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Old 10-11-2012, 02:10 PM   #9586
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The Electric Company reference escaped me, but I was fairly confident of the other three.

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Old 10-11-2012, 02:16 PM   #9587
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Take a strip of paper.

Give it a half twist and join the ends. You now have a Möbius strip - a piece of paper where the flat area is all one side.

Join the ends without a half twist and you just have a loop, with the flat area is two sides.

How can you turn a strip of paper into something with three flat sides? (No cutting, and with just one join, as in the previous two examples.)
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Old 10-11-2012, 05:32 PM   #9588
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Fold it into a triangle?
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Old 10-11-2012, 05:40 PM   #9589
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Roll it up like you would an old rug.
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Old 10-11-2012, 06:14 PM   #9590
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Quote:
Originally Posted by BaenSidhe View Post
Fold it into a triangle?
No, not that kind of side. Three sides. If you start colouring in a mobius strip, stopping at edges, you cover the whole surface with one colour. With a loop you finish colouring, and then have another side to colour.

With this object, you have three sides to colour. (Although you can't see all three at once.)

Triangles do come into it though.
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Old 10-11-2012, 06:15 PM   #9591
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Roll it up like you would an old rug.
Still only two sides.
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Old 10-11-2012, 06:21 PM   #9592
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Fold it lengthways, then join into a mobius strip? One side on the "outside", two sides on the "inside".
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Old 10-11-2012, 06:33 PM   #9593
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Quote:
Originally Posted by orlok View Post
Fold it lengthways, then join into a mobius strip? One side on the "outside", two sides on the "inside".
An interesting idea, but you have a problem joining the ends. Because of half twist, you're trying to join a mountain fold to a valley fold.


I will say that while the Möbius strip and the simple loop both still resemble the original strip of paper, this construction makes it harder to see that it is one single strip of paper.
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Old 10-11-2012, 07:13 PM   #9594
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If you roll it up you have three sides. Two ends and a center (picture a rolled up rug), but I look forward to seeing your answer pdurrant.
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Old 10-11-2012, 08:02 PM   #9595
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I hope you have a picture.
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Old 10-12-2012, 05:11 AM   #9596
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Quote:
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If you roll it up you have three sides. Two ends and a center (picture a rolled up rug), but I look forward to seeing your answer pdurrant.
Ah! I see. OK, that's a pretty good answer. The edge between the 'sides' isn't really like the edge of a piece of paper, but it could be argued that my solution is flawed too, so I think I shall declare a winner.

My solution is the trihexaflexagon.

No, it's not as bad as it sounds, really. Here's how you make it. Take a strip of paper that's at least 6.5 times longer than it is wide.

Now fold it up in equilateral triangles. You want to end up with ten triangles.

Now fold the strip into a hexagon. One triangle sticks out (the tenth), along with any bits of extra paper.

You can fold and stick the extra bits of paper past the first and last triangles onto the first and last triangles, or just cut them off. Then stick the first and last triangles together.

Now you have your trihexaflexagon. Fold both ways across all three corner-to-corner lines. Now fold from the centre into valley/mountain/valley/mountain/valley/mountain.

There are two ways to do this. One way is dull. The other way: as you push in the valleys to make to mountain folds as sharp as possible, the centre opens up, and you get back to your hexagon. And you can keep doing this. fold open, open from centre, fold up, open from centre.

And where do the three sides come in. Well, if we colour the two sides of the hexagon of paper, say red and blue, and then do the folding/opening from the centre, we find another side that we haven't coloured yet. It will go red/white/blue/red/white/blue!

But you really need a video to see it working. Or make one yourself!

Here's a video (not me!):
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Last edited by pdurrant; 10-12-2012 at 05:15 AM.
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Old 10-12-2012, 05:18 AM   #9597
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As I said (I'm repeating in case it got lost in my previous post) I'm declaring Dathi the winner. It hadn't occurred to me that we could get two 'sides' by rolling up the paper tightly.

Over to you, Dathi!
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Old 10-12-2012, 05:53 AM   #9598
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I only got the 3 sided concept by watching the video. A bit of a 'cheat' (), but a clever and interesting question all the same.
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Old 10-12-2012, 11:12 AM   #9599
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I have to get a project done at work today, so if anyone else would like to go then have at it.

P.S. those trihexaflexagon things are pretty cool. It reminds me of when I was a kid the girls used to make similar things out of paper that they could open different ways displaying different interiors. They would write stuff on the various panels and figure out who their next boyfriend was or something.
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Old 10-12-2012, 11:44 AM   #9600
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