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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
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
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.