[math-fun] star polygon area
(Very elementary stuff.) When n>2d, (d,n)=1, connecting every dth vertex of a regular n-gon makes an n-pointed star {n/d} consisting of d-2 concentric rings of kites surrounding a ring of isosceles triangles surrounding a regular n-gon. Working inward, successive rings of kites have winding numbers W=1,2,3,...; the triangles have W=d-1, and for the polygon, W=d. For unit edges, the area is %pi d cot(-----) n n n A(-) = ------------, d 4 where the component polygons are weighted by their winding numbers, i.e, W dx dy integrated over the whole plane. This correctly gives n n A(-----) = - A(-), n - d d i.e., mirror imaging negates the area by winding clockwise. The %pi d csc(-----) n circumradius = ----------, and the 2 %pi d cot(-----) n inradius = ----------. 2 For unit circumradius instead of unit sides, the area is %pi d sin(-----) n n n a(-) = ------------, d 2 which, for fixed n is clearly maximal for d=n/4. I.e., the star with greatest weighted area is the one with most nearly right-angled vertices, since the vertex angles are pi - 2 pi d/n. E.g., the star of Lakshmi (two squares) is maximal among the octagrams. Removing just the outermost edges that form the vertex angles leaves an {n/(d-1)} star polygon, if when (n,d-1)=g we interpret this as g {n/g,(d-1)/g} stars superposed. Interestingly, for such improper stars, continued edge deletion recovers the proper ones with lesser d. E.g., {9/4} -> {9/3} (star of Goliath = three triangles) -> {9/2}. This tells us all the interior angles. The circumradius formula for consecutive d provides the proportions of the inner stars, and hence the sizes of all the kites, etc. Note also that the area formulas work for improper stars: A((gn)/(gd)) = g A(n/d). --rwg I keep warm peanut butter in my pockets AND I VOTE
I prattled: >The circumradius formula for consecutive d provides the
proportions of the inner stars, and hence the sizes of all the kites, etc. Duh, I meant inradius. You get the side ratios by scaling the {n/d} and {n/e} inradii to be equal.
Now from the elementary to the mysterious. Recall the odd (mod 3) squarefree sequence d(k) = 0 1 2 1 0 1 2 0 2 1 0 2 0 1 2 1 0 1 2 0 1 0 2 1 2 0 2 1 0 ... defined as the left-to-right alternating sum of the nonzero trits of k. Running sums of cis(%pi (k+2d(k)/3)) draw self-avoiding polygonal approximations to a continuous map S(t) of a real interval onto the usual triangular Sierpinski gasket (fractal dimension lg 3). Besides this crunchy application, d(k) also appears in a squishy identity for the Fourier coefficients of S(t). Specifically, let S map {-1,0,1} to {-i, sqrt(3), i} respectively. Furthermore, let S_m map [-1,2m-1] onto m copies of S arranged around a regular m-gon of side 2 centered at 0. Then the Fourier series is inf ==== \ 1 S (t) = > a(t, - + k), m / m ==== k = - inf where %i %pi t x 2 %e f(2 %pi x) a(t, x) := -------------------------, %pi x and f is defined four different ways(!) by the matrix product [ 1 x %pi ] inf [ - - cos(-- - ---) ] /===\ [ 2 n 3 ] [ f(x) - f(x) ] | | [ 3 ] [ ] = | | [ ]. [ - f(- x) f(- x) ] | | [ x %pi 1 ] n = 1 [ - cos(-- + ---) - ] [ n 3 2 ] [ 3 ] Negative m flips the gaskets into the inside of the m-gon. E.g. m=-6 makes a nice doily. Now define k k x 2 %pi d(k) (- 1) cos(--- + ----------) n 3 3 b := b (x,n) := ---------------------------- . k k n 2 Claim: n 3 + 1 ------ 2 ==== \ f(x) = limit > b n -> inf / k ==== k = 0 n n 3 - (- 1) ----------- 4 ==== \ = 2 limit > b n -> inf / 2 k ==== k = 0 n n 3 - (- 1) ----------- 4 ==== \ = 2 limit > b n -> inf / 2 k + 1 ==== k = 0 n - 1 3 - 1 ---------- 2 ==== \ = 4 limit > b n -> inf / 3 k + 1 ==== k = 0 n - 1 3 - 1 ---------- 2 ==== \ = 4 limit > b n -> inf / 3 k + 2 ==== k = 0 n - 1 3 - 1 ---------- 2 ==== \ = 2 limit > b . n -> inf / 3 k ==== k = 0 Only the last seems obvious, since d(3k) = d(k). Other dissections of the series, e.g., sum(b[4k+a]) or sum(b[6k+a]) seem to produce f(x) times weird step functions, or converge to different values for odd and even n. I showed earlier how to evaluate S for rational t, which gives us exotic closed forms for the Fourier sums. --rwg Aristotelian retaliations
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R. William Gosper