},{0.0678814435159129245391272084020537468261602861014556523811`30.,0.3090169943749474241022934171828190588601545899028814310677`30.}},{{-0.23655287340841685924089709777999313562963995857612992108`30.,-0.2101002885970218574421243118872125173744343257400015456754`30.},{0.2729161064971163890868861731107248094862895015374730163342`30.,0.1600505920038566750545485284627665669513665532959264303898`30.}},{{-0.3148691457953567400818197815283479592951189565633237611983`30.,0.0309324136211959008431580543544143741884788568011921385423`30.},{0.3148691457953567400818197815283479592951189565633237611983`30.,0.0309324136211959008431580543544143741884788568011921385423`30.}},{{-0.2729161064971163890868861731107248094862895015374730163342`30.,0.1600505920038566750545485284627665669513665532959264303898`30.},{0.23655287340841685924089709777999313562963995857612992108`30.,-0.2101002885970218574421243118872125173744343257400015456754`30.}}}

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]]}