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JosLeys
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« on: August 15, 2010, 08:56:30 PM » |
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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...
<![CDATA[<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 » |
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This looks pretty cool  Could you do 4D transforms on this and project it in different ways?  |
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teamfresh
Fractal Lover
Posts: 246
nothing is everything
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« Reply #2 on: August 15, 2010, 09:35:14 PM » |
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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 » |
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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)
<![CDATA[<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 » |
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Amazing JosLeys, would love to see more of this...  |
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Direct2Brain
Forums Newbie
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« Reply #5 on: August 16, 2010, 06:05:39 PM » |
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Very nice videos Jos.
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kram1032
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« Reply #6 on: August 16, 2010, 06:13:27 PM » |
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Looks really nice |
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bib
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« Reply #7 on: August 16, 2010, 09:15:05 PM » |
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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 Valéry)
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Jesse
Download Section
Fractal Schemer
Posts: 1013
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« Reply #8 on: August 20, 2010, 10:53:42 PM » |
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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 » |
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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 » |
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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)
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Jesse
Download Section
Fractal Schemer
Posts: 1013
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« Reply #11 on: August 21, 2010, 10:25:35 PM » |
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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 |
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knighty
Fractal Iambus
Posts: 819
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« Reply #12 on: September 08, 2010, 01:30:18 AM » |
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Those 4D mandelbulb are great!  |
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Tglad
Fractal Molossus
Posts: 703
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« Reply #13 on: September 08, 2010, 04:28:13 AM » |
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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 » |
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Jesse, you asked about the formula. Here it is : (it's basically the same as my 3D 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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