The generalized quadrangle

The generalized quadrangle (2,2) has 15 lines and 15 points with 2+1=3 points on each line and 2+1=3 lines through each point:

Without loose ends:

The 5 dark blue points are the 5 pairs of antipodal vertices of two antipodal 5-cells. They are in the sections 0₀ ,12₀ ,18₀ and 30₀ of the vertex first projection of a 120-cell. Each pair of vertices belongs to a different copy of a set of 5 inscribed 600-cells. The light blue points are the centres of both the 10 (pairs of) faces of the two 5 cells and the centres of the 10 edges of each. Each plane contains 7 points of which the 3 dark blue lie in 6 planes and the light blue points in 4 planes. Checking: 10x(3/6 + 4/4)=15 points.

Producing 12 of the lines starting from the light blue points we stop at the 12 vertices of section 6₃ of the cell first projection of the 600-cell. Connecting these points by 12 lines we get another 18 points of intersection with the 15 lines of the  generalized quadrangle (2,2). in this way we have got the  generalized quadrangle (4,2) having 4+1=5 points on each line and 2+1=3 lines through each point.

The 12 new lines have a double six configuration: each line intersects 5 lines and is skew to the 6 other lines. Here they are also 12 lines coinciding with edges of a 600-cell.

The 30 points of intersection of the double 6 appear to be the 6, 12 and 12 vertices of resp. the sections  3₃, 4₃ and 6₃ of the cell first projection of the 600-cell. 

In the next POV-Ray image the 30 yellow points have changed color: those in section 3₃  brownred,  4₃  violet  and 6₃ red.

Why did I use 2 colors for the double six? The answer is given after a change of affine choice:

Adding more vertices of the 600-cell in vertex first projection by zooming out:

There are 6 yellow intersection points at infinity which results in 6 sets of 3 parallel lines (a red, a blue and a black one).

I made a copy in POV-Ray where our point of view is only slightly different:

The 5+10 fuzzy points are the vertices of the generalized quadrangle(2,2). The 15 lines of the generalized quadrangle (2,2) by themselves:

 The 6 red lines lie on the edges of this small stellated dodecahedron:

The 6 blue lines lie on the edges of this great stellated dodecahedron:

Which lines? The 12 lines are the fat lines in the next picture:

With POV-Ray:

The 30 points of intersection can be found on the icosahedron (6), dodecahedron (12) , the small stellated dodecahedron (6) and 6 at infinity.

The 5 dark blue points are in sections 1₃ and 15₃ (2x1), 5₃ and 11₃ (2x3), 7₃ and 9₃ (2x1) of the cell first projection of a 120-cell.

Returning to the affine situation we started with, we can give each line a symbol of two digits (call this a duad):

Then we add the double 6:

If we add a c to the 2 digits of the black lines then c_ij intersects a_i, b_j, a_j, b_i. Compare this to the 27 lines on the Clebsch cubic surface:

The connection between the the two is simple: the 12 lines a_i, b_j are identical and the 3 lines c_ij c_kl c_mn of the generalized quadrangle together with the 3 lines c_ij c_kl c_mn of the Clebsch cubic surface make the edges and (pairs of) vertices (the 1+3 dark blue points) of a 16-cell. This 16-cell is inscribed in a 600-cell which is inscribed in a 120-cell that is dual to the 600-cell that has the double six as edges and the 30 yellow points as 30 of its vertices. The line c_ij of the generalized quadrangle together with the line c_ij of the Clebsch cubic surface form a Hopf linked pair of orthogonal lines.

The symbols of the lines in a generalized quadrangle in the second affine choice (not all points are visible:1 dark blue and 3 light blue points are outside the picture):

The line c₁₄ is parallel to b₁ and a₄. The line c₁₆ is parallel to b₆ and a₁. The line c₂₃ is parallel to b₂ and a₃. The line c₂₅ is parallel to b₅ and a₂. The line c₃₅ is parallel to b₃ and a₅. The line c₄₆ is parallel to b₄ and a₆. Of these 6 lines c_ij the line c_ij is orthogonal to c_kl if one of the indices is equal. 

The lines on the Clebsch cubic in the second affine  situation:

The lines c_ij are colored to distinguish 5 tritangents c₁₂ c₃₄ c₅₆ ; c₁₃ c₂₆ c₄₅ ; c₁₄ c₂₅ c₃₆ ; c₁₅ c₂₃ c₄₆ ; c₁₆ c₂₄ c₃₅ . Each tritangent of the Clebsch cubic combined with the corresponding tritangent of the quadrangle make the edges of a 16-cell whose vertices lie on one of the 5 600-cells inscribed in the 120-cell. In the quadrangle these 5 sets of 3 lines meet in the 5 dark blue points. Call the set of 3 lines that meet in a vertex a syntheme. Then call a syntheme through a dark blue point a true cross and a syntheme through a light blue point a skew cross. In the spherical view you can see why the dark blue points are rightly named true cross although these names have their origin elsewhere:

The 3 lines are mutually perpendicular at the dark blue points. At the light blue points they lie in one plane making angles of 120°. 

The names for the crosses comes from the icosahedron and its rotation group. The identity element and 3 twofold rotations are represented by the dark blue points in the next picture:

Then we make 5 copies of it in different directions to get all 15 twofold rotations:

The 5 copies represent the 5 true crosses because the 3 duads correspond to 3 planes through opposite edges of the icosahedron that are mutually perpendicular. The skew crosses correspond to 3 planes that are not perpendicular. By means of a Clifford displacement or double rotation on the sphere S3 we can change the 5 octahedra with one shared centre into 5 disjointed 16-cells.   

Only the 15 lines of  the Clebsch cubic:

The 5 16-cells in spherical view:

To see some sections draw some polytopes:

There are also another 2 tetrahedra (one with yellow and one with light blue vertices), 2 octahedra (one with light blue and one with dark blue vertices) and a cuboctahedron with yellow vertices.

Now in projective space:

Separately 15 lines of the generalized quadrangle (2,2) :

Separately 12 of the 15 lines of the Clebsch surface (3 are at infinity):

We have a configuration of 30 lines and 40 points (9 at infinity) with 3 lines through each point and 4 points on each line. 

In the second affine choice:

Returning to the generalized quadrangle we investigate sets of synthemes. In the following table the first column and row marked by A represent the 5 true crosses.

In the other columns and rows B, C, D, E and F one finds one true cross and 4 skew crosses. Such a set of 5 crosses is called a pentad. There are 6 pentads A, B, C, D, E and F. Here is a previous picture again, but now the vertices, representing the synthemes, are marked by the letters of the pentads to which the lines belong that pass through it :

The pentads A and B share the syntheme 12.34.56 and AB is a short notation for the syntheme 12.34.56 . Likewise AB.CD.EF is a notation for the line 12. The permutation (AB) equals the permutation (12)(34)(56). One can check that (12) equals (AB)(CD)(EF) . One can make a similar table in which  letters and numbers are exchanged emphasizing the symmetry between letters and numbers. This symmetry is seen in the picture as a symmetry between lines and points.