Previous Up Next

stereographic 24-cell  

 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 is a four-dimensional figure with remarkable symmetry properties. It is to be compared with the ordinary three-dimensional polyhedra: tetrahedron, cube, octahedron, icosahedron and dodecahedron. These are the so-called Platonic polyhedra. It can also be compared to the cuboctahedron, that has 6 quadrangles and 8 triangles as bounding planes. The cuboctahedron can be made by cutting off a octahedron just halfway the edges. A 24-cell can be made by a similar process: by cutting off the four-dimensional analogue off the octahedron, the so-called 16-cell. Contrarily to the cuboctahedron, having two kinds of bounding planes, the 24-cell is completely regular having only one can kind of 3-dimensional cells. These cells are octahedra. Here, this 24-cell is shown in a very symmetrical projection.

The 24-cell has 24 vertices, 96 edges, 96 triangles and 24 octahedral cells.

A stereographic projection projects each point of a 4-dim sphere (S3) into the usual 3-dim space. One draws a line from the "north pole" of the sphere that intersects the sphere in another point and intersects the 3-dim subspace in a third point. This latter point is the projected point of the former point of the sphere. Below a picture is shown of the stereographic projection of a point Q lying on a three-dimensional sphere (S2), onto a point P lying in the equatorial plane (in red). The line joining Q and P is passing through the north pole N of the sphere. Points on the northern hemisphere are projected outside the equator and points on the southern hemisphere are projected inside the equator. Points closer to the north pole than Q lie further from the origin in the plane than P. The only point that is not projected in the plane is the north pole itself.

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.

If we join the 24 vertices of spherical 24-cell by straight lines instead of circle segments the figure becomes more "readable".

The colouring code is the same as in the second spherical case.

With this colouring we recognize a symmetry of the 24-cell that's known as TRIALITY. We can discern a red , a (light) blue and a green hypercube (4-dim cube) that share 8 of their 16 vertices with one other copy. We can see immediately that triality corresponds to three dimensions of space.

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.

Let's single out the red hypercube:

The blue and green hypercube have the same projected form but are rotated about 90.

A chain of 4 cells can look like this (to be compared with this picture):

Let's fill up the 4 cells:

 

More artistically go here.

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.

The remaining 4 circles all lie on one torus and contain the 16 vertices of a hypercube (4-dim cube) with red edges. The 32 red edges of this hypercube are also edges of the 24-cell.

Now we rotate all points on their grey circles over 45 degrees. This is a Hopf flow. The result is shown below.

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 !

If you want to see a movie of the Hopf flow by which the 24-cell transforms to its dual then click on METAMORPHOSIS (about 400 kB).

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 blue hypercube is:

The green hypercube looks similar:

When combining the two mutually dual 24-cells, we get after making the DUAL smaller by a factor 1/2*SQR(2) :

(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.

This compound of two 24-cells in a different projection (not stereographic) looks like this. .

The 6 grey circles were said to form body diagonals of the 24 cells. The 6 circles form only 6x4=24 diagonals, while there must be three diagonals in each of the 24 cells, so 3x24=72 in total. To complete the number of diagonals draw two other Hopf-fibrations, but now with their singular circle (RP1 instead of a circle S1) in the two remaining directions perpendicular to vertical one and perpendicular to each other. In total we have got 96 edges and 72 diagonals, summing up to 168 circle segments (or line segments).

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:

Together with dual hypercube:

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

Previous Up Next