600-cell cross-section sequence

In Peek, all aspects of the polytope can have color and opacity information assigned to them. In the 600-cell used to create the images below, the tetrahedral cells are colored according to their position along the fourth-dimensional axis (traditionally called w), ranging from red at one extreme to blue at the other. In the cross-section, the faces' color is inherited from the cells that they are a cross-section of, so the colors on the surface of the cross-sections indicate where the 600-cell was cut.

All of the small thumbnail images below are links to full-size images.

Each vertex is shared by the 20 tetrahedra which are clustered around it. Because they are arranged symmetrically, if we cut through the 600-cell near that vertex, we cut through the 20 tetrahedra equally, producing an icosahedron. As we cut closer to the center of the 600-cell, we are cutting deeper and deeper into the 600-cell, so the icosahedrons get bigger.

As soon as we get past the first 20 cells we start cutting into a new region of cells. Although the basic shape is still an icosohedron, its vertices and edges have become faceted. Note how the faces of the icosohedron have turned from red to dark orange: the dark orange tetrahedra are adjacent to the red ones but they point in opposite directions. So while the red triangles in previous cross-sections grew bigger as we cut further along, these dark orange triangles are going to shrink.

We are getting to the end of the orange cells, and then at one point the cross-section becomes a stellated dodecahedron: on each of the 12 pentagonal sides there is a set of five triangles which bulge out slightly. So far we have seen three different orientations of cell, and each produces a different series of faces on the cross-section. We've cut from vertex to opposite face (red, and dark orange cells), or from edge to opposite edge (orange cells), or along an edge (golden cells).

Again, when we start impinging on a new region, edges and vertices become faceted again, but this time it happens differently. Note how some of the the edges have expanded into two yellow faces. Here is an instance where we can see that at each (1-dimensional) edge of the 600-cell, there are 5 tetrehedra, for instance: two yellow cells, two golden cells, and one orange cell. Eventually, in the ninth cross-section, we get something where around each of the 12 vertices of an icosohedron, there are 10 narrow triangles.

In this latest region we have seen a new orientation of cell: the very light lime green cells have been cut along one of their triangular faces, so the lime green faces started as triangles but then became quadrilaterals.

By the twelvth cross-section, we are half-way though the 600-cell, so this is the largest cross-section we'll ever get. Here, there are 12 pentagonal clusters of faces centered at the vertices of of an icosohedron, and there are 20 (equilateral) triangular faces centered at the vertices of a dodecohedron; making for 80 faces all-together. As in all of these cross-sections, some of the vertices of the cross-section are actual vertices of the original 600-cell, and some of the vertices are created by generating the cross-section. In this central section, the vertices that are from the original 600-cell are the 30 vertices that define the equilateral triangular faces.

Now, the sequence basically reverses itself, and the geometry of these cross-sections is the same as the previous ones, but since we are cutting closer toward the other end of the w-axis, the cross-sections are getting more and more blue.