Classifying clothing: the quest for the non-orientable tank top.

When I walk down the street, I create patterns in how I walk, often by controlling my stride length so I will step on cracks every third sidewalk square, or whatnot. If I were a true master, my stride length would be incommensurable with respect to the sidewalk length–surely this was the problem that forced irrationalities upon the Greeks…

Anyway, I was also happy to realize (at a recent retreat) that clothing is nicely categorized by how many disks must be removed from a sphere to produce the particular clothing item. For some examples, consider:

• A sock or a hat is a sphere minus a disk.
• A headband (or tube top) is a sphere minus two disks.
• Jeans are a sphere minus three disks (the fabled “pair of pants”).
• A shirt is a sphere minus four disks (the “lantern”).
• A bathing suit is a sphere minus five disks.
• A fingerless glove might be a sphere minus six disks.
• Two fingerless gloves connected by a band is a sphere minus 11 disks.

Another lovely example is that of some scarves, which are a projective plane minus a disk (i.e., a Mobius strip), and therefore sit flat against one’s neck. I would be very interested in owning more non-orientable clothing (someone, somewhere, must own a non-orientable tank top–though perhaps that mythical object would be too annoying to be allowed to exist).