# Charles Gunn and Thomas Banchoff

We are both excited by the use of computer graphics to help reveal mathematical structures, and have been since the days of punch cards. A semester together at TU Berlin in 2012 led to our first collaboration, and the work shown here. Banchoff was studying a particular higher-dimensional manifold (see description below); Gunn observed that his new, panoramic rendering techniques provided an ideal tool for studying a 3-dimensional slice of this space, the so-called 10-cell in the 3-sphere (see description below). Our artistic aim is, by astute choice of view, form, and color, to allow this space, particularly its rich set of symmetries, to reveal itself. We are also working on a movie devoted to this theme.

A line in the 3-sphere has a unique polar line, which is its orthogonal complement with respect to the spherical metric. In this image the 10 central symmetry lines from the previous image have been replaced by their 10 polar lines, which are also symmetry axes of the pattern. Each polar line carries a screw motion of order-6 with pitch 1/3 -- the rotation is 3 times faster than the translation. The three colored stripes on the line identifies which polar pairs of tetrahedra are permuted by this symmetry, and the twisting of the stripes corresponds to the pitch of the screw motion.

Halfway through a five-dimensional cube, perpendicular to the long diagonal, we encounter a four-dimensional ball with boundary consisting of ten three-dimensional truncated tetrahedra arranged in five antipodal pairs. Spreading this configuration outward from the center of the ball to the three-sphere in four-space gives a symmetrical arrangement of ten lines, each containing two antipodal pairs of polyhedra. Shrinking these polyhedra and coloring the five antipodal pairs makes it possible to color-code the ten lines, each with two corresponding colored stripes. The submitted image is a conformal projection of the full viewable sphere seen by an insider in the 3-sphere viewing all 10 polyhedra and 10 symmetry axes.