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Julia sets of the burning ship are wild

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9/19/17

8 Photos - View album

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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.

#Fractal #Math #Space #Art #Spiral #Iteration

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.

*__*#Fractal #Math #Space #Art #Spiral #Iteration

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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

#Parametric #Math #Art #Space #Trigonometry

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

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

*__*#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:

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

#Fractal #Parametric #Math #Art #Space

*______________*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

*__*#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

(

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}

(

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*)

(* 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

4 Photos - View album

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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

#Fractal #Parametric #Helix #Spiral #Petal #Math #Art

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

*__*#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} ]

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

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The knot factors as a movie:

https://archive.org/details/TheKnotFactor

https://archive.org/details/TheKnotFactor

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