Klein Configuration

Author

Eric W. Weisstein
December 15, 2023

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

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

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

Sources

https://demonstrations.wolfram.com/TheKleinConfiguration/

Construction

https://arxiv.org/abs/2010.08863

In an abstract projective space, there are 60 points and 60 planes such that 15 points are on each plane and 15 planes go through each point. The configuration is built from what Klein called the 15 fundamental tetrahedra, which define 60 distinct planes. An abstract presentation is here.

The following can be considered the points OR planes:

points={{0,0,1,1},{0,0,1,I},{0,0,1,-1},{0,0,1,-I},{0,1,0,1},{0,1,0,I},{0,1,0,-1},{0,1,0,-I},{0,1,1,0},{0,1,I,0},{0,1,-1,0},{0,1,-I,0},{1,0,0,1},{1,0,0,I},{1,0,0,-1},{1,0,0,-I},{1,0,1,0},{1,0,I,0},{1,0,-1,0},{1,0,-I,0},{1,1,0,0},{1,I,0,0},{1,-1,0,0},{1,-I,0,0},{1,0,0,0},{0,1,0,0},{0,0,1,0},{0,0,0,1},{1,1,1,1},{1,1,1,-1},{1,1,-1,1},{1,1,-1,-1},{1,-1,1,1},{1,-1,1,-1},{1,-1,-1,1},{1,-1,-1,-1},{1,1,I,I},{1,1,-I,I},{1,1,I,-I},{1,1,-I,-I},{1,-1,I,I},{1,-1,I,-I},{1,-1,-I,I},{1,-1,-I,-I},{1,I,1,I},{1,I,1,-I},{1,-I,1,I},{1,-I,1,-I},{1,I,-1,I},{1,I,-1,-I},{1,-I,-1,I},{1,-I,-1,-I},{1,I,I,1},{1,I,-I,1},{1,-I,I,1},{1,-I,-I,1},{1,I,I,-1},{1,I,-I,-1},{1,-I,I,-1},{1,-I,-I,-1}};

Length[points]

Each point is on 15 planes, and vice-versa:

planes=Table[Select[points, points[[k]].# ==0&],{k,60}]

Tally[Length/@planes]

LeviGraph[planes]//RecognizeGraph

Original construction

Ed Pegg, Jr.
Dec 15, 2022

In an abstract projective space, there are 60 points and 60 planes such that 15 points are on each plane and 15 planes go through each point. The configuration is built from what Klein called the 15 fundamental tetrahedra, which define 60 distinct planes. An abstract presentation is here.

More recently, Piotr Pokora, Tomasz Szemberg, Justyna Szpond, “Unexpected properties of the Klein configuration of 60 points in (ℙ^3)” found a less abstract set of points. arxiv.

The following can be considered the points OR planes:

points={{0,0,1,1},{0,0,1,I},{0,0,1,-1},{0,0,1,-I},{0,1,0,1},{0,1,0,I},{0,1,0,-1},{0,1,0,-I},{0,1,1,0},{0,1,I,0},{0,1,-1,0},{0,1,-I,0},{1,0,0,1},{1,0,0,I},{1,0,0,-1},{1,0,0,-I},{1,0,1,0},{1,0,I,0},{1,0,-1,0},{1,0,-I,0},{1,1,0,0},{1,I,0,0},{1,-1,0,0},{1,-I,0,0},{1,0,0,0},{0,1,0,0},{0,0,1,0},{0,0,0,1},{1,1,1,1},{1,1,1,-1},{1,1,-1,1},{1,1,-1,-1},{1,-1,1,1},{1,-1,1,-1},{1,-1,-1,1},{1,-1,-1,-1},{1,1,I,I},{1,1,-I,I},{1,1,I,-I},{1,1,-I,-I},{1,-1,I,I},{1,-1,I,-I},{1,-1,-I,I},{1,-1,-I,-I},{1,I,1,I},{1,I,1,-I},{1,-I,1,I},{1,-I,1,-I},{1,I,-1,I},{1,I,-1,-I},{1,-I,-1,I},{1,-I,-1,-I},{1,I,I,1},{1,I,-I,1},{1,-I,I,1},{1,-I,-I,1},{1,I,I,-1},{1,I,-I,-1},{1,-I,I,-1},{1,-I,-I,-1}};

From https://en.wikipedia.org/wiki/Levi_graph:
From a collection of points and lines in an incidence geometry or a projective configuration, we form a graph with one vertex per point, one vertex per line, and an edge for every incidence between a point and a line.
...
Levi graphs may also be defined for other types of incidence structure, such as the incidences between points and planes in Euclidean space.

Denote point vertices as above, place vertices as 2*point vertices,  and create edges when points lie on a plane:

edges=Flatten[Table[UndirectedEdge[2points[[k]],#]&/@Select[points,points[[k]].#==0&],{k,60}],1]

vertices=Union[Flatten[List@@@edges,1]]

Length[vertices]

Union[points,2points]===vertices

kc=IndexGraph@Graph[vertices,edges]

kc2=Graph[VertexList[kc],Sort[Sort/@EdgeList[kc]]]

kc2=;

VertexCount[kc]

RecognizeGraph@kc

Factor[CharacteristicPolynomial[AdjacencyMatrix[kc],x]]

DistanceRegularGraphQ@kc

DistanceTransitiveGraphQ@kc