An untwisted ribbon and a Moebius strip are topologically distinct shapes. Paths that wrap around the ribbon live entirely on one side. But a Moebius strip doesn't have sides, and a path needs to make two revolutions to get back to where it started. The only way to get from one shape to the other is to tear the ribbon, add a twist and glue the two sides back together. (Credit: S. Kelley/NIST)