Saturday, 23 April 2011

Constructructable Polygons, and X^17 = 1

A while back I wrote a blog about Gauss' proof that the 17 gon (heptadecagon) was constructable.  Dave Renfro then set himself the goal of writing a clear and complete explanation such that a really good high school student (or more particularly, a somewhat slow high school teacher like me) could understand.  After very many hours of labor and multiple "final" drafts he has done a super job. It is a document well worth the attention of every upper level HS math student and teacher.  Thanks, Dave.

  I have posted his complete document. A Detailed and Elementary Solution to x^17=1  ,
and below is his introduction.

This manuscript was written during a three week period in March-April 2011 and it was motivated by Pat Ballew's 20 March 2011 blog entry Gauss and Constuctable Polygons <>. It occurred to me that, in all of the many dozens of expositions about solutions to x^17 = 1 I had encountered in 30+ years, none gave all the details for obtaining one of the lengthy square root expressions one often sees exhibited. This includes the huge number of class notes and unpublished manuscripts that have populated the internet in the past 15 years, although that may no long be the case after this manuscript circulates. I suppose a case could be made that this is done in Klein [44] (Article 6, pp. 29-32) and in Barnard/Child [4] (pp. 174-175). However, I have not encountered any treatment where sufficient detail was given so that someone competent in high school algebra and trigonometry, but otherwise not very mathematically sophisticated, could be reasonably expected to follow. With this in mind, there are three main ways that I have tried to distinguish the present treatment from the many already in existence.

First, I have tried to be extremely detailed and explicit in the exposition, especially in carrying out algebraic manipulations, in rewriting numerical expressions, and in giving precise explanations for how the manipulations and rewriting are done.

Second, in the case of obtaining explicit square root expressions related to the equation x^17 = 1, no computer algebra systems or calculators are used. In fact, I believe the only instance in which I needed to multiply something out by hand using grade school arithmetic methods was (34)(170) = 578 at one point, although even this could have been avoided, as will be evident when I obtain a more compact square root expression for Cos(2 Pi/17) .

Third, I have paid as much attention to the completeness and accuracy of the bibliography as I have with the mathematics. Thus, author names, journal titles, page citations, and the like are detailed and explicit. I have also given careful thought to the items included, with the exception of the x^257=1 entries (whose novelty and rarity merit the exception). My intended audience consists mainly of English readers who are high school students and their teachers, faculty at small colleges, and other math enthusiasts who do not have access to the resources of a research university library (an extensive book and journal collection, JSTOR and other expensive digital collections, etc.). Thus, the items in the bibliography were mostly selected for being freely available on the internet and for assuming relatively little mathematical background knowledge. I made a few exceptions to these principles, but this was done when I thought an item might be especially useful to someone interested in exploring the subject further. Also, I have noted which books have been reprinted by Dover Publications, because such a reprinting (even if no longer in print) makes it much more likely that the book can be found on the shelves of a public library or a small college library.


Steven Colyer said...

The link didn't work for me. But yeah, cool! Is there an arXiv paper in this? Renfro-Ballew-Colyer? Or don't I get props for getting you to think of it, which got Dave to think of it, and actually do it?

OK, just call it Renfro-Ballew then, I don't care. What's cool is it was done, and points out that there are always new problems to be solved in Mathematics, and new questions to ask! Too many students think Maths is a done deal; that we know everything. No. Barry Mazur taught me that in a book introduction I read by him. Good guy.

And remember Pat, Hermann Grassmann was "just" a German high school math teacher, but he came up with Grassmann algebras, and all THAT did was change Mathematics, and physics, and so, the world. People can make a contribution to the field at any age as well. Euler's life proved that. ;-)

Pat's Blog said...

Steven, Thanks, I'll check and fix the link... And actually I was just the middle man, so maybe the Colyer-Refro paper...although writing a paper with Dave is a goal for me since I can raise my Euler number above infinity... maybe when I'm back in the US full time you and I can do one too, there must be SOMEONE who would publish it.

Dave said...

This is fantastic, and your timing is amazing. I've been spending the past couple of weeks reading about (and struggling through) Gauss' cyclotomy work. Dave's right---it is difficult to find a good source on this. I found the Hadlock book to be very helpful (and of course you can go back to Gauss's original work, which has been translated into English).

So, while I wish you'd published this two weeks ago, I am still very much looking forward to reading it.


Pat's Blog said...

Dave, glad to help out from me to you once. Seems I owe you several..
David R is a wonderful researcher and fastidious writer (as opposed to my shoot from the hip, misspelled cacophony.. *don't know the equivalent term for bad writing, cacographony?) .
Do you have links where Gauss is translated that are available to the peons who don't have access to Jstor? I know, lazy, I should search this out.

Dave said...

You blog is great, Pat. I love it.

Gauss's work on cyclotomy is in book 7 of his Disquisitiones Arithmeticae, which is translated and is a Springer book. It is on Google books (, but unfortunately, the relevant pages are blocked. Here's a scan (that I found on the 'net) of some of the pages of interest:

It cuts off before my favorite part, which is Gauss's acknowledgment of the converse to his constructibility theorem, but his omission of a proof:

"The limits of the present work exclude this demonstration here, but we issue this warning lest anyone attempt to achieve geometric constructions for sections other than the ones suggested by our theory (e.g., sections into 7, 11, 13, 19, etc. parts) and so spend his time uselessly."

Steven Colyer said...

OK thanks, Pat, the link went through. Wow, wow, whoa, awesome work, very clear. I must research this Renfro chap. Job well done. And since he does reference you in the very first line, all is love. Thanks.

If you permit me to talk Euler for a bit, I recently learned that there are still Euler papers waiting to be translated into English! Wow, what language did he write in mostly? German, Latin, English, or Russian?

Dave said...

Steven, there are many untranslated Euler articles. They're collecting them at the Euler Archive, which was a project started by some grad students at Dartmouth, but have now graduated and moved on to teaching jobs. The Euler archive just moved to the MAA

Speaking of Euler, he solved the equations x^n-1=0 for n<11 (that is, gave solutions using radicals) You can read about it here:

Steven Colyer said...

Thanks Dave (Richeson). Yes, it was at Division by Zero, your weblog, that I learned that. Thanks bunches for that link, which you link to yt again today on your tombstone article. Sweet.

Does anyone know WHICH Russian works Julian Barbour (The End of Time author) translates into English, in order to pay his rent, when FQXi isn't giving him grants? Hmmm, maybe Euler. But there are so many great Russian mathematicians to choose from.