The 24-cell in isometric projection
- 24-cell in isometric projection
-The gem of the modular universe
The 24-cell is a fourdimensional figure with remarkable symmetry properties. It is to be compared with the ordinary threedimensional polyhedra: tetrahedron, cube, octahedron, icosahedron and dodecahedron. These are the socalled 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 fourdimensional analogue off the octahedron, the socalled 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. Below, this 24-cell is shown in a projection that gives a very symmetrical picture. However, your position as an observer of this object is without realistic perspective. If you want to see the view if your position is in the middle of a cell, click on this link or search for the middle of the cell with green vertices in the picture and click on it. If you want the view from a vertex, click on this link or search for a (green) vertex in the picture and click on it.
The 24-cell has 24 vertices, 96 edges, 96 triangles and 24 octahedral cells.
Three of the 24 cells are shown in the next picture ( there is a total of 12 of these):
Cells are also discernable in this projection (there are 12 of these):
There are also cubes to be found and two opposite ones are shown here:
Together they form one fourdimensional hypercube if corresponding vertices are linked together by 8 line segments in the fourth direction:
The 8 new edges together with 2 squares of the connected cubes form 2 new cubes. This results in a chain of 4 cubes linked together by squares.
But there are more cubes than these 4 in the hypercube:
which are in a more familiar projection. Take another one:
Take these two together:
They can be joined by 8 horizontal line segments in the fourth direction to get the hypercube again. They result in another chain of 4 cubes. The 2 chains of 4 cubes share all 32 edges and 8 squares.
For the hypercube we get 16 vertices, 32 edges, 24 squares and 8 cells.
The same "hypercube game" can be played with the edges in green and in blue ( as a consequence of triality).
The 24 octahedral cells have 24x3=72 diagonals. If we choose 24 of them we can construct a 16-cell that consists of 16 tetrahedral cells:
This one is the dual polytope of the hypercube in red:
The 16-cell has its vertices in the midpoints of the cubic cells of the hypercube. In the next picture one can see that the vertices of this 16-cell are the alternating vertices of the green hypercube:
Eight of the tetrahedral cells are in the eight cubic cells of the green green hypercube so that their edges of the tetrahedron are diagonals of the squares of the cube. For instance see two cells that are linked by an edge :
The other 8 tetrahedra are in the 8 cubes of the light blue hypercube.
We can picture three 16-cells with their 3x8 vertices matching the 24 vertices of the 24-cell:
We can show the 24-cell with all its diagonals:
We can see that 4 of 6 diagonals that emerge from each of the 12 outer vertices are hidden by edges in this projection. From the inner vertices 2 of the 6 diagonals are hidden.
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:
You can see 24 lines from each vertex to the centre, but in the centre are 3 points projected and each vertex is linked to 2 of them. So, each line counts twice.
Click on the WEYL page of Tony Smith to compare with the projection of this configuration of 27 points.
We're climbing up the A-D-E ladder !
If two diametrically opposite cells (like these ones) are omitted one is left with a cuboctahedron. Its vertices all lie in a 3-dimensional subspace:
More elegantly is this cuboctahedron that emerges by leaving out 2 other cells ( like this one):
One can see chains of 4 cells that are linked together by means of one common vertex :
(Note: The 4 points inside the red squares do not intersect the blue or green edge, so 20 points are in this chain.)
A chain of 4 cells of this kind is seen in the next figure:
The four points by which they are connected are the ones that are only reached by blue and green edges. You can count the edges in this chain and arrive at a number of 48. The first chain above also contains 48 edges. Together they contain exactly all 96 edges of the 24-cell.
So, one chain contains 20 vertices, 48 edges, 32 triangles and 4 cells.
Yet another type of chains is present:
The cells are linked by triangles.
The 6 vertices that are not involved in this chain are the ones apparently intersected by 2 edges.
Another chain of six:
So, 18 of the 24 vertices are involved in such a chain. There are 54 edges, 42 triangles and 6 cells used.
A compound of two 24-cells looks like this (the coloring is like the one used in this figure):
