## Another 600-cell cross-section sequence

This time, instead of translating the 600-cell along the w-axis, I'm doing a rotation of the y and w axes, while keeping the position along w-axis fixed. Again, since the colors of the faces on the cross-section are inherited from the cells that they are a cross-section of, we can use the colors as an indication of orientation.

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

There is not a lot that I can say about this sequence of images, because I don't know how to account for each and every face or explain the symmetries that occasionally arise in them. Rather, the one basic fact that is pretty obvious, and which I can explain, is that as the rotation progresses, we see a greater and greater range of colors appear on the faces of the cross-section. This is because the rotation is gradually bringing points that originally lay on the w-axis to lie in the x-y-z hyperplane. As the magnitude of their w-coordinates is being lowered down to zero, more and more of the color map which extends along the w-axis is made visible by cutting along the hyperplane where w=0. At the end of the rotation through 90 degrees, these antipodal points have a zero w-coordinate, and we see the broadest range of color. Also, note that points which had x or z coordinates of zero are unchanged by this rotation, so the medium green faces which form a vague band around the final cross-section are from cells which lie close to or in the the x-z plane.

Note that the geometry of the last cross-section is the same as that of the first, though the orientation in space is different. What does that say about a symmetry of the 600-cell?

Here is another view of the 8th cross-section shown above (rightmost one on the third row):