# The Klein Configuration

The Klein Configuration

Take three points , , on one line and three points , , on another line. Join them with six lines and define three points , , . Pappus's theorem states that , , lie on a line. These nine points and nine lines represent a "configuration". Each point is on three lines, and each line goes through three points. This is called the Pappus configuration.

A

B

C

A'

B'

C'

A'''=BC'⋂CB'

B'''=AC'⋂CA'

C'''=AB'⋂BA'

A'''

B'''

C'''

9

3

The extreme case is the Klein configuration. 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.

60

15

The construction starts by defining lines coded as all possible pairs of integers 1–6: 12, 13, 14, 15, 16, 23, …, 56 and their opposites 21, 31, 41, 51, 61, 32, …, 65. The line and its reverse are opposite edges of a fundamental tetrahedron. Each such pair contains opposite edges of three fundamental tetrahedra.

ab

ba

A point is a triple of lines such that is an odd permutation. A point is a vertex of a fundamental tetrahedron.

ab-cd-ef

abcdef

A plane is also a triple of lines, but now is an even permutation. A plane is a face of a fundamental tetrahedron.

uv-wx-yz

uvwxyz

The graphic is only a kind of schematic representation; it shows 15 of the 60 vertices of the buckyball, which are not coplanar. A perfect 3D representation may be impossible. Still, the icosidodecahedron has more than 7,071,672 stellations, and there are four 60-faced icosahedra, each with an unknown number of stellations. What is the nicest or densest point and plane packing in 3D space? It might be the Klein .

60

15