Wolf Molkentin Posted January 1, 2015 Share Posted January 1, 2015 Posting a pic of a nonagon so that not everyone has to google it individually And here's a heptagon. 220px-Regular_polygon_7_annotated.svg.png Wow! The happy medium between the ordinary hex concertina and a fancy Aeola! Link to comment Share on other sites More sharing options...
JimLucas Posted January 1, 2015 Share Posted January 1, 2015 Except that 360 degrees divided over eight sides, does yield a whole number, 45 degrees. Yes, 360 degrees divided over nine sides yields 40 degrees, which is great when looked at digitally, but difficult to construct geometrically because it is messy in radians. That 45 degrees is a very neat Pi/4 radians, but 40 degrees is a tricky 2Pi/9 radians. What that means in practical carpentry terms is you have to divide a circle into thirds, and then divide into thirds again, which gets a bit messy. but 45 degrees is easy to do by simply dividing in half twice, and the 60 degrees of a hexagon is also easy to do, as the sides are the same length at the radius of the circle. So a Nonagon is a pain to build, although I expect such an instrument would look very nice once completed by a skilled craftsman. Personally, I'm intrigued with seven sides, but that would be even harder to create geometrically, and is a mess digitally as well. It would be a step farther from the danger of rolling away than nine sides of course, but then again the good old hexagon has both of them beat on that score. Tradition has it that a ship's wheel (for steering) should have seven spokes, because it's not possible to divide a circle into seven equal segments by mechanical means (i.e., with a straightedge and drawing compass). It would have to be done by eye, and as with music, it's far easier to perceive deviations from perfection than to produce perfection. Thus, if the carpenter with his eye and hand could construct a visibly even 7-spoked wheel, then one could trust his ability to shape and fit other parts properly, and trust that the ship he built would be safe and secure on the ocean. I know that division into 4, 5, 6, 8, and 10 can be done mechanically. I'm not sure about 9, but I suspect it can, too. The next "impossible" division would be 11 sides or pie slices. (And then any further prime numbers or any number divisible by a prime number greater than 5. Anybody want a concertina with 31-sided ends... or 77?) But computing and cutting patterns under computer control, while not infinitely precise, can still give greater precision for division by any number at all than can be distinguished by the human eye or constructed by hand-steered tools, so it should certainly be "good enough". Link to comment Share on other sites More sharing options...
Chris Ghent Posted January 1, 2015 Share Posted January 1, 2015 A NONAGONAL shape always seemed to make sense to me. With 360 degrees 8 sides does not result in a whole number. One of my protractors is permanently set to 67.5°... Link to comment Share on other sites More sharing options...
JimLucas Posted January 1, 2015 Share Posted January 1, 2015 One of my protractors is permanently set to 67.5°... 5-1/3 sides? Link to comment Share on other sites More sharing options...
Chris Ghent Posted January 2, 2015 Share Posted January 2, 2015 One of my protractors is permanently set to 67.5°... 5-1/3 sides? When you make octagons from segments there are more angles involved than just 360/8..! Link to comment Share on other sites More sharing options...
Wolf Molkentin Posted January 2, 2015 Share Posted January 2, 2015 One of my protractors is permanently set to 67.5°... 5-1/3 sides? When you make octagons from segments there are more angles involved than just 360/8..! F.i.: 67,5 * 2 (resp. 4) + 45 (resp. 90) = 180 (resp. 360)? Link to comment Share on other sites More sharing options...
Chris Ghent Posted January 2, 2015 Share Posted January 2, 2015 Oops, not sure what you mean, I am not very maths savvy and I suspect things are expressed differently where you come from. Perhaps this will explain... Oops again, I can't post an image, the forum only wants a URL. Has the procedure changed? There is something called "My Media library" but no way to upload there either... Link to comment Share on other sites More sharing options...
Wolf Molkentin Posted January 2, 2015 Share Posted January 2, 2015 (edited) I wouldn't guess to have expressed myself following any strict convention. I was just trying to figure out the circumstances under which an angle as mentioned would occur. Assuming that a given (rectangular) workpiece would "have" a total of 180° (or 360°) at any event, 67,5° twice and one square angle would add up to 180°. Maybe this is misleading... Edited January 2, 2015 by blue eyed sailor Link to comment Share on other sites More sharing options...
Chris Drinkwater Posted January 2, 2015 Share Posted January 2, 2015 Going back to ten-sided concertinas, I went to a friends new year's eve musical party a few years ago and one of the other guest musicians was playing what I thought was a twelve-sided Edeophone English concertina. During a break in the playing, I went up to him to talk to him about his lovely Edeophone, only to discover that it had ten sides and had been made by Colin Dipper a few years earlier. I now wish I had been in a position to take a photo of this rare instrument. Chris Link to comment Share on other sites More sharing options...
JimLucas Posted January 2, 2015 Share Posted January 2, 2015 Going back to ten-sided concertinas, I went to a friends new year's eve musical party a few years ago and one of the other guest musicians was playing what I thought was a twelve-sided Edeophone English concertina. During a break in the playing, I went up to him to talk to him about his lovely Edeophone, only to discover that it had ten sides and had been made by Colin Dipper a few years earlier. I now wish I had been in a position to take a photo of this rare instrument. It sounds as if Colin has made several 10-sided concertinas. One of the few positive consequences of decimalisation? Link to comment Share on other sites More sharing options...
Geoffrey Crabb Posted January 2, 2015 Share Posted January 2, 2015 One of my protractors is permanently set to 67.5°... 5-1/3 sides? When you make octagons from segments there are more angles involved than just 360/8..! 67.5 is the angle of the mitred ends of the side pieces for an 8 sided shape. See attached. Geoffrey Link to comment Share on other sites More sharing options...
Little John Posted January 2, 2015 Share Posted January 2, 2015 "I know that division into 4, 5, 6, 8, and 10 can be done mechanically. I'm not sure about 9, but I suspect it can, too." Evariste Galois constructed a regular 17-gon mechanically (i.e. with only a straight edge and compasses) when he was 19. It was a discovery that led him to decide to become a mathematician. The branch of mathematics he founded, later called Galois Theory, was subsequently used to show that the only prime numbers p for which a p-gon can be constructed mechanically are 2, 3, 5, 17, 257 and 65537. Since you can bisect any angle mechanically that means that a polygon with p times a power of two can also be constructed in this manner, e.g. 4, 6, 8, 10. However, Galois also showed that an arbitrary angle cannot be trisected, so a nonagon (nine sides) can't be constructed mechanically. Incidentally, Galois also showed that the classical problem of squaring the circle is impossible. (That is, producing a square with the same area as a given circle using straight edge and compasses alone.) I understood all this at one time, but never thought I'd make use of it ... Link to comment Share on other sites More sharing options...
Chris Ghent Posted January 3, 2015 Share Posted January 3, 2015 One of my protractors is permanently set to 67.5°... 5-1/3 sides? When you make octagons from segments there are more angles involved than just 360/8..! 67.5 is the angle of the mitred ends of the side pieces for an 8 sided shape. Elegantly expressed and much easier than posting a .jpg from an Acad .dxf. Thanks Geoff and compliments of the quickly departing festive season to you and Jean..! "I know that division into 4, 5, 6, 8, and 10 can be done mechanically. I'm not sure about 9, but I suspect it can, too." Evariste Galois constructed a regular 17-gon mechanically (i.e. with only a straight edge and compasses) when he was 19. It was a discovery that led him to decide to become a mathematician. The branch of mathematics he founded, later called Galois Theory, was subsequently used to show that the only prime numbers p for which a p-gon can be constructed mechanically are 2, 3, 5, 17, 257 and 65537. Since you can bisect any angle mechanically that means that a polygon with p times a power of two can also be constructed in this manner, e.g. 4, 6, 8, 10. However, Galois also showed that an arbitrary angle cannot be trisected, so a nonagon (nine sides) can't be constructed mechanically. Incidentally, Galois also showed that the classical problem of squaring the circle is impossible. (That is, producing a square with the same area as a given circle using straight edge and compasses alone.) I understood all this at one time, but never thought I'd make use of it ... Using an engineering drawing program (it does not groan as I would if asked to create a 17 sided polygon) and cnc neatly steps around most maths issues but I would love to understand the details of your examples. Sadly, so far no indications of the issuing of a second lifetime, so pure maths will have to go in the junk pile along with building a yacht and reading the Rosetta Stone... Link to comment Share on other sites More sharing options...
David Barnert Posted January 5, 2015 Share Posted January 5, 2015 While we're comparing polygons, the Nott Memorial on campus at Union College in Schenectady, NY is reputed to be tho only 16-sided regular polygonal building in the western hemisphere. Link to comment Share on other sites More sharing options...
Robin Harrison Posted January 5, 2015 Share Posted January 5, 2015 I wonder if this one counts as having only one side ? Robin Round lachenal 2.bmp Link to comment Share on other sites More sharing options...
Tradewinds Ted Posted January 6, 2015 Share Posted January 6, 2015 Hmmm, maybe an infinite number of corners? Link to comment Share on other sites More sharing options...
Daniel Hersh Posted January 7, 2015 Share Posted January 7, 2015 These five-sided ones are all from this old c.net thread . Link to comment Share on other sites More sharing options...
Chris Ghent Posted January 7, 2015 Share Posted January 7, 2015 Hmmm, maybe an infinite number of corners? I was thinking less than one side, because if you start looking for a corner as you go around you will never get to it... Link to comment Share on other sites More sharing options...
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