#91041
01/04/2003 7:52 PM
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Joined: Jan 2003
Posts: 2
stranger
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stranger
Joined: Jan 2003
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Like most people, I am familiar with the names for common polygons, such as pentagon, hexagon and octagon for polygons of 5, 6 and 8 sides, respectively. I now find myself in need of a word to describe a 15-sided polygon. Does anybody here know if there is such a word? Is there a rule of thumb for the naming of polygons of various numbers of sides?
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#91042
01/04/2003 8:14 PM
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Joined: Jun 2002
Posts: 7,210
Carpal Tunnel
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Carpal Tunnel
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formerly known as etaoin...
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#91043
01/04/2003 8:31 PM
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Joined: Jun 2002
Posts: 7,210
Carpal Tunnel
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Carpal Tunnel
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formerly known as etaoin...
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#91045
01/04/2003 9:31 PM
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Joined: Jun 2002
Posts: 7,210
Carpal Tunnel
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Carpal Tunnel
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that's a good site too, Bill. if you scroll down on my second link, it lists them all up to 50, and then by tens, etc. 
formerly known as etaoin...
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#91046
01/04/2003 10:33 PM
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Joined: Jan 2001
Posts: 13,858
Carpal Tunnel
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Carpal Tunnel
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#91047
01/05/2003 1:12 AM
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Joined: Jun 2002
Posts: 1,624
Pooh-Bah
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Pooh-Bah
Joined: Jun 2002
Posts: 1,624 |
And here is a site about how to construct one:
You're way out in geek territory there, Bill. Why not just draw a circle and call it a pentadecagon? Or even Jim or Bob, for that matter. Who'd notice?
- Pfranz
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#91048
01/05/2003 1:32 AM
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Joined: Jul 2000
Posts: 3,467
Carpal Tunnel
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Carpal Tunnel
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Many years ago I was sitting looking at a styrofoam coffee cup, which is basically a conic section. I can;t remember what possessed me to do this, but I started cutting the coffee cups into rings about an inch deep and using paper clips to see how they went together.
The inner rings didn't work really well, but the largest rings, which were just a tad thicker than the rest of the cup, when paperclipped together, yielded a globe about 20 inches in diameter. The pattern was quite complex, since there were places where instead of six circles around a central circle the natural geometry of the circles required a pentagonal formation, much like the patterns you see on geodesic domes.
Eventually, I figured out that I could put a white globe light source in the middle of this larger globe, and I made several of them as lamps for friends of mine.
Mike Hill, when he moved from the DC area to (coincidentally) here in the Denver area, had one trashed during the shipment of his household goods and the moving company actually paid him $200 for the damage! And this was perhaps 25 years ago.
The things were actually quite attractive if you didn't get right up close to them, and made interesting hanging lights.
And everyone's wondering where the pun is. Sorry to disappoint you!
TEd
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#91049
01/05/2003 2:33 AM
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Joined: Jan 2001
Posts: 13,858
Carpal Tunnel
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Carpal Tunnel
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Dear Pfranz: If I had devised that solution, or could prove it, you'd be entitled to call me a geek.
I used to enjoy doing originals in geometry, but can't imagine how to construct an angle of
twenty four degrees. I don't have enough smarts left to even think about it.
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#91050
01/05/2003 3:19 AM
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Joined: Jan 2001
Posts: 13,858
Carpal Tunnel
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Carpal Tunnel
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Hey, Pfranz: This Internet is really something. Just for the hell of it, I searched for and found a way to construct an angle of 24 degrees. An equilateral triangle has 120 debrees. A regular pentagon has an angle of 72 degrees. The difference is 48 degrees. Half of that is 24 degrees. I found a site to construct a regular pentagon. It is fairly short, but I haven't mastered it yet. http://mathforum.org/dr.math/faq/formulas/faq.regpoly.html#floorNot quite enough to pat myself on the back, and claim I'm a geek. Bill
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#91051
01/05/2003 2:34 PM
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Joined: Jan 2003
Posts: 2
stranger
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stranger
Joined: Jan 2003
Posts: 2 |
Thanks to everyone who responded. The second post by etaoin gave me a direct answer. I'm going to use pentakaidecagon but will allow that pentadecagon is also correct.
Thanks again,
John Whittier
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#91052
01/05/2003 2:56 PM
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Joined: Jan 2001
Posts: 13,858
Carpal Tunnel
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Carpal Tunnel
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Any geometers on the Board? I was able to repeat the construction, but as yet do not see
how to prove it. That will be something to work on while the board is quiet.I'd welcp,e some
competition.
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#91053
01/05/2003 4:12 PM
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Joined: Jan 2001
Posts: 13,858
Carpal Tunnel
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Carpal Tunnel
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The motto of Plato's Academy was:. "Let no one ignorant of geometry enter here.:
Have we no geometers here?
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#91054
01/05/2003 4:46 PM
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Joined: Dec 2002
Posts: 122
member
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member
Joined: Dec 2002
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>I found a site to construct a regular pentagon. It is fairly short, but I haven't mastered it yet.Albrecht Durer in his book Anderweysung der messung mit der zirckel und richtscheyt in Linien ebnen und gantzen corporen durch Albrecht Durer zusamen getzogen und zu nutz aller kunstliebhabenden mit zugehorigen figuren in truck gebracht im jar MDXXV  presented very simple and elegant way how to form a regular pentagon only with a ruler and one base of a circle (hope it means what i wanted to say ). i've read that the construction is not perfect but for practical goals - good enough.
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#91055
01/05/2003 5:38 PM
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Joined: Mar 2000
Posts: 6,511
Carpal Tunnel
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Carpal Tunnel
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im jarShouldn't that be "jahr"?
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#91056
01/05/2003 5:46 PM
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Joined: Jan 2001
Posts: 13,858
Carpal Tunnel
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Carpal Tunnel
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Shouldn't that be "jahr"?
Ja.
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#91057
01/05/2003 5:51 PM
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Joined: Jan 2001
Posts: 13,858
Carpal Tunnel
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Carpal Tunnel
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Here's a URL that just with compass step hy step constructs a regular pentagon. Down at bottom of the green screen the is a circle that can be moved by clicking on it twice for each step, and the put back to zero to do it again. I still can't explain it or prove it. http://poncelet.math.nthu.edu.tw/disk3/cabrijava/c-pentagon.html
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#91058
01/05/2003 9:32 PM
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Joined: Dec 2002
Posts: 122
member
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member
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>Shouldn't that be "jahr"? as far as i know - yes, it should. but who knows that eccentric Durer? suppose nobody, cause he's dead :P
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#91059
01/05/2003 9:36 PM
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Joined: Dec 2002
Posts: 122
member
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member
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>I still can't explain it or prove it.
it would be much easier to understand and prove if there weren't so many circles :P
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#91060
01/05/2003 9:45 PM
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Joined: Aug 2001
Posts: 11,074 Likes: 2
Carpal Tunnel
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Carpal Tunnel
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Calling all pragmatists: how to construct a regular pentagon without even a compass?
Start with a ribbon of paper. (How you construct that is a separate problem.)
Fold it into a simple overhand knot. Gently pull tight and flatten. The knot is a regular pentagon. !
(Q: What does "pragmatist" mean?
A: I don't know, but it doesn't really matter as long as I can use it properly.)
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#91061
01/06/2003 4:41 PM
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Joined: Jan 2001
Posts: 13,858
Carpal Tunnel
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Carpal Tunnel
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#91063
01/07/2003 6:09 AM
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Joined: Mar 2000
Posts: 315
enthusiast
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enthusiast
Joined: Mar 2000
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Me!
Well, the constructions for the pentagon are not difficult to explain, but not so immediate to prove. I was used to prove one of them in a third year class of university course in mathematics, and it required to master complex numbers ;indeed, the key fact here is that the vertices of the regular polygon with n edges can be seen as the n-roots of 1, i.e.they correspond to complex numbers z= (a + b i) such that z times z times z ... n times gives 1. Here you should know the i times i gives -1.
The important fact here is to undestand that we are talking about "precise" constructions made just with ruler and compass.
There is a famous theorem of Gauss stating that just few polygons are constructible that way - I remember 3,5, 17,255... there is a rule...
the other construction you can find are approximate, in the sense that the error is so small that they are good for applications (for example, building gears)
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#91064
01/07/2003 10:47 AM
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Joined: Jul 2000
Posts: 3,467
Carpal Tunnel
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Carpal Tunnel
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What about a hexagon? That one's trivial.
TEd
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#91065
01/07/2003 12:13 PM
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Joined: Mar 2000
Posts: 315
enthusiast
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enthusiast
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yes, indeed.
I was speaking just about the possible values of n which are prime numbers.
The real statement is that the "constructible" polygons have n = 2 times 2...times 2 (any number of times) times again p1 times p2...times pw, where p1,p2,...,pw are distinct primes of the kind I was saying before. I could write the rule, if I had a mathematical editor.
So, 6 = 2 times 3 is ok.
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#91066
01/07/2003 12:20 PM
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Joined: Mar 2000
Posts: 315
enthusiast
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enthusiast
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The only known Primes of this form are 3, 5, 17, 257, and 65537.
A clear - I suppose- statement is well written in
http://hades.ph.tn.tudelft.nl/Internal/PHServices/Documentation/MathWorld/math/math/c/c615.htm
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#91067
01/07/2003 1:56 PM
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Joined: Jan 2001
Posts: 13,858
Carpal Tunnel
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Carpal Tunnel
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" A Portia come to judgement!"
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#91068
01/07/2003 2:24 PM
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Joined: Dec 2000
Posts: 13,803
Carpal Tunnel
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Carpal Tunnel
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The only known Primes of this form are 3, 5, 17, 257, and 65537.
I note that these are all primes of the form 2^n+1, where n is a power of two.
OK I looked at the web page and I see that my little insight was explicitly stated.
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#91069
01/07/2003 2:46 PM
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Joined: Jan 2001
Posts: 13,858
Carpal Tunnel
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Carpal Tunnel
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I still remember that e to the pi i = minus one. But only that.
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#91070
01/07/2003 5:28 PM
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Joined: Aug 2001
Posts: 11,074 Likes: 2
Carpal Tunnel
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Carpal Tunnel
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The only known Primes of this form are 3, 5, 17, 257, and 65537.
I note that these are all primes of the form 2^n+1, where n is a power of two.
Are these the Mersenne Primes? Something like (2^(2^n))+1? I think I remember something about the series failing at n>3, with the counterexample for n=4 being publically and painstakingly rendered on the blackboard in front of the assembled multitudes, one of the factors being 671...Does that sound familiar to anyone? But I don't recall ever coming across the numbers in the context of constructing regular polygons.
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#91071
01/07/2003 5:52 PM
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Joined: Dec 2000
Posts: 13,803
Carpal Tunnel
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Carpal Tunnel
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#91072
01/07/2003 5:55 PM
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Joined: Mar 2000
Posts: 315
enthusiast
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enthusiast
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the previous ones are the Fermat primes, indeed of this kind(2^(2^n))+1
The problem is that , given n, it is not known usually if the corresponding Fermat number is prime (usually not)
The Mersenne primes are
2^n - 1 ( I was not aware that this way of writing was almost universally understood) - when they are prime themselves.
n has to be prime indeed (Faldage, you were right) because if n = ab, then the above number cannot be prime
In that case, you could write
(2^a)^b - 1^b =
[2^a - 1 ] times { something else}
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#91073
01/07/2003 6:00 PM
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Joined: Dec 2000
Posts: 13,803
Carpal Tunnel
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Carpal Tunnel
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almost universally understood
The ^ is a common operator for exponentiation in computer languages such as BASIC.
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#91074
01/07/2003 6:05 PM
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Joined: Mar 2000
Posts: 315
enthusiast
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enthusiast
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I know, of course, but I didn't assume that this knowledge is spread around...
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#91075
01/07/2003 6:16 PM
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Joined: Mar 2000
Posts: 6,511
Carpal Tunnel
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If it's any consolation, Emanuela, I certainly didn't know!
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#91076
01/07/2003 6:21 PM
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Joined: Dec 2000
Posts: 13,803
Carpal Tunnel
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Carpal Tunnel
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spread around
Perhaps "scattered around" would be a better phrase. That is, it's widely known but not by that many people.
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