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Author Topic: 4D Mandelbulb  (Read 5423 times)
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JosLeys
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« on: August 15, 2010, 08:56:30 PM »

There is also a 4D version of spherical coordinates.
The little film belows shows the 4D (degree eight) Mandelbulb, projected into 3D when the value of the fourth coordinate changes from -1 to +1...



<a href="http://www.youtube.com/v/8sojcs5yhHI&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/8sojcs5yhHI&rel=1&fs=1&hd=1</a>

(done in Ultrafractal)
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kram1032
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« Reply #1 on: August 15, 2010, 09:01:29 PM »

This looks pretty cool cheesy

Could you do 4D transforms on this and project it in different ways? smiley
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teamfresh
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« Reply #2 on: August 15, 2010, 09:35:14 PM »

that was cool! it would be good to magnify slightly as it decreases in size.....
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JosLeys
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« Reply #3 on: August 16, 2010, 08:59:49 AM »

As to transformations in 4D, here is a rotation in 4D around the x-y plane, which changes the shape of the bulb.
(the rotation around the vertical axis is just a simple rotation in 3D)

<a href="http://www.youtube.com/v/Ktqm_vYy5K8&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/Ktqm_vYy5K8&rel=1&fs=1&hd=1</a>"
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KRAFTWERK
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« Reply #4 on: August 16, 2010, 10:23:53 AM »

Amazing JosLeys, would love to see more of this...  afro
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Direct2Brain
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« Reply #5 on: August 16, 2010, 06:05:39 PM »

Very nice videos Jos. 
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kram1032
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« Reply #6 on: August 16, 2010, 06:13:27 PM »

Looks really nice smiley
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bib
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« Reply #7 on: August 16, 2010, 09:15:05 PM »

Very nice videos, I like the style : simple yet rich graphics to show some mathematical features.
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Between order and disorder reigns a delicious moment. (Paul ValÚry)
Jesse
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« Reply #8 on: August 20, 2010, 10:53:42 PM »

Is this the birth of the 4d power8 bulb?
Great!
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JosLeys
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« Reply #9 on: August 20, 2010, 11:03:40 PM »

If this is the birth of the 4D bulb, I'm afraid the baby looks just like it's 3D relatives...
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JosLeys
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« Reply #10 on: August 21, 2010, 12:23:26 PM »

The 4D version does offer some new views.
Here is a fourth-power Julia, that I do not think can be reproduced by the 3D formulas. (the Julia constant has a value for the fourth coordinate)


* 4D_quat_009S.jpg (157.68 KB, 800x800 - viewed 475 times.)
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Jesse
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« Reply #11 on: August 21, 2010, 10:25:35 PM »

Maybe your 3D versions have some parameters to get these 4D variants, i can't do them.
Nice Julia, btw.
I am still eager to find a minkowski 4d variation, so how much must i pay or beg to get the formula  cheesy
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knighty
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« Reply #12 on: September 08, 2010, 01:30:18 AM »

Those 4D mandelbulb are great! grin
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Tglad
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« Reply #13 on: September 08, 2010, 04:28:13 AM »

It is possible to render full 4d objects (not just slices) by projecting all 4 dimensions onto the view frustum...
It would be good for viewing these things as for instance you would never see disconnections in a set that is fully connected.
You sometimes see what looks like self intersection, but that is just overlap due to viewing angle.
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JosLeys
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« Reply #14 on: September 08, 2010, 09:22:29 AM »

Jesse, you asked about the formula. Here it is :
(it's basically the same as my 3D code)
Code:
//initalize
// zx,zy and zz =coordinates of point on the ray, zw=4D slice value
R=sqrt(zx*zx+zy*zy+zz*zz+zw*zw)
theta3=atan2(zx+i*zy)
theta2=atan2(i*zz*sin(theta3)+zy)
theta1=asin(zw/R)
RRdz=1
//c=1 (Mandelbrot) or c=0 (Julia)
//theta1dz,theta2dz and theta3dz=0

//loop
//derivative for distance estimate
 dzx=@pow*R^(@pow-1)*RRdz*cos((@pow-1)*theta1+theta1dz)*cos((@pow-1)*theta2+theta2dz)*cos((@pow-1)*theta3+theta3dz)+c
 dzy=@pow*R^(@pow-1)*RRdz*cos((@pow-1)*theta1+theta1dz)*cos((@pow-1)*theta2+theta2dz)*sin((@pow-1)*theta3+theta3dz)
 dzz=@pow*R^(@pow-1)*RRdz*cos((@pow-1)*theta1+theta1dz)*sin((@pow-1)*theta2+theta2dz)
 dzw=@pow*R^(@pow-1)*RRdz*sin((@pow-1)*theta1+theta1dz)
 
 Rdz=sqrt(dzx*dzx+dzy*dzy+dzz*dzz+dzw*dzw)

 theta3dz=atan2(dzx+i*dzy)
 theta2dz=atan2(i*dzz*sin(theta3dz)+dzy)
 theta1dz=asin(dzw/Rdz)

// main iteration
 zx=R^@pow*cos(@pow*theta1)*cos(@pow*theta2)*cos(@pow*theta3)+cx
 zy=R^@pow*cos(@pow*theta1)*cos(@pow*theta2)*sin(@pow*theta3)+cy
 zz=R^@pow*cos(@pow*theta1)*sin(@pow*theta2)+cz
 zw=R^@pow)*sin(@pow*theta1)+cw

R=sqrt(zx*zx+zy*zy+zz*zz+zw*zw)

theta3=atan2(zx+i*zy)
theta2=atan2(i*zz*sin(theta3)+zy)
theta1=asin(zw/R)

// distance estimate after bailout
DE=f*log(R)*R/Rdz //choose some f<1
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