## Profile

## Stream

### Roger Bagulaowner

fractal art (pictures, poems) - {Cos[t]/(Sqrt[2] - Cos[p - t]), Cos[p]/(Sqrt[2] - Cos[p - t]),

Cos[p + t]/(Sqrt[2] - Cos[p - t])}

### Roger Bagulaowner

fractal art (pictures, poems) - I tried a complete “ring-out” solution, but

that doesn’t look at all good.

This solution uses the wavefunction orthogonality (3rd condition)

to tie off two of the ends {xp,zp} and you get a space turn -pike

looking surface!

The yp solved variable has singular “ends” that meet at Infinity.

(* Mathematica*)

(* figure eight knot*)

a = 2; b = 3;

x = (2 + Cos[a*t])

**Cos[b*t]**

**y = (2 + Cos[a*t])Sin[b*t]**

**z = Sin[2*a*t]**

**w = {x, y, z}**

**FullSimplify[ExpandAll[w.w]]**

**Clear[xp, yp, zp, wp]**

**wp = {xp, yp, zp}**

**( dot zero :first condition**)

w.wp

zp = Cos[4*t]

(* wavefunction orthogonality condition*)

Integrate[

zp Sin[4 t], {t, -Pi, Pi}]

Solve[xp (2 + Cos[2 t]) Cos[3 t] + yp (2 + Cos[2 t]) Sin[3 t] +

zp Sin[4 t] == 0, yp]

yp = -((Csc[3 t] (2 xp Cos[3 t] + xp Cos[2 t] Cos[3 t] + Cos[4 t] Sin[4 t]))/(

2 + Cos[2 t]))

Integrate[y*yp, {t, -Pi, Pi}]

(* 2nd condition Cross product is constant: checking*)

xp = (2 +

Sin[2 t]) Sin[3 t]

Integrate[x*xp, {t, -Pi, Pi}]

Simplify[Expand[Cross[w, wp].Cross[w, wp]], Trig -> True]

N[(33905 + 57910 Cos[2 t] + 36635 Cos[4 t] + 17850 Cos[6 t] + 8295 Cos[8 t] +

4740 Cos[10 t] + 2640 Cos[12 t] + 940 Cos[14 t] + 57 Cos[16 t] -

138 Cos[18 t] - 91 Cos[20 t] - 22 Cos[22 t] - Cos[24 t] +

12588 Sin[2 t] + 16480 Sin[4 t] + 11948 Sin[6 t] + 5354 Sin[8 t] +

1552 Sin[10 t] + 396 Sin[12 t] + 152 Sin[14 t] + 54 Sin[16 t] -

4 Sin[18 t] - 20 Sin[20 t] - 12 Sin[22 t] -

2 Sin[24 t])/(32 (2 + Cos[2 t])^2 (1 + 2 Cos[2 t])^2) /. t -> 0]

N[(33905 + 57910 Cos[2 t] + 36635 Cos[4 t] + 17850 Cos[6 t] + 8295 Cos[8 t] +

4740 Cos[10 t] + 2640 Cos[12 t] + 940 Cos[14 t] + 57 Cos[16 t] -

138 Cos[18 t] - 91 Cos[20 t] - 22 Cos[22 t] - Cos[24 t] +

12588 Sin[2 t] + 16480 Sin[4 t] + 11948 Sin[6 t] + 5354 Sin[8 t] +

1552 Sin[10 t] + 396 Sin[12 t] + 152 Sin[14 t] + 54 Sin[16 t] -

4 Sin[18 t] - 20 Sin[20 t] - 12 Sin[22 t] -

2 Sin[24 t])/(32 (2 + Cos[2 t])^2 (1 + 2 Cos[2 t])^2) /. t -> Pi]

Sqrt[%]

Simplify[Expand[w.wp], Trig -> True]

ParametricPlot3D[{N[wp], w}, {t, -Pi + 0.01, Pi - 0.01}]

xa = (8 + Sin[p])

**xp;**

**ya = (8 + Sin[p + Pi/4])yp;**

**za = (8 + Sin[p + 3*Pi/4])*zp;**

**ww = {xa, ya, za};**

**gg = Import["EscherSkyandWaterredblueCrop.jpg"]**

**ParametricPlot3D[N[ww], {t, -Pi + 0.01, Pi - 0.01}, {p, -Pi, Pi},**

**Boxed -> False, Axes -> False, Lighting -> "Neutral", PlotPoints -> 200,**

**PlotStyle -> {LightBlue, Specularity[White, 10], Texture[gg]},**

**TextureCoordinateFunction -> (Normalize[{#1, #2, #3}] &),**

**Background -> Black, ImageSize -> 1000, Mesh -> False]**

**( end**)

### Roger Bagulaowner

Discussion - ### Roger Bagulaowner

fractal art (pictures, poems) - ### Communities

27 communities### Roger Bagulaowner

2d Julias and Mandelbrots - with a 33rd circle transform:

### Roger Bagulaowner

2d Julias and Mandelbrots - ### Roger Bagula

Shared publicly -### Roger Bagulaowner

fractal art (pictures, poems) - - UCLA
- SDSU

- fractals and surfaces (current)
- All things mathematical and fractal (current)

About me? A Natural Philosopher is what they call people like me. I Program computers with Mathematica... I like and do quantum mechanics and cosmology. I have some reviews for Amazon in Mathematics, Physics, fiction and Movies. I'm a big Sci fi movie fan. I've spent the last 30 years as an Fractal activist and am currently a global warming news activist...a "Green". With my groups activity on the web, I have a hobby of paleoarchaeology and the study of mathematical influences on history. I organized my high school reunion 11 years ago and then, didn't go. I write poetry, short stories and fractal based jazz. I still have my draft card. Just got a gift card from the local library as runner up in their poetry contest. Have been making 3d printing models for the last few years.

https://plus.google.com/110803890168343196795/about

http://siggrapharts.ning.com/profile/RogerBagula

http://www.sculpteo.com/en/s/rlbagulatftn/

Old guy who uses Mathematica: started 3d on a Radio Shack color computer in basic about 1979. I'm an Inorganic Chemist by training with physical/ quantum Chemistry. I have been doing fractals since I was using a Commodore 64 for graphics in the 80's. In the 90's I had an Amiga and discovered ray tracing and Phong shading. I'm on a couple of Macs right now and I'm getting old. TFTN was a fractal news letter I published.

http://rogerbagula.brandyourself.com/

Born in National city. Lived childhood mostly in La Mesa, Ca. High School in Lakeside, Ca.1964. Chemistry degree from UCLA 1968. About two years drafted in the Army and 2 semesters graduate school SDSU. Computer programming self taught 1979 on. Fractal newsletter 1993 -2003 and web page design. Number theory work 2002 to present with OEIS entries. Was a Amazon Reviewer / Vine since about 2006. Interest groups at Yahoo, Linked In and Google+ communities. Over 40 years genealogy research since the late 1960's. Researching the Bagula/ Bagola/ Plowka names from Drachhausen, Hochza in Cottbus, Brandenburg,East Germany. I've composed fractal music and done some graphic art with pen and ink and mixed media painting.

- Bmftgmath programming, present

Phone | 619-5610814 |

- roger.bagula@gmail.com
| |

Address | Lakeside |

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