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. |

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. |

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. |

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 2*n* + 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. |

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. |

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