It was almost thirty years ago that I carved a bug on a band (patterned after a sculpture seen as a boy at the Museum of Science and Industry in Chicago). It was carved from a solid piece of white pine, to illustrate that a Moebius band is one-sided. Once around a circle and the bug is upside down; twice around and it is right-side up again. Such a band (or surface) is called non-orientable. This band also has only one edge and is called a spanning surface of that edge, which is itself homeomorphic to a circle (or unknot) in three space.
Only recently, upon seeing a computer stereo image of a spanning surface for a trefoil knot (or three-knot) in an article on heart arrythmia [1], I again resorted to a wood carving to gain an understanding of the surface - this time an orientable one. This carving reminded me to return to the Topological Picture Book by George Francis [2], and later to the book Knots, by Livingston [3] for more recent information on knots and spanning surfaces.
In the meantime I was teaching third semester calculus, where we were encountering the Frenet frame determined via the tangent, velocity, and acceleration vectors to a curve in 3-D (such as our trefoil knots). Hence, it seemed natural to twist a band centered on this knot - just to see how much the Frenet frame twists in space for this knot on a torus. After viewing several widths of such bands (with aid from my colleague Thomas Hern using a graphics package called "movie.byu," out of Brigham Young University), we observed that expanding the width of the surface would allow it to self-intersect in a new surface. The twisted trefoil knot surface shown was created on a 3D Systems stereolithographic machine along with a wood carving of the new collapsed surface. Note that a parameterized version of this surface is not easy to obtain - hence the lack of a quick computer graphic representation.
It is known that every knot has at least one spanning surface. Many of these are stable enough that they can be generated using soap film and wire frame models of the knots. These minimal surfaces are often quite beautiful, but short-lived. After many dippings and viewings, the figure-eight knot (four-knot) managed to yield a surface similar to this wooden sculpture - an orientable spanning surface of the four knot. Returning briefly to the spanning surface of the trefoil knot, it is clear that this surface is not a minimal surface as it bulges out.
The question arises as to the possibility of finding a similar spanning surface of the figure-eight knot. The proud sculptor is shown with his favorite sculpture in this photo, taken with the Apple QuickTake 150 digital camera. The picture is captured in a PICT graphics format, and stored either on disc or directly to the Mac. It can be converted easily to postscript for printing on laserprinters. An mpeg movie resulting from storing 36 views of this sculpture at 10 degree intervals can be viewed on the author's home page, thanks to his son, Andy Long, now teaching at Ripon College.
The eight knot basis for this non-orientable sculpture was formed using piecewise special Bezier quartic curves [4] defined as an interpolating curve for the eight points shown here, which stem from lines drawn from the center to the vertices of a regular tetrahedron. There are of course lots more surfaces which edge on this and other figure-eight knots [2].
The author's most recent surface carving provides for a clear understanding and physical image of Hoffman's surface [5], "the first complete, embedded minimal surface of a finite topology to be found in nearly 200 years." [6]
The carvings shown here have significantly improved the author's understanding of the surfaces indicated and the mathematics behind them, as well as instilled interest in further study of related topics. One need not carve in wood: bring out the modeling clay and turn yourself loose. You may be amazed at what you'll learn and what you'll create for use by others as well as yourself.
You may even be lucky enough to attend a Mathematics and Art Conference at SUNY Albany and participate in a stone carving session with mathematicians and master carvers Helaman Ferguson[7] and Nate Friedman[8]. My experience with them led to the carving, Trinity, (in white alabaster) of the previously mentioned orientable spanning surface of a trefoil knot.
