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Icosians and ADE

Icosians and E8, E7, E6, D5 en D4

 

The 240 vertices of two 600-cells represent the roots of E8. One 600-cell has its vertices on a sphere of unit radius and the other on a sphere of radius -sigma=tau^-1 where tau=1,6180339... is the golden mean. The vertices of a vertex first projection of the 600-cell represent the 120 roots of the binary icosahedral group.

Figure 1

The 120 vertices of one 600-cell are the 24 roots of the binary tetrahedral group and 96 vertices of 4 24-cells made by dividing the edges of one 24-cell in the the golden mean 0,6180339....:

Figure 2

The 24 roots of the binary tetrahedral group:

Figure3

The 96 vertices of 4 24-cells made by dividing the edges of one 24-cell in the the golden mean 0,6180339....:

Figure 4

Then 24 vertices in the neighbourhood of the central octahedral cell of the 24-cell:

Figure 5

Then 48 vertices in the neighbourhood of the cuboctahedron in the section of the 24-cell with fourth coordinate x4=0:

Figure 6

Then 24 vertices in the neighbourhood of the outermost octahedral cell of the 24-cell:

Figure 7

The 96 vertices fall apart in 24 tetrahedra around each vertex of the 24-cell that has its edges divided.

A possible choice of the roots of E7 consists of 2 of the 4 vertices of each tetrahedron, a complete set of 24 roots of the tetrahedral group and another 6 vertices with fourth coordinate x4=0 (the rotations around twofold symmetry axes). This makes 2x48 + 24 + 6 = 126 roots of E7. Which ones?

On the sphere with radius -sigma there are 54:

In figure 3 the 6 green vertices of a (edgeless) octahedron.

In figure 5 the 12 inner vertices and in figure 7 the 12 outer vertices that make the vertices of a icosahedron. 

In figure 6 the 24 vertices on the edges of the cuboctahedron.

On the sphere with radius 1 there are 72:

All 24 vertices in figure 3.

In figure 5 the 12 outer vertices and in figure 7 the 12 inner vertices that make 12 of the 20 vertices of a dodecahedron.  

In figure 6 the 24 vertices on the edges of the cuboctahedron.

These 126 root vectors are all perpendicular in 8-dim space to the root corresponding to the central vertex (unit element of the icosahedral group) on the sphere with radius -sigma. We now choose a vertex of the dodecahedron and choose the straight line through the centre as the line on which the 6 vertices lie that correspond to a A2 subgroup of E8. All root vectors perpendicular to these 6 root vectors make the 72 root vectors of a E6 subgroup.

Now for the roots of E6, we keep up with those vertices that lie in sets of 6 on equatorial (or great or geodesic) circles. On 12 of these circles we find the 72 roots of E6. One of them is the straight line in the same direction as the one of the A2 subgroup but now on the sphere with radius 1. In the plane perpendicular to this line lies a circle whose 6 vertices have x4=0. The other 9 circles lie in sets of 3 on 3 tori with different radius but with the same symmetry axis which is the same straight line. All 12 circles are fibres of one Hopf fibration of the 3-sphere.

The short roots (on sphere of radius -sigma) on 3 circles correspond to vertices of opposite faces of the 2 icosahedra (that are inscribed in the the edges of 2 opposite octahedral cells of the 24-cell) and opposite triangular faces of the icosidodecahedron. Together with a circle of 6 edges of the (spherical) cuboctahedron they make 24 vertices and 4x6=24 edges of the spherical 24-cell. The remaining 72 edges are shortened or elongated because of the 4-dim double rotation that caused the transport of 24 vertices along the 4 circles of the 24-cell to a position on its edges dividing the edges in the golden mean. The 24 vertices are therefore of a deformed 24-cell. The 3 circles through the vertices of the icosahedra lie on a fatter torus ( like an inflated tyre).

As for the long roots (on the sphere of radius 1), 4 circles pass through the 24 vertices of the 24-cell corresponding to the tetrahedral group. The other 4 circles are but for one, i.e. the straight line, the same as those of the tetrahedral group but then rotated over an angle of 1/2*arccos(1/4). The cubic cells of the hypercubes are 2 of 5 cubes of a compound of 5 cubes in a dodecahedron. The 2*6 vertices form 2 octahedra that are 2 of 5 octahedra in a compound inscribed in the 30 vertices of a icosidodecahedron. but the 4th circle is not the straight line but the circle with x4=0 in the plane perpendicular to it. Therefore it should be regarded as a deformed 24-cell, dual to the one of the tetrahedral group, with the 3 circles on a deflated torus.

For D5, we keep the 24 vertices of the tetrahedral group but of each triangle with its plane perpendicular to the straight axis we skip 2 of the 3 vertices. To put it another way, we skip 4 of the 6 vertices on each circle. The short roots lie in 2 sets of 4 vertices on 2 circles that have 10 vertices and 10 edges of the spherical 600-cell on it. The long roots lie in 2 sets of 4 vertices on 2 circles that have 6 vertices and 6 edges of inscribed 24-cells on it. These 16 vertices lie on 2 deformed 16-cells, one with short roots and one with long roots. Let me remind you that each 24-cell has 3 inscribed 16-cells on its 3x8 vertices.

It should be remarked that a 600-cell does contain more then 3 regular 16-cells and hypercubes and more than 1 24-cell that are not deformed.

In figures 5.2 and 6.2 below we mark each vertex with 2 digits: the left one gives the lowest dimension in which the vertex survives as representation of a long root of the ADE-ladder E8-E7-E6-D5-D4. The one at the right gives the lowest dimension in which the vertex survives as representation of a short root of the ADE-ladder E8-E7-E6-D5-D4. 

Figure 5.2

Figure 6.2

For example, in each figure there are 24 sevens indicating 48 of the 54 roots in E7 but not in E6. Together with the 6 green vertices of the edgeless octahedron in figure 3 they are twice the 27-dim representation of E6 representing the 27 lines on the cubic surface. 

Each tetrahedron (of the 24) has four 8's , two 7's and at most 2 of the digits 5 and 6. 

For a notation of the roots of E6 and E7 by the 36 double sixes and the 27 lines look at this page (to be published).

Finally an illustration of 3 sections of the 600-cell with edges.

Sections 7o, 8o en 9o

 

References:

1)  A Highly Symmetric Four-Dimensional Quasicrystal Veit Elser and Sloane

2)  J. H. Conway and J. Sloane       Sphere packings, Lattices and Groups                 Springer Verlag

3) Coxeter H S M, Regular Polytopes , Dover

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