SB3D (Tour/Horizon)

"Math Horizon" is so named because it appeared as the cover image for an article in the journal "Math Horizons", published by the Mathematical Association of America. The object under investigation is an immersion of the two-dimensional sphere into four-space in such a way that there is exactly one point where the surface intersects itself. In this sense, it represents an analogue of the figure-eight, an immersion of the one-sphere into two-space self-intersecting at a single point. In neither case can the immersion be deformed into an embedding, with no self-intersections, without introducing local singularities such as cusps. The particular image is obtained by projecting centrally from a point on the three-sphere so that the observer appears to be inside the object. Color depicts different circles of latitude on the original two-sphere. The north and south poles are both mapped to the origin in four-space and no other pair of points is mapped to a common point in four-space. Projecting into three-space does introduce a curve of double points, including the image of the origin. The example was originally introduced in a research/expository article on the geometry of characteristic classes for surfaces in four-space written together with Frank Farris.

When this image was used as the cover for the journal, it was rotated ninety degrees from how it appears in this show. The placement on the exhibit wall was voted on by a group of artists at the Art Club. This orientation was preferred since it more closely suggested a sunset over the water. One artist strongly wanted the image to be hung upside down from the position finally chosen, precisely because it presented the same geometrical picture but with an inversion of the expected sunset color values. There is, of course, no right way to hang such an abstract image. One exhibitor at the Providence Art Club expressed this ambiguity in his show by mounting some pieces on rotating discs but this does not work for a basically rectangular piece.

Surfaces Beyond the Third Dimension Last modified: 08 Oct 2000 08:14:23 Comments to: Thomas F. Banchoff (thomas_banchoff@brown.edu)

The Mathematics of Math Horizons Math Horizon