tag:blogger.com,1999:blog-2433841880619171855.post4466473492876507681..comments2024-03-27T21:09:44.320+00:00Comments on Pat'sBlog: Gauss and Constuctable PolygonsUnknownnoreply@blogger.comBlogger2125tag:blogger.com,1999:blog-2433841880619171855.post-46717351185710258342011-03-24T15:40:39.954+00:002011-03-24T15:40:39.954+00:00When I get a chance to write it up (by tomorrow, m...When I get a chance to write it up (by tomorrow, maybe later today), I'll send you an algebraic way to obtain square root expressions for the 17th roots of 1.<br /><br />For the time being, it may be useful to others to indicate what "constructible" means from an algebraic standpoint. Roughly speaking, the constructible numbers are those that can be obtained by solving at most quadratic equations (in series or in parallel, to use electrical circuit terminology), beginning with the integers. In this sense, the rational numbers are those that can be obtained by solving linear equations, beginning with the integers. [In the case of rational numbers, you can always manage with at most one linear equation, something that is not the case with the constructible numbers.]<br /><br />Here are two examples, presented without proof.<br /><br />One way to show that cos(pi/17) can be obtained by solving at most quadratic equations, beginning with the integers, is: Let b be the positive solution of 4x^2 + x - 1 = 0. Let c be the positive solution of x^2 + 4bx - 1 = 0. Let d be the negative solution of 4x^2 + (16b + 4)x - 4 = 0. Then cos(pi/17) is the negative solution of x^2 + cx + d = 0, divided by -2.<br /><br />One way to show that cos(2*pi/17) can be obtained by solving at most quadratic equations, beginning with the integers (besides using appropriate trig identities with the previous sequence of steps), is: Let p, n be the positive and negative solutions of x^2 + x - 4 = 0. Let e be the positive solution of x^2 - px - 1 = 0. Let f be the positive solution of x^2 - nx - 1 = 0. Then cos(2*pi/17) is the greatest solution of x^2 - ex + f = 0.<br /><br />Incidentally, I can think of two natural algebraic ways to define what it means for a complex number to be constructible. One way is exactly as above, which among other things implies that we can make use of any known constructible complex numbers as coefficients of quadratic equations in showing that some specified complex number is constructible. The other way is to use the above to define what it means for a real number to be constructible, and then define a + bi (where a, b are real numbers) to be constructible if and only if both a and b are constructible real numbers. It can be shown that these two definitions lead to the same collection of complex numbers.Dave L. Renfrohttps://www.blogger.com/profile/00863074796446784081noreply@blogger.comtag:blogger.com,1999:blog-2433841880619171855.post-17030975134593608442011-03-20T13:30:56.554+00:002011-03-20T13:30:56.554+00:00I assume he is talking about Gauss' discovery...<i>I assume he is talking about Gauss' discovery that the heptadecagon was constructable with the classic tools of Greek Geometry.... hope that's it anyway.</i><br /><br />Yes that's it, and thank you very much Pat.<br /><br />Currently I'm studying Fourier Analysis and Fourier Transforms, you know, the Calculus that explains how you can "schmeer" the discrete with the continuous, and vice versa, that is to say for one example: in equations (analysis), how one can express sines/cosines, sawtooth waves, and square waves in the same body of Mathematical tools. We studied this in Engineering school, but I forgot so it's a nice review.<br /><br />I mention that because no matter how far one goes into analysis, geometry is the best way to hook interest in math (textbooks are the worst ... it's almost like if you want to kill a young student's love for a subject, force him to read a textbook on same), and Geometry will always be my greatest love.Steven Colyerhttps://www.blogger.com/profile/10435759210177642257noreply@blogger.com