Split Strips

"Möbius bands (or strips) are beautiful as objects of art, and their mysterious qualities fascinate those who discover or encounter them," sculptor Larry Frazier wrote in an article titled "Möbius strips of wood and alabaster," published in the Journal of Mathematics and the Arts.

"I'm not a mathematician, but as a sculptor, I have been fascinated by the myriad forms that a Möbius band can take, especially when interpreted in a beautiful piece of wood or stone," he continued.

In recent years, Frazier has often brought his gracefully carved artworks to the annual Joint Mathematics Meetings, where he displays and sells them.

Particularly intriguing examples arise from slicing a Möbius strip along its length. Slicing it lengthwise down the middle produces a single, longer band. Slicing a Möbius strip about a third of the way in from its edge produces two linked bands. In effect you've sliced off its edge to produce a narrower version of the original band. The outer piece has two half-twists, and the inner piece has one half-twist, just as the original, uncut band did.

Doubleslice, by Larry Frazier.

Frazier performs this trick in wood. Unlike paper strips, the two wood components can be easily reassembled into the original Möbius band.

Larry Frazier displays his split Möbius strip.

"It's especially nice in wood, because you can put the pieces back together, and you can see the Möbius it came from," Frazier said. "Making the same two cuts in a paper Möbius just produces two loops of paper, and you can't see the original Möbius they came from."

A reassembled sliced Möbius.

Sculptor Keizo Ushio performs similar magic in split stone (see "Sculpting with a Twist").

References:

Frazier, L., and D. Schattschneider. 2008. Möbius strips of wood and alabaster. Journal of Mathematics and the Arts, 2(No. 3):3, 107-122.

Peterson, I. 1991. Fragments of Infinity: A Kaleidoscope of Math and Art. Wiley.

Photos by I. Peterson