My little brother, who is ‘captain funny-sayings,’ said something absurd the other day. It made me laugh really hard, but then it got me to thinking, which is dangerous. He said, “I feel like a mobius with 3 sides.” Hmmm… A mobius by definition is one-sided right? A mobius strip is the symbol for infinity, right? So, I got to thinking, can you have a “mobius” with three sides? The answer I came up with is yes… sort of. I don’t know if you can still call it a mobius, though.
If you take a hoola-hoop that has six distinct planes, cut it and twist it clockwise three sides, then you have a three sided “mobius,” right? If you twist it two sides, then you get a two sided “mobius.” I don’t know. What do you think? What would you call these things?
AND if you have more than one mobius, do you have mobii? These are the questions that kept me out of the really good schools!
First, the grammar of this – you would not call the simplest version of the construct you are talking about “a mobius”, you would call it “A Mobius Strip” – Mobius is an adjective (the name of the mathematician that came up with the idea). And the o should have an umlaut over it, but I’m not that picky. Perhaps the next poster that comes along will be.
Onwards. You can classify more complicated structures in several ways, but the ones that would interest you are:[ul][li]How many distinct sides does it have? (without crossing an edge), and What is the cross section?[/ul][/li]
A standard Mobius strip (take a strip of paper, give it a half twist, and tape) has a line segment as a cross section, and one side.
How about this for a “three-sided mobius”, or something like it: Take a triangular prism made out of clay, give it a twist of a third of a circle, and connect it. One side, one edge, triangular cross section.
Your hula-hoop idea (hexagonal cross section) has some interesting possibilities. 60 degree twist – one side, one edge, you have to circle the opening 6 times to come back to where you start. Whee!
But: a 120 degree twist gives you two distinct sides and two distinct edges, with the same cross section, and you circle the opening 3 times!
Do 180 degree twist, and you have 3 sides and edges, circling twice.
sevenwood:
A mathematician named Klein
Thought the Mobius strip was divine.
Said he, “If you glue
The edges of two
You’ll get a weird bottle like mine!”
My high school math teacher used to like to read to us from a collection of math-related short stories called Fantasia Mathematica…two pertinent ones are The No-Sided Professor by Martin Gardner, and And He Built a Crooked House by Heinlein (or was it Bradbury…?)
The only resemblance your hula hoop bears to a Mobius strip is that you twist and connect. This is not what makes a Mobius strip special. So I would not call this a “three sided ‘mobius’”. In order to compare your hoop to a Mobius strip, you have to count both the outside and inside surfaces, of which there are six total. That is, you have to count both sides of your ribbon-like surface. The cool thing about a Mobius strip is that when you do this, you get an odd number.
If it’s solid, it’s an entirely different problem. All together. It’s trivial to make a solid with an odd number of “faces”. A sphere springs to mind, with one. What NE Texan describes is known as Umbilic Torus NC, and it also has one.
