Acme Posted July 6, 2014 Author Share Posted July 6, 2014 (edited) Okaly dokaly. I have my answer. First I should say I made a mistake earlier when I said: Band with 2 twists cut in half yields 2 bands interlaced, each band with 1 twist. ... It is 2 bands interlaced, but with 2 twists each just as the Wiki article stated as the general rule for bands with even numbers of 1/2 twists. That out of the way, I managed to put together a copy of imatfaal's ring and then cut it. The result is a single ring with 2 twists. As imatfaal said, cutting it into a strip again and flattening, it looks like the starting strip but with some lumpiness. 'Course I haven't imatfaal's experience or skill with paper & tape and my starting ring is rather lumpy. So without further ado, here's a link to my video of the cut. (I started untwisting the wrong way at the end and decided to end the recording so as not to appear more of an idiot than I already do. ) Edit: Apparently Flickr has a 3 minute video length limit, so you don't get to see my fumblage. Video of Prismatic Ring Cut: >> https://flic.kr/p/ods5We Edited July 6, 2014 by Acme 1 Link to comment Share on other sites More sharing options...
Janus Posted July 8, 2014 Share Posted July 8, 2014 Sometimes, when I get to playing around with an idea, I just don't know where to stop. To wit: Cousins of the prismatic ring. The original idea, if considered as a solid is pretty simple. Take a triangular prism, twist it by 120 degrees and then join the ends to form a loop. However, you can also give it a 240 degree twist before joining the ends and still end up with a shape with one continuous side. A 360 degree twist will however get you back to 3 unconnected sides. Neither are you restricted to a triangular prism. you can do the same trick with a rectangular prism. With the rectangular prism a 90 degree and 270 degree twist will get you one continuous "side" while a 180 twist results in two continuous sides. A pentagonal prism will give you single continuous sides with twists of 72, 144, 216, and 288 degrees. Here are how they end up looking. The coloring is done as a continuous spectrum; From red to orange to yellow to green to blue to purple and back to red. at the top left is the 240 twist triangular prismatic ring. Below it are the two rectangular types Down the right are the four pentagonal ones You could keep going with hexagonal etc., though the hexagon version only has 2 single continuous side forms. 4 Link to comment Share on other sites More sharing options...
imatfaal Posted July 9, 2014 Share Posted July 9, 2014 I have the "triangular" with a 240 twist in paper - but it was not so nicely executed; the tolerances were too small Okaly dokaly. I have my answer. First I should say I made a mistake earlier when I said:It is 2 bands interlaced, but with 2 twists each just as the Wiki article stated as the general rule for bands with even numbers of 1/2 twists.That out of the way, I managed to put together a copy of imatfaal's ring and then cut it. The result is a single ring with 2 twists. As imatfaal said, cutting it into a strip again and flattening, it looks like the starting strip but with some lumpiness. 'Course I haven't imatfaal's experience or skill with paper & tape and my starting ring is rather lumpy. ... Better than my first attempt! ==== The "Hexagonal" forms - you get one with only two external sides? (or one or three?) ie. no twists gives you six external sides each only making a loop, one twist gives you one external side making six laps around the ring, two twists give you 2 sides making three laps each, three twists gives you three sides each making 2 laps, 4 same as 2, five same as one, and six same as zero. ==== Just a pretty picture really. How many sides / cuts / joins do ya reckon? 1 Link to comment Share on other sites More sharing options...
Janus Posted July 9, 2014 Share Posted July 9, 2014 Another pretty picture. Like the Mobius strip, this one has only one surface. 2 Link to comment Share on other sites More sharing options...
Acme Posted July 9, 2014 Author Share Posted July 9, 2014 That out of the way, I managed to put together a copy of imatfaal's ring and then cut it. The result is a single ring with 2 twists. As imatfaal said, cutting it into a strip again and flattening, it looks like the starting strip but with some lumpiness. 'Course I haven't imatfaal's experience or skill with paper & tape and my starting ring is rather lumpy. Better than my first attempt! Thank you! High praise indeed. Just a pretty picture really. How many sides / cuts / joins do ya reckon? closeup.png 42? . Link to comment Share on other sites More sharing options...
moth Posted July 9, 2014 Share Posted July 9, 2014 (edited) Thank you for the visualization, Janus. i was trying to do this in my head, but i was ending up with crossover Mobius' strips.Awesome photo Imatfaal Edit:sp. Edited July 9, 2014 by moth Link to comment Share on other sites More sharing options...
Acme Posted July 9, 2014 Author Share Posted July 9, 2014 Another pretty picture. Like the Mobius strip, this one has only one surface. Nice! I am reminded of Escher's 1965 print Knots. (The work is copyright so I can only direct readers to a link.) >> Escher's Knots Of course his Möbius prints are perennial favorites. Mobius strip I Mobius strip II Link to comment Share on other sites More sharing options...
imatfaal Posted July 9, 2014 Share Posted July 9, 2014 Another pretty picture. Like the Mobius strip, this one has only one surface. I love that. Any chance you could do one that is a bit more complex. If you could maybe do the above in a single colour - then you could merge it with one that is just the chiral mirror (ie the twist is in the other direction) Link to comment Share on other sites More sharing options...
Acme Posted July 10, 2014 Author Share Posted July 10, 2014 Aha! The [hasty] Möbius hole. One surface, two boundaries. 1 Link to comment Share on other sites More sharing options...
moth Posted July 10, 2014 Share Posted July 10, 2014 (edited) Aha! The [hasty] Möbius hole. One surface, two boundaries. Is it a 2-d analog of a Klein bottle? Edited July 14, 2014 by Phi for All Fixed editing Link to comment Share on other sites More sharing options...
Acme Posted July 10, 2014 Author Share Posted July 10, 2014 Is it a 2-d analog of a Klein bottle? [For some reason, my question appears as part of the qoute. But the edit option has faded.] Got it anyway. I would answer no, but I'm no topologist. The idea just came to me on account of trying to keep up with Janus & imatfaal. I can't say I have seen it before, but that's not saying much either. I'm trying to cook up some variations, and in the mean time I think I can only say it's maybe possibly a non-orientable surface. Definitively a disorienting surface though. ajb? Got your ears on? Klein bottle @ wiki In mathematics, the Klein bottle is an example of a non-orientable surface; informally, it is a surface (a two-dimensional manifold) in which notions of left and right cannot be consistently defined. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a surface with boundary, a Klein bottle has no boundary (for comparison, a sphere is an orientable surface with no boundary). ... Link to comment Share on other sites More sharing options...
moth Posted July 10, 2014 Share Posted July 10, 2014 If you where able to understand forward and backward along a line that makes a trip around the Mobius and returns to its starting point, and depth, a dimension orthagonal to that, but not the dimension orthagonal to both (the width of the strip) . The picture seems analogous to time being represented by the width of the strip. But I've had a couple beers... Link to comment Share on other sites More sharing options...
imatfaal Posted July 10, 2014 Share Posted July 10, 2014 Got it anyway. I would answer no, but I'm no topologist. The idea just came to me on account of trying to keep up with Janus & imatfaal. I can't say I have seen it before, but that's not saying much either. I'm trying to cook up some variations, and in the mean time I think I can only say it's maybe possibly a non-orientable surface. Definitively a disorienting surface though. ajb? Got your ears on? taking inspiration from Acme passing a twisted loop through a hole - I tried with more than one 2 Link to comment Share on other sites More sharing options...
Acme Posted July 10, 2014 Author Share Posted July 10, 2014 If you where able to understand forward and backward along a line that makes a trip around the Mobius and returns to its starting point, and depth, a dimension orthagonal to that, but not the dimension orthagonal to both (the width of the strip) . The picture seems analogous to time being represented by the width of the strip. But I've had a couple beers... I plan on having a beer or two when it warms up later today so I'll get back to you on this in good time(s). I will be cutting the strip with pinking shears some time before that however. . taking inspiration from Acme passing a twisted loop through a hole - I tried with more than one two twists.jpg I knew I shouldn't have slept. Love the form, love the precision, love the paper. Coffee. Now there's a beverage! Here's the Möbius hole pinked. Here's a Möbius 3-Hole. 1 Surface, 4 Boundaries. 1 Link to comment Share on other sites More sharing options...
Ophiolite Posted July 11, 2014 Share Posted July 11, 2014 I've found the opinions in this thread rather one-sided. 2 Link to comment Share on other sites More sharing options...
moth Posted July 11, 2014 Share Posted July 11, 2014 I've found the opinions in this thread rather one-sided. The conversation may seem flat in some places, but there are some interesting twists. 2 Link to comment Share on other sites More sharing options...
Acme Posted July 14, 2014 Author Share Posted July 14, 2014 Spotty Knotty Möbius 2-hole Link to comment Share on other sites More sharing options...
Phi for All Posted July 14, 2014 Share Posted July 14, 2014 Spotty Knotty Möbius 2-hole A little naughty, too! A one-sided relationship? 1 Link to comment Share on other sites More sharing options...
moth Posted July 14, 2014 Share Posted July 14, 2014 Hamlet's question. Performed by a Mobius. 3 Link to comment Share on other sites More sharing options...
Acme Posted July 15, 2014 Author Share Posted July 15, 2014 A little naughty, too! A one-sided relationship? +one Recall that strips with more than a 1/2 twist are termed 'paradromic rings'. Further recall that any paradromic ring with an odd number of 1/2 twists shares the property of Möbius strips in having 1 surface and 1 boundary. Further further recall that each addition of a hole adds another boundary. Spotty Knotty Potty Paradromic 1 1/2 Twist 2-hole Ring @Moth The undiscoverd country from whose bourn No traveller returns, puzzles the will... 1 Link to comment Share on other sites More sharing options...
moth Posted July 15, 2014 Share Posted July 15, 2014 The undiscoverd country from whose bourn No traveller returns, puzzles the will... +1. I have always suspected that was the answer . Fardels Link to comment Share on other sites More sharing options...
sunshaker Posted July 29, 2014 Share Posted July 29, 2014 For those who like more cream cheese on their "mobius bagels". http://spring-of-mathematics.tumblr.com/post/91043165514/mathematically-correct-breakfast-how-to-slice-a 3 Link to comment Share on other sites More sharing options...
Phi for All Posted July 29, 2014 Share Posted July 29, 2014 For those who like more cream cheese on their "mobius bagels". http://spring-of-mathematics.tumblr.com/post/91043165514/mathematically-correct-breakfast-how-to-slice-a This is the kind of thing they should be teaching at Einstein Bros. To promote a brainy brand, you need something intellectually challenging. Link to comment Share on other sites More sharing options...
Unity+ Posted July 29, 2014 Share Posted July 29, 2014 This thread should be renamed "Mobius porn" Also, here is something since I can't post my own: http://mathworld.wolfram.com/MoebiusStrip.html 1 Link to comment Share on other sites More sharing options...
Acme Posted July 30, 2014 Author Share Posted July 30, 2014 This thread should be renamed "Mobius porn"We try and probe the hole subject most thoroughly. Also, here is something since I can't post my own: http://mathworld.wolfram.com/MoebiusStrip.html Trey kewl figure, though I suspect impossible to physically build and operate. 1 Link to comment Share on other sites More sharing options...
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