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 10-11-2012, 01:10 PM #9586 issybird o saeclum infacetum     Posts: 8,550 Karma: 92124435 Join Date: Oct 2010 Location: New England Device: H2O, Glo HD, Aura One The Electric Company reference escaped me, but I was fairly confident of the other three. If anyone's got a question, go for it. I'm posting on the fly.
 10-11-2012, 01:16 PM #9587 pdurrant The Grand Mouse 高貴的老鼠     Posts: 49,710 Karma: 170522855 Join Date: Jul 2007 Location: Norfolk, England Device: Kindle Voyage 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.)
 10-11-2012, 04:32 PM #9588 BaenSidhe Groupie   Posts: 150 Karma: 1087916 Join Date: Mar 2010 Device: PRS-500 and PRS-650 Fold it into a triangle?
 10-11-2012, 04:40 PM #9589 Daithi Publishers are evil!     Posts: 2,418 Karma: 36205264 Join Date: Mar 2008 Location: Rhode Island Device: Various Kindles Roll it up like you would an old rug.
10-11-2012, 05:14 PM   #9590
pdurrant
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Quote:
 Originally Posted by BaenSidhe 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.

10-11-2012, 05:15 PM   #9591
pdurrant
The Grand Mouse 高貴的老鼠

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Quote:
 Originally Posted by Daithi Roll it up like you would an old rug.
Still only two sides.

 10-11-2012, 05:21 PM #9592 orlok THE END. Yes!     Posts: 9,397 Karma: 256304892 Join Date: Jan 2011 Location: UK Device: Kindle Oasis, Kindle Fire HDX 7, Kindle 6" Fold it lengthways, then join into a mobius strip? One side on the "outside", two sides on the "inside".
10-11-2012, 05:33 PM   #9593
pdurrant
The Grand Mouse 高貴的老鼠

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Quote:
 Originally Posted by orlok 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.

 10-11-2012, 06:13 PM #9594 Daithi Publishers are evil!     Posts: 2,418 Karma: 36205264 Join Date: Mar 2008 Location: Rhode Island Device: Various Kindles 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.
 10-11-2012, 07:02 PM #9595 SneakySnake I'm watching you!     Posts: 6,113 Karma: 22344652 Join Date: Mar 2010 Location: Sunny Coast Qld, OZ Device: Sony PRS-900 I hope you have a picture.
10-12-2012, 04:11 AM   #9596
pdurrant
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Quote:
 Originally Posted by Daithi 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!):
Attached Thumbnails

Last edited by pdurrant; 10-12-2012 at 04:15 AM.

 10-12-2012, 04:18 AM #9597 pdurrant The Grand Mouse 高貴的老鼠     Posts: 49,710 Karma: 170522855 Join Date: Jul 2007 Location: Norfolk, England Device: Kindle Voyage 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!
 10-12-2012, 04:53 AM #9598 orlok THE END. Yes!     Posts: 9,397 Karma: 256304892 Join Date: Jan 2011 Location: UK Device: Kindle Oasis, Kindle Fire HDX 7, Kindle 6" 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.
 10-12-2012, 10:12 AM #9599 Daithi Publishers are evil!     Posts: 2,418 Karma: 36205264 Join Date: Mar 2008 Location: Rhode Island Device: Various Kindles 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.
 10-12-2012, 10:44 AM #9600 Kumabjorn Basculocolpic     Posts: 4,302 Karma: 20180655 Join Date: Jul 2010 Location: Sweden Device: Kindle 3 WiFi, Kindle 4SO, Kindle for Android, Sony PRS-350 and PRS-T1 Like the intro to Community.