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Topologically Equivalent To A Donut: 10 Unusual Shapes (PHOTOS, VIDEO)

The Huffington Post     First Posted: 01/04/2012 1:39 pm EST   Updated: 04/02/2013 12:18 pm EDT

There's an old joke among mathematicians that topologists (who study deformations of space) can't tell a coffee cup from a donut. This is because, as far as the objects' topological properties are concerned, they're equivalent; they both have a single hole and can be deformed from one into another without tearing or gluing.

It turns out that the donut, called a 'torus' in topology, is a pretty basic surface, and there are a lot of other things besides coffee cups that can be made to transform into one if you use your imagination.

In the slideshow below, we investigated some other unlikely objects that are (more or less) topologically equivalent to a donut, from Rose McGowan's tube top to the 'A' guy in a 'YMCA' dance. See if you can imagine the transformations that would have to happen to make these objects into donuts, and add your own in the comments.

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  • Coffee Cup A coffee cup is the go-to example of something topologically equivalent to a donut. The GIF above shows the transformation (or homeomorphism, the function that gets you from one to the other), and you can see that getting from a donut to a coffee cup would require making a depression into one side of the donut to make the cup and shrinking the donut's hole to make the cup's handle. This is how you can think about all the rest of the objects in this slideshow transforming--but not tearing or forming a new connection--into their true, donutty form.

  • Tube Top In the photo above, actress Rose McGowan must be pointing out the fact that her tube top is topologically equivalent to a donut. What can I say? Math is hot. Image: Getty File

  • The 'A' In 'YMCA' If you use your imagination a bit, the fellow on far right in this picture is (temporarily) topologically equivalent to a donut. It only works if you imagine him with fused fingers, closed bodily openings and completely impermeable skin, but it's close enough. Image: Darryl Ring/Wikimedia Commons

  • Krazy Straws In the commercial above, we're told that drinking from a Krazy Straw is "just like playing a game." We're not so sure about that, but we are sure that no matter how much you deform and bend a straw it will always be (more or less) a straw. And, after all, what's a donut but a short, fat, impractical, delicious straw?

  • Isamu Noguchi's 'Cube' This sculpture, which sits across the street from Occupy Wall Street's old encampment at Zuccotti Park, fits the bill. Noguchi sculpted <a href="https://www.google.com/search?q=isamu+noguchi&um=1&ie=UTF-8&hl=en&tbm=isch&source=og&sa=N&tab=wi&ei=lcv4TtyzO-LliAL3ic2pDg&biw=1222&bih=639&sei=l8v4Tv7wFcGXiAK8iIGqDg" target="_hplink">a number of donut-friendly shapes</a>, although this one may be the most prominent. Photo: Flickr: <a href="http://www.flickr.com/photos/endymion120/" target="_hplink">Vincent Desjardins</a>

  • Earthworms They'd make for a pretty unappealing donut, but it could be done. Earthworms have one hole on either end of their body. Need we say more? Image: Flickr: Daves Portfolio

  • Ancient Chinese Coins Simple enough, but one has to wonder: if you buy a donut with these coins, does the transaction count as a homeomorphism? Photo: mc559/Wikimedia Commons

  • CCTV Headquarters, Beijing, China This structurally complicated building, nicknamed "dà kùchǎ" (big boxer shorts) by observers who can't see the base of the building, is actually donut-shaped. Image: Flickr: <a href="http://www.flickr.com/photos/richardsummers/" target="_hplink">Banalities</a>

  • Sewing Needle Warning: Do not eat sewing needles. They are not donuts. This is just a thought experiment. Image: Didier Descouens/Wikimedia Commons

  • Hurricanes Or, anyway, idealized hurricanes with no gaps and fissures. We're not sure if atmospheric phenomena count, but we already used the YMCA guy, so your imagination should be primed for this sort of thing. Photo: NASA

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