# Sculpting with a Twist

There’s more than one way to slice a bagel.

 A bagel (or a doughnut) can serve as a physical model for a mathematical surface called a torus.

You can slice it horizontally (or longitudinally) so that you end up with two halves, each containing a hole. That’s great for making sandwiches because the cut exposes a reasonably large annular surface on each piece.

 A horizontal slice through a torus.

Or you can perform a vertical slice to get two equal pieces. This cut exposes two circular disks on each piece. That’s nice for dunking.

 A vertical slice through a torus.

What if you twisted the knife as you sliced so the cut isn’t strictly vertical or horizontal? That’s a little tricky to do, but such twisting cuts are the basis for some intriguing sculptures created by Japanese artist Keizo Ushio.

 Keizo Ushio with a split band sculpture.Courtesy of Keizo Ushio

Keizo starts with a massive granite ring having a hole width equal to the thickness of the ring. He then drills into the granite to slice it longitudinally, not the way you would normally slice a bagel to get two halves, but with a cut that makes a 180-degree twist during its travel around the ring.

In effect, such a cut creates a space that can be considered a Möbius strip. A Möbius strip is a one-sided, one-edged surface. You can make a model of this unusual form by joining the ends of a long strip of paper after giving one end a 180-degree twist.

 An example of a Möbius strip.

If Keizo makes a 360-degree twist, the twisting cut separates the torus into two equal, but interlocked parts. In a remarkable demonstration of his prowess, Keizo created such a sculpture over the course of about a week, during ISAMA 99 (First Interdisciplinary Conference of the International Society of the Arts, Mathematics, and Architecture), held in San Sebastián, Spain. He called it Oushi Zokei 1999.

Several conference participants photographed Keizo’s effort. You can see various stages of the process of creating the sculpture at http://www.cs.berkeley.edu/~sequin/SCULPTS/KEIZO/, http://www.mtholyoke.edu/courses/jmorrow/keizo.html, http://torus.math.uiuc.edu/jms/Photos/99Jun/keizo.html. Computer scientist Carlo Séquin of the University of California, Berkeley, provides images of models showing how to produce such a sculpture at http://www.cs.berkeley.edu/~sequin/SFF/FDM_parts/fdm_keizo.html.

 Keizo’s Oushi Zokei at Sukaingawa Park, Japan.Courtesy of Keizo Ushio

There are versions of Keizo’s Oushi Zokei in Sydney, Australia, Sukaingawa Park in Japan, and several other locations.

Many of Keizo’s dramatic sculptures involve the equivalent of starting with a twisted band, then splitting it along its center line.

 One of Keizo’s split bands.Courtesy of Keizo Ushio

You can explore some of the possibilities yourself. For example, when a Möbius strip is cut in half along a line down its middle, the result is not two bands but a single larger band. Surprisingly, the new band produced by this “bisection” is two-sided and two-edged.

In general, joined strips made with an odd number of half-twists are one-sided and one-edged. Joined strips made with an even number of half-twists are two-sided and two-edged. The new band produced by a splitting a Möbius band made with n half-twists, where n is an odd number, has 2n + 2 half-twists. So, when n = 1, the new strip has four half-twists.

A band with an even number of half-twists always produces two separate bands when it’s cut down the middle. Each of the new bands is identical with the original except for being narrower. Each has n half-twists, and the two bands are linked n/2 times.

So, when n is 2, the cut produces two bands, each with two half-twists, and they are joined together like links in a chain. When n is 4, one band is looped twice around the other.

 Keizo’s Dream Lens.Courtesy of Keizo Ushio

One of Keizo’s most recent sculptures features an intricately carved and split ring. Unveiled last May at Kobe Bridge, Japan, the sculpture is called Dream Lens (see http://www2.memenet.or.jp/~keizo/0128.htm and http://www2.memenet.or.jp/~keizo/0130.htm). The structure actually consists of three interlocked rings (see http://www.cs.berkeley.edu/~sequin/SCULPTS/KEIZO/).

 Keizo’s split band sculpture at Mihama, Japan.Courtesy of Keizo Ushio

Keizo’s fascinating sculptures provide a vivid introduction to the unsuspected intricacies of slicing bagels and cutting Möbius bands.