       New kind of Weierstrass fractal scaling: Eulerian scales,MacMahon

• To: mathgroup at smc.vnet.net
• Subject: [mg95190] New kind of Weierstrass fractal scaling: Eulerian scales,MacMahon
• From: Roger Bagula <rlbagula at sbcglobal.net>
• Date: Fri, 9 Jan 2009 06:24:17 -0500 (EST)
• References: <gk1rls\$os8\$1@smc.vnet.net>

```This idea is a consequence of the Bernoulli types using the
general Pascal row sum products.
The Pascal row sum is 2^n which is a regular Self-Similar Weierstrass or
Bescovitch-Ursell function.
Using n! or Gamma[1+n] or (n+1)! or Gamma[2+n]
scales are like scales 2^n and 3^n.
3=2^s
gives ( Sierpinski gasket self-similar dimension):
s=Log/Log
That kind of calculation/ conversion isn't as easy in this new
type of  scales:
FindRoot[Gamma - Gamma[1 + s] == 0, {s, 5/3}]
s=2
FindRoot[3*Gamma - 2^s*Gamma[1 + s] == 0, {s, 5/3}]
{s -> 1.81782}

Clear[c, s, s0, x]
s0 = 3/5;
c[x_] = Sum[Cos[2^n*x]/2^(s0*n), {n, 0, 20}];
s[x_] = Sum[Sin[2^n*x]/2^(s0*n), {n, 0, 20}];
ParametricPlot[{c[x], s[x]}, {x, -Pi, Pi}, PlotPoints -> 1000, Axes ->
False]

http://www.geocities.com/rlbagulatftn/2weier.gif

Eulerian scale Weierstrass fractal with n! scale:
Clear[c, s, s0, x]

s0 = 3/5;
c[x_] = Sum[Cos[Gamma[1 + n]*x]/Gamma[1 + s0*n], {n, 0, 20}];
s[x_] = Sum[Sin[Gamma[1 + n]*x]/Gamma[1 + s0*n], {n, 0, 20}];
ParametricPlot[{c[x], s[x]}, {x, -Pi, Pi}, PlotPoints -> 1000, Axes ->
False]

http://www.geocities.com/rlbagulatftn/2eulerian_weier.gif

MacMahon type scaling:
Clear[c, s, s0, x]
s0 = 3/5;
c[x_] = Sum[Cos[2^n*Gamma[1 + n]*x]/(2^(s0*n)*Gamma[1 + s0*n]), {n, 0, 20}];
s[x_] = Sum[Sin[2^n*Gamma[1 + n]*x]/(2^(s0*n)*Gamma[1 + s0*n]), {n, 0, 20}];
ParametricPlot[{c[x], s[x]}, {x, -Pi, Pi}, PlotPoints -> 1000, Axes ->
False]

http://www.geocities.com/rlbagulatftn/2weier_macmahon.gif

Mew kind of Bescovitch-Ursell fractal as Eulerian:
Clear[f, g, h, k, ff, kk, ll]
f[x_] := 0 /; 0 <= x <= 1/3
f[x_] := 6*x - 2 /; 1/3 < x <= 1/2
f[x_] := -6*x + 4 /; 1/2 < x <= 2/3
f[x_] := 0 /; 2/3 < x <= 1
ff[x_] = f[Mod[Abs[x], 1]];
s0 = Log/Log;
kk[x_] = Sum[ff[(k + 1)!*x]/Gamma[2 + s0*k], {k, 0, 20}];
ll[x_] = Sum[ff[(k + 1)!*(x + 1/2)]/Gamma[2 + s0*k], {k, 0, 20}];
ga = Table[{kk[n/10000], ll[n/10000]}, {n, 1, 10000}];
ListPlot[ga, Axes -> False, PlotRange -> All]

http://www.geocities.com/rlbagulatftn/biscuit_eulerianvonkoch.gif

The implication of this type of result is that there are more types of
fractal scaling than just the straight integers scales of traditional
fractal theory.
These ideas are a result of the application of Pascal
type Modulo two Sierpinski resy\ults to higher symmetries of combinatorial
triangles and the scaling that the row sums ( products) provide.

Respectfully, Roger L. Bagula