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The Devil's curves

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  • Roger Bagula
    I call them the Devil s curves after tthe Devils s staircase. The idea that the Pascal s ( Sierpinski gasket) triangles: (x+1)^n and (1-x)^n are fractal isn t
    Message 1 of 1 , Feb 1, 2009
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      I call them the Devil's curves after tthe Devils's staircase.
      The idea that the Pascal's ( Sierpinski gasket) triangles:
      (x+1)^n
      and
      (1-x)^n
      are fractal isn't new. Combinations as in Binomial[n,m]
      are always subsets of n!.
      The set of polynomial given by ChebyshevT and ChebyshevU
      are Hilbert space levels that are topolgically covering for two space.
      ( they are A_n or SU(n+1) like in group theory terms).
      I noticed last night that the polynomial :
      p(x,n)=(x+1)^n+(1-x)^n
      and
      q(x,n)=(x+1)^n-(1-x)^n
      behave very much like cosine and sine in being even and odd.
      tc[n_, m_] = Binomial[n, m] + (-1)^m*Binomial[n, m]
      ts[n_, m_] = Binomial[n, m] - (-1)^m*Binomial[n, m]
      What if we subtracted the fractal Pascal triangle curves from the
      Hilbert space
      Chebyshev curves?
      The result should be a set of Cantor dust like curves.
      To make them better, make them symmetrical by adding the toral or reverse
      curves to them.
      p[x_, n_] = -(ChebyshevT[n, x] - ((x + 1)^n + (1 - x)^n))
      q[x_, n_] = -(ChebyshevU[n, x] - ((x + 1)^n - (1 - x)^n))
      sp[x_, n_] = p[x, n] + x^n*p[1/x, n]
      sq[x_, n_] = q[x, n] + x^n*q[1/x, n]
      The do show a cusp like behavior when plottted against each other:
      a = Table[ParametricPlot[{sp[x, n], -sq[x, n]}, {x, -1, 1}], {n, 2, 12}]
      Show[a, PlotRange -> All]
      The fractals seem to be dust like ( disconnected):
      Clear[a]
      a = Table[CoefficientList[FullSimplify[ExpandAll[sp[x, n]]], x], {n, 2,
      32}];
      b = Table[If[m ≤ n, Mod[a[[n]][[m]], 3], 0], {m, 1, Length[
      a]}, {n, 1, Length[a]}];
      ListDensityPlot[b, Mesh -> False]

      The fractals:
      Clear[a]
      a = Table[CoefficientList[FullSimplify[ExpandAll[sq[x, n]]], x], {n, 2,
      32}];
      b = Table[If[m ≤ n, Mod[a[[n]][[m]], 3], 0], {m, 1, Length[
      a]}, {n, 1, Length[a]}];
      ListDensityPlot[b, Mesh -> False]

      %I A155994
      %S A155994 2,3,8,3,6,10,10,6,17,16,24,16,17,30,4,52,52,4,30,63,24,56,80,
      %T A155994 56,24,63,126,22,234,10,10,234,22,126,257,32,488,224,480,224,488,
      %U A155994 32,257,510,8,1096,328,420,420,328,1096,8,510,1023,40,2244,480
      %V A155994
      -2,-3,8,-3,-6,10,10,-6,-17,16,24,16,-17,-30,4,52,52,4,-30,-63,24,56,80,
      %W A155994
      56,24,-63,-126,22,234,-10,-10,234,22,-126,-257,32,488,224,-480,224,488,
      %X A155994
      32,-257,-510,8,1096,328,-420,-420,328,1096,8,-510,-1023,40,2244,480
      %N A155994 A triangle of polynomial coefficients: p(x,n)=-(ChebyshevU[n,
      x] - ((x + 1)^n - (1 - x)^n)); sp(x,n) = p(x, n) + x^n*p(1/x, n).
      %C A155994 Row sums are:
      %C A155994 {-2, 0, 2, 8, 22, 52, 114, 240, 494, 1004, 2026,...}.
      %F A155994 p(x,n)=-(ChebyshevU[n, x] - ((x + 1)^n - (1 - x)^n));
      %F A155994 sp(x,n) = p(x, n) + x^n*p(1/x, n).
      %e A155994 {-2},
      %e A155994 {},
      %e A155994 {-3, 8, -3},
      %e A155994 {-6, 10, 10, -6},
      %e A155994 {-17, 16, 24, 16, -17},
      %e A155994 {-30, 4, 52, 52, 4, -30},
      %e A155994 {-63, 24, 56, 80, 56, 24, -63},
      %e A155994 {-126, 22, 234, -10, -10, 234, 22, -126},
      %e A155994 {-257, 32, 488, 224, -480, 224, 488, 32, -257},
      %e A155994 {-510, 8, 1096, 328, -420, -420, 328, 1096, 8, -510},
      %e A155994 {-1023, 40, 2244, 480, -1232, 1008, -1232, 480, 2244, 40, -1023}
      %t A155994 p[x_, n_] =-(ChebyshevU[n, x] - ((x + 1)^n - (1 - x)^n));
      %t A155994 sp[x_, n_] = p[x, n] + x^n*p[1/x, n];
      %t A155994 Table[FullSimplify[ExpandAll[sp[x, n]]], {n, 0, 10}];
      %t A155994 Table[CoefficientList[FullSimplify[ExpandAll[sp[x, n]]], x],
      {n, 0, 10}]; Q Flatten[%]
      %K A155994 sign,tabl
      %O A155994 0,1
      %A A155994 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 01 2009


      %I A155993
      %S A155993 2,1,1,3,0,3,2,9,9,2,5,0,40,0,5,14,5,40,40,5,14,27,0,90,0,90,0,
      %T A155993 27,62,21,154,14,14,154,21,62,125,0,400,0,40,0,400,0,125,254,9,
      %U A155993 648,288,180,180,288,648,9,254,507,0,1410,0,120,0,120,0,1410,0,507
      %V A155993 2,1,1,3,0,3,-2,9,9,-2,-5,0,40,0,-5,-14,5,40,40,5,-14,-27,0,90,0,90,0,
      %W A155993 -27,-62,21,154,14,14,154,21,-62,-125,0,400,0,-40,0,400,0,-125,-254,9,
      %X A155993 648,288,-180,-180,288,648,9,-254,-507,0,1410,0,120,0,120,0,1410,0,-507
      %N A155993 A triangle of polynomial coefficients: p(x,n)=-(ChebyshevT[n, x] - ((x + 1)^n + (1 - x)^n)); sp(x,n) = p(x, n) + x^n*p(1/x, n).
      %C A155993 Row sums are:
      %C A155993 {2, 2, 6, 14, 30, 62, 126, 254, 510, 1022, 2046,...}.
      %F A155993 p(x,n)=-(ChebyshevT[n, x] - ((x + 1)^n + (1 - x)^n));
      %F A155993 sp(x,n) = p(x, n) + x^n*p(1/x, n).
      %e A155993 {2},
      %e A155993 {1, 1},
      %e A155993 {3, 0, 3},
      %e A155993 {-2, 9, 9, -2},
      %e A155993 {-5, 0, 40, 0, -5},
      %e A155993 {-14, 5, 40, 40, 5, -14},
      %e A155993 {-27, 0, 90, 0, 90, 0, -27},
      %e A155993 {-62, 21, 154, 14, 14, 154, 21, -62},
      %e A155993 {-125, 0, 400, 0, -40, 0, 400, 0, -125},
      %e A155993 {-254, 9, 648, 288, -180, -180, 288, 648, 9, -254},
      %e A155993 {-507, 0, 1410, 0, 120, 0, 120, 0, 1410, 0, -507}
      %t A155993 p[x_, n_] = -(ChebyshevT[n, x] - ((x + 1)^n + (1 - x)^n));
      %t A155993 sp[x_, n_] = p[x, n] + x^n*p[1/x, n];
      %t A155993 Table[FullSimplify[ExpandAll[sp[x, n]]], {n, 0, 10}];
      %t A155993 Table[CoefficientList[FullSimplify[ExpandAll[sp[x, n]]], x], {n, 0, 10}]; Q Flatten[%]
      %K A155993 sign,tabl
      %O A155993 0,1
      %A A155993 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 01 2009




      --
      Respectfully, Roger L. Bagula
      11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :http://www.geocities.com/rlbagulatftn/Index.html
      alternative email: rlbagula@...
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