Contents

**Overview of Project**

With this piece, I set out to create an umbilical torus through the carving of a block of wood. While I recognized that this would be a very difficult endeavor, I found that after breaking the process down into more basic steps, it came together quite nicely. This leap into complex geometry lead me to encounter a variety of issues, including an utter failure of my differential geometry skills that resulted in having to restart entirely; however, after overcoming some of the obstacles, I was very happy with the piece I created and learned a great deal about woodworking, art, and geometry in the process.

**Mathematics**

In this piece, a variety of mathematics are at play, both in the final product and the creation of said product. The name of this shape is an umbilical torus, and it has become a relatively popular tangible example of calculus over the years, whether on the covers of calculus textbooks or as a statue (a rather famous one is located at Stony Brook University). The simple explanation of this piece is that it is a single edged three-dimensional shape, something which at first thought might sound impossible. Naturally, to go along with this lone edge that wraps around the body of the figure three times uniquely (and infinitely over the same track) there is only a single side to this figure as well. These traits make it a particular fascination in the field of singularity theory.

The mathematics that apply to this rather complex shape are equally complex if not more. While I can take no credit for the personal discovery of such things, the math behind the figure is as follows:

x = sin u (7 + cos(u/3 – 2v) + 2cos(u/3 + v)

y = cos u (7 + cos(u/3 – 2v) + 2cos(u/3 + v)

z = sin(u/3-2v) + 2sin(u/3-v)

for the interval:

-3.14 < u < 3.14

-3.14 < v < 3.14

These mathematics work together to form what is (in layman’s terms) a quadrilateral traveling through a three-dimensional space while orbiting a point while rotating at 90° for every orbit, meaning that each edge of the quadrilateral lines up perfectly with its adjacent side’s path following a full orbit around the center point.

As for the mathematics of creating it, there were many complications that ought to be noted. First, a proficiency in the way that the shape operates was fundamental to planning this project as it is a remarkably difficult shape to imagine when drawn two-dimensionally. However, after viewing enough angles, I felt I had a good grip on the process I would need to take (it should be noted that this shape was extremely difficult to orient without the back side). I had to take measurements of the wood block I used in order to ensure that the thickness of the “band” I was creating would be circular and devoid of serious irregularities, and then I simply had to begin carving until I had created a donut shaped piece. Following this, I had to find the circumference of the circle orbiting the center point and find the arc length of one fourth of it. Knowing that this arc length would constitute one orbit around the center point, I had figure out how much the rotation would be at smaller intervals, for example, how much rotation would occur in 1/12 of an orbit around the center point. Once I had this figured out, I connected the lines and continued this process three times around the circle until I had come back to my beginning point. After this, I grinded, whittled, carved, and sanded off the wood until the lines I had created represented the edge.

This shape has gained popularity due to its bizarre characteristics, but also its aesthetic draw, as it leads the viewer to engage in the mathematics of it as well. Upon my first viewing, I pondered whether it was even possible to create such a figure in a three-dimensional form, or whether it was simply an illusion. However, as this piece, and hundreds of other recreations of this geometric marvel have shown, it is possible, fascinating, and a remarkable example of the interaction between mathematics and aesthetics.

**Last Thoughts**

It is deeply unfortunate that we find ourselves in a set of circumstances wherein the three-dimensionality of this shape is lost. One of the most interesting aspects of this shape is tracing the side around the object, and simply viewing it from a variety of different angles. This being said, I hope that the curiosity of this shape is not lost in the finitude of a photograph, and I hope you can enjoy it visually and mathematically as much as I do.