Wikipedia:Reference desk/Archives/Mathematics/2022 June 16

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June 16

Understanding the procedure to divide a circle in 5 parts

I can follow the procedure, like describe here for example: [1]

I just want to know why does it work. Why you take half the radius + GH and get GJ. And why is GJ == 365/5. 360/5.

--Bumptump (talk) 18:18, 16 June 2022 (UTC)

Several constructions of a regular pentagon are given at Pentagon#Geometrical constructions, but, while similar, I don't think they're quite the same as the one in the link. (This section of the article isn't very clear IMO.) In that procedure, Step 2 is to create a line through O perpendicular to AB. Step 3 is to bisect OB at H. If the radius of the circle is 1 then OH is 1/2 and GH is √5/2 by the Pythagorean theorem. In step 4, JH = GH = √5/2, so OJ = (√5-1)/2. Applying the Pythagorean theorem again gives GJ = √((5-√5)/2). (I don't know where you're getting 365/5 from.) This turns out to be 2 sin(π/5) which is the side of the pentagon. In general, the side of a regular n-gon inscribed in a circle of radius 1 is 2 sin(π/n). That 2 sin(π/n) = √((5-√5)/2) is trickier, but it follows from 2 cos(π/n) = (√5+1)/2 and there are several ways to show this. The procedure given is incomplete since it doesn't justify that the result is correct; a justification is required for the procedure to be valid. I think the cut-the-knot page has essentially the same procedure but with more detail. --RDBury (talk) 21:17, 16 June 2022 (UTC)

Here is an algebraic way to establish the trigonometric values. Define

$C={\sqrt {\frac {3-{\sqrt {5}}}{8}}}=0.309...\,,$

$S={\sqrt {\frac {5+{\sqrt {5}}}{8}}}=0.952...\,.$

It is easily verified that $C^{2}+S^{2}=1$, which means that for some angle $\alpha$, $\cos \alpha =C$ and $\sin \alpha =S$. The numerical approximations of $C$ and $S$ show that we can take $\alpha$ to satisfy ${\tfrac {1}{4}}\pi <\alpha <{\tfrac {1}{2}}\pi .$ Using the trigonometric multiple-angle formulae, we calculate that

$\cos 5\alpha =C^{5}-10C^{3}S^{2}+5CS^{4},$

$\sin 5\alpha =5C^{4}S-10C^{2}S^{3}+S^{5}.$

After an elementary computation and simplification of the right-hand sides we obtain:

$\cos 5\alpha =1,$

$\sin 5\alpha =0.$

(A shortcut: the rhs for $\sin 5\alpha$ can be written as $S(5C^{4}-10C^{2}S^{2}+S^{4}),$ so it is sufficient to show that $5C^{4}-10C^{2}S^{2}+S^{4}=0.$ It follows then that $|\cos 5\alpha |=1$; that the value is positive follows from the bounds on $\alpha$.) So, for some value of $k$, $5\alpha =2k\pi .$ From the bounds on $\alpha$ we see that $k=1$, and therefore $\alpha ={\frac {2\pi }{5}},$ the angle of a slice of a circular pie if it is cut up in 5 exactly equal pieces. Relating the radical expressions for $C$ and $S$ to the geometric construction is tedious but elementary.  --Lambiam 22:38, 16 June 2022 (UTC)

The Dirichlet kernel can be used as well. From (see List of trigonometric identities#Lagrange's trigonometric identities)

$1+2\cos(x)+2\cos(2x)={\frac {\sin {\tfrac {5}{2}}x}{\sin {\tfrac {1}{2}}x}},$

we have

$1-2\cos {\tfrac {\pi }{5}}+2\cos {\tfrac {2\pi }{5}}=1+2\cos {\tfrac {2\pi }{5}}+2\cos {\tfrac {4\pi }{5}}={\frac {\sin \pi }{\sin {\tfrac {\pi }{5}}}}=0.$

If φ=2 cos π/5 then φ² = 4 cos² π/5 = 2 (cos 2π/5 + 1). Applying the previous equation, φ² - φ - 1 = 0 and φ can be determined by the quadratic formula. (It is, in fact, the Golden ratio.) This can be generalized to produce algebraic equations for ω_n for any odd n.

ω₇³-ω₇²-2ω₇+1=0

ω₉³-3ω₉-1=0

ω₁₁⁵-ω₁₁⁴-4ω₁₁³+3ω₁₁²+3ω₁₁-1=0

--RDBury (talk) 02:43, 17 June 2022 (UTC)

This won't be helpful but in geometry class I remember learning how to derive the construction of the regular pentagon using Galois theory. The instructor said that it was much harder than the hexagon, octagon, etc. because while (like in any classical construction) you could only make quadratic extension fields, the pentagon was the first figure where you had to actually do multiplication in the extension field rather than just addition. Sorry if that makes no sense. I can't make that much sense of it either by now, but the explanation somewhat stuck with me. I remember never managing to do the pentagon or 7-gon while fooling with a compass as a kid. The 7-gon is impossible, but the pentagon is doable given fancier methods than I had available at the time. 2601:648:8202:350:0:0:0:90B2 (talk) 06:26, 17 June 2022 (UTC)