    24-cell in stereographic projection

-24-cell in stereographic projection

-hypercube

-16-cell

-Triality

-Duality

-its diagonals

You can look on the page that describes the 24-cell in isometric projection:

- 24-cell in isometric projection

-hypercube

-16-cell

-The gem of the modular universe The 24-cell has 24 vertices, 96 edges, 96 triangles and 24 octahedral cells.  I've chosen for this projection for two reasons:

- We're not very accustomed to looking in 4 dimensions.

- The stereographic projection has to my mind a fundamental physical meaning: it's the separation between observer and object, it's perception itself causing the coming into existence of space and time.

Below is given a stereographic projection of the 4-dimensional 24-cell. The same figure in a different colouring code looks like this: The different colourings illustrate different properties of the 24-cell. The first figure gives in green a spherical version of the cuboctahedron. In dark blue one recognizes a small spherical version of a octahedron and in red one can recognize with some more effort another spherical octahedron. The red octahedron is much bigger than the blue one which means that the fourth coordinate is positive for the red one and negative for the blue one. (made with POV-Ray)

Both the edges and vertices are colored in triality code. The 8 vertices of one color are of one 16-cell. The 32 edges of one color are of one hypercube. The colouring code is the same as in the second spherical case.

The colouring code of the vertices is:

Light blue and dark blue for vertices that lie in 3-dim subspace.

Red for vertices that have a negative fourth coordinate.

Green for vertices that have a positive fourth coordinate. The blue and green hypercube have the same projected form but are rotated about 90°.  Let's fill up the 4 cells: It's important to note that the space occupied by the 4 cells is in yellow. The big one is thus considerably larger than you might have thought. It's as large as the rest of the 3-dimensional space when we take out the grey area. The grey area is occupied by the 20 cells not contained in this 4-chain. The centre of the big one is the "north pole" of the 4-dim sphere.

Another typical chain of 4 cells (to be compared to this picture) : Or, with some filling up:  Now let's search for another property called DUALITY.

We can divide the 24 vertices in 6 groups of 4 and put these four on one circle. The 6 circles are part of a so-called Hopf-fibration of the 4-dim sphere. After stereographic projection we can fill three-dimensional space completely with circles in such a way that each point of space lies on one and only one circle. Each circle is divided in 4 segments by the 4 points. Each circle segment acts as body diagonal of a cell (=octahedron). The 4 segments of one circle thread 4 cells into a string of 4 "beads". The 4 cells have one vertex in common with a neighbouring cell on the string.

In the following figure some vertices have changed their color. The sequence of the 4 points on a (grey) circle is dark blue, green, light blue, red. One of the circles has changed into a straight line because it intersects the "north pole", the centre of the projection, between the dark blue and the green point. The circle in the horizontal plane hasn't changed at all by the projection. On these two circles lie 8 of the 24 vertices. They act as the vertices of a 16-cell (4-dim equivalent of a octahedron). The edges of this 16-cell are not part of the 24-cell and so aren't represented in the picture. There are only 23 points left in the finite realm. The green point on the straight line has become the projection centre, it lies on the "north pole" of the 4-dim sphere. The 8 edges in green and blue intersect the point at infinity. What we have got is the DUAL figure of the previous one. Each vertex of the dual figure is in the middle of a cell of the original 24-cell. And each centre of each cell of the dual figure coincides with a vertex of the original 24-cell.

The vertex at infinity in the dual figure thus corresponds with a cell of the original figure that has an infinite volume. Actually, when you look at the original 24-cell you find yourself within this 24th cell !

The spherical version of the dual 24-cell looks like this:  (made with POV-Ray)

Both the edges and vertices are colored in triality code. The 8 vertices of one color are of one 16-cell. The 32 edges of one color are of one hypercube.

Each of the three colors still represents a complete hypercube.

The coloring of the 24 vertices is the same as in this picture.

The red hypercube alone:  The green hypercube looks similar:   (made with POV-Ray)

A compound of 2 mutually dual spherical 24-cells where the vertices of one are in the centres of the cells of the other. Here are the 72 diagonals:  (made with POV-Ray)

The 16 yellow and 8 black vertices belong to one 24-cell and the 12 red and 12 blue vertices belong to another 24-cell.

The edges of the 16-cells are the diagonals of the cells of both the 24-cell and its dual:  The 16-cell 16-cell has 24 diagonals as its edges: Straightening out the edges and using one color gives:  These 24 diagonals are diagonals of the green hypercube: Another choice of 24 diagonals delivers a second 16-cell: Straightened out and in one color: Its dual hypercube is the light blue one. And together: A chain of 4 cubes is formed in the red hypercube: The fourth cube is the blue exterior of the big cube and you are in it while watching.

A chain of 4 tetrahedral cells that correspond to these 4 cubes: In the above picture the fourth cell is the green exterior of the big tetrahedron.

Another chain of 4 cells in the hypercube with the one in front a bit transparent: The corresponding chain of 4 tetrahedra: In return, we can find the 16-cell in this projection (achievable by a Hopf flow but also by rotation along the 6 circles of this set,which means 3 intertwining Hopf flows): Together with its dual hypercube: The blue hypercube with its dual 16-cell: The green hypercube with its dual 16-cell: All diagonals: The 24-cell with 96 edges and its 72 diagonals(compare with this projection: Through each vertex pass 8 edges (4 of one color) and 6 diagonals (dark blue), making up 14 line segments. Remember that we project a 3-sphere (a 4-dimensional hypersphere) into 3-dimensional space. We can regard this hypersphere as the intersection of 3 4-spheres sharing the 24 points of the 24-cell.

We can add three points and connect all vertices of each of the three hypercubes with one of these three points. This makes 3x16=48 line segments. In total we now have 96+72+48=216 line segments connecting 27 points. Each of the 27 points intersects 8+6+2=16 line segments. This configuration corresponds to the 27-dimensional representation of E6 and the 27 lines on a cubic surface. The Gem of the Modular Universe . The stereographic projection of a cuboctahedron that lies entirely in the 3-dimensional subspace looks like this: The 12 vertices all lie on one sphere. Another way to divide the 24 points is in 4 groups of 6 points. The 6 points are lying on a circle and the four circles are again part of one Hopf fibration. One can choose one of the circles to be the equator in a 3-dimensional space that is orthogonal to the axis on which the pole N is lying. A second possibility is choosing the pole N on one of the circles.  If we let the points move on their Hopf circles we get this "movie" of 450 kB: rotating dual 24-cell   