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Julia sets of the burning ship are wild
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9/19/17
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Here is an older Julia fractal of mine:

https://plus.google.com/101799841244447089430/posts/8j1pjQ4TWzK

using one my my newer color monitor algorithms. The per-pixel iteration formula is:
________
z = (std::pow(z, 1.3) - (1.0 / (std::log(z) + 0.11))) + 0.42457;
double slow = 1.2 + (1 / i);
z /= slow;
________

where i is the actual integer iteration count starting at 1 to avoid a divide-by-zero condition. Iirc, this has some very interesting formations inside it. Will go for a new zoom.

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#Fractal #Math #Space #Art #Spiral #Iteration
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Here is a zoom at point (-3.05, 0) with radius .072 from:

https://plus.google.com/101799841244447089430/posts/PSavhGfuGQJ

Imvvho, it has a sort of "natural" look in the "cells"... ;^)

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#Fractal #Space #Math #Art #Tree #Cell #Nature #Biology
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A better picture maybe:
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Using +Roger Bagula's (2*Pi-s) as a radius gives this 2-ary circle twister funneling down. Thanks Roger. Thinking. Well, I am going to have more time to working on this later on today. Humm..

https://plus.google.com/110803890168343196795/posts/4ky5MkotmqE

https://plus.google.com/101799841244447089430/posts/S5AP4aaj13w

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#Parametric #Math #Art #Space #Trigonometry
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Here is a 6-ary twisting tubes with a little "wrinkle" in the z function for the parametric here, using +Roger Bagula's radius:
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ParametricPlot3D[{
Cos[t] * Abs[Cos[t*3 + s]] * [2*Pi-s],
Sin[t] * Abs[Cos[t*3 + s]] * [2*Pi-s],
Sin[s - Pi] + s * 2},
{t, 0, 2 * Pi},
{s, 0, 2 * Pi}]
______________

z is Sin[s - Pi] + s * 2 here.

https://plus.google.com/101799841244447089430/posts/b2Q6GMxmE8m

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#Fractal #Parametric #Math #Art #Space
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Factoring the 4d bi-helix:
(* Mathematica*)
x = Cos[t]*Abs[Cos[s + t]]
y = Sin[t]*Abs[Cos[s + t]]
z = s
w = 2*Pi - s
{x, y, z, w} /. t -> p /. s -> t
(Clifford torus projection of bi-Helix)
ParametricPlot3D[{x, y,
z}/(3*Pi - w), {s, 0, 2*Pi}, {t, -Pi, Pi}, ImageSize -> {740, 580},
ColorFunction -> "Pastel", Axes -> False, Boxed -> False,
Mesh -> False, PlotPoints -> 60, PlotStyle -> Opacity[0.5]]
Clear[a, b, c, d, e, f, v, w]
a = Cos[p]; b = Sin[p];
v = {a, b, c}
w = {d, e, f}
vcw = Cross[v, w]
vdw = v.w
vv = {vcw[[1]], vcw[[2]], vcw[[3]], vdw}
v4 = {Abs[Cos[p + t]] Cos[p], Abs[Cos[p + t]] Sin[p], t, 2 \[Pi] - t}
(solving for the two vectors)
ww = {c, d, e, f} /.
Solve[Table[vv[[i]] - v4[[i]] == 0, {i, 4}], {c, d, e, f}]
(* 4d torus cylinder*)
ww1 = Flatten[FullSimplify[ExpandAll[ww]]]
www = Table[ww1[[i]], {i, 2, 4}]
(* cylinder*)
g1 =
ParametricPlot3D[{Cos[p], Sin[p], -(Abs[Cos[p + t]]/t)}, {t, -Pi,
Pi}, {p, -Pi, Pi}, PlotPoints -> 200, PlotStyle -> Opacity[0.50],
ColorFunction -> "Rainbow"]
(* torus spiral*)
g2 =
ParametricPlot3D[{t (((2 \[Pi] - t) t Cos[p])/(
t^2 + Abs[Cos[p + t]]^2) - Sin[p]),
t (Cos[p] + ((2 \[Pi] - t) t Sin[p])/(t^2 + Abs[Cos[p + t]]^2)), (
t (-2 \[Pi] + t) Abs[Cos[p + t]])/(
t^2 + Abs[Cos[p + t]]^2)}, {t, -Pi, Pi}, {p, -Pi, Pi},
PlotPoints -> 60, PlotStyle -> Opacity[0.25],
ColorFunction -> "Pastel"]
Show[{g1, g2}, PlotRange -> {{-10, 10}, {-10, 10}, {-4, 4}}]
(* end*)
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9/19/17
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Here is a 6-ary helix. They are not circles damn it:

ParametricPlot3D[{ Cos[t] * Abs[Cos[t*3 + s]], Sin[t] * Abs[Cos[t*3 + s]], s}, {t, 0, 2 * Pi}, {s, 0, 2 * Pi}]

I think there is a way to decompose my semi-circle parametric to get n-circles. Humm... Need to think here..

https://plus.google.com/101799841244447089430/posts/6D5f8AqCFRX

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#Fractal #Parametric #Helix #Spiral #Petal #Math #Art
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Here is a crude 3d parametric surface plot of my semi-circle wave function, v is elongated to 5 * Pi.

ParametricPlot3D[ {Cos[u] + Floor[Sin[v + u]] * 2 + 1, Sin[u], v }, {u, 0, 2 * Pi}, {v, 0, 5 * Pi} ]

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