im no math genius or nothing but what does a 4d julia fractal supposed to represent or be used for? and what is the 4th dimension, cos i know you got x y and z? is the 4th time?
@DJBigz1988 While we can only perceive 3 spacial dimensions, such limitations doesn't apply to math and physics (or the imagination ;P). Time is independent of geometry.
Fractals have a number of uses, although not all are direct or even obvious, it is a field of science that is somewhat abstract. It can, for example, be used to understand why and how a (simple) system can give rise to chaotic behavior and the dynamics of it.
This fractal is an example of the system Q->Q*Q+C in 4D space.
@ImpoliteFruit makes a bit more sense i guess but i cant personally imagine in my head how it would look(guess thats what the videos are for? XD) so if you initially make q=4 and im guessing C means constant. can you make that any number? well for the sake of my guessing make it 1. Q->4*4+1 so then Q =17|-17? and so on and so forth? and you just graph the 2 Q's on the Rside of the equation and the 2 possible Q's on the Lside? i have a concept of basic algebra. so not sure i understand. @_@
@DJBigz1988 replypart1: You are on the right track on how the equation works, each new Q is feed back into it. So if Q=0 and C=1 the sequence is Q->0*0+1->1*1+1->2*2+1->...->infinity (it can be shown that Q always goes towards infinity if the magnitude is larger than 2). The interesting thing is how the equation behaves depending on the start values for Q and C, there are broadly speaking three behaviours; goes towards infinity, enters a cycle (try Q=0 and C=-2) and goes towards a fixed number.
@DJBigz1988 Part 2: If Q-> infinity it is outside the fractal, Q->fixed number or cycle it is either on the edge or inside the fractal. The point were it gets really interesting is when you move from normal numbers (1 dimension) to (hyper)complex numbers which are 2 dimensional or higher.
A complex number (2D) is given by Q=a+i*b, if b is zero it behaves just like the normal 1D numbers you are familiar with, the "magic" happens when b isn't zero. i is the square root of -1. See the Mandelbrot.
You can add and subtract complex numbers as normal Q1=a1+i*b1, Q2=a2+i*b2, Q1+Q2=(a1+a2)+i*(b1+b2). However when multiplying it becomes Q1*Q2=(a1*a2-b1*b2)+i*(a1*b2+a2*b1).
The hypercomplex (4D+) numbers have more complex parts and there are different kinds, whose behavior depends on the complex i,j,k.
If you look at the Julia fractal, it shows how the fractal behaves when start conditions are; Q is value from -2-i2 to 2+i2 and C is a fixed (complex) number. Inside if |Q|<4
A hypercomplex is done the same way, but instead of a plane (2D) one have to trace lines through a volume and since we can't perceive more than 3D, one of the dimensions in a 4D fractal is kept at a fixed value or depend on one of the other 3. I tend to set the 4th at a fixed value. What you see here is a 3D cut of a 4D space (think of a very thin slice of cheese as a 2D slice of a 3D space ;)).
Sorry for the wall of text, I tried to be as brief as possible.
Sort of variation on quaternions? My skill with this kind of thing has faded due to non-use over the decades -- is the algebra formed by these rules isomorphic to quaternions?
I was wondering if you can provide details about the software, the formula used, etc.
They are both hypercomplex numbers. For comparison, a quaternion is given by:
ii=jj=kk=ijk=-1
ij=-ji=k
jk=-kj=i
ki=-ik=j
The software is written by me, it is capable of raytracing any hypercomplex number, either as a Julia or Mandelbrot fractal.
Basically it is ray traced the same way you would a quaternion or a 3D object; Checking if the ray "hits" the set. If it does, find the surface it hits and the vector normal, which is used for light and surface texture (not used in this video).
im no math genius or nothing but what does a 4d julia fractal supposed to represent or be used for? and what is the 4th dimension, cos i know you got x y and z? is the 4th time?
DJBigz1988 1 year ago
@DJBigz1988 While we can only perceive 3 spacial dimensions, such limitations doesn't apply to math and physics (or the imagination ;P). Time is independent of geometry.
Fractals have a number of uses, although not all are direct or even obvious, it is a field of science that is somewhat abstract. It can, for example, be used to understand why and how a (simple) system can give rise to chaotic behavior and the dynamics of it.
This fractal is an example of the system Q->Q*Q+C in 4D space.
ImpoliteFruit 1 year ago
@ImpoliteFruit makes a bit more sense i guess but i cant personally imagine in my head how it would look(guess thats what the videos are for? XD) so if you initially make q=4 and im guessing C means constant. can you make that any number? well for the sake of my guessing make it 1. Q->4*4+1 so then Q =17|-17? and so on and so forth? and you just graph the 2 Q's on the Rside of the equation and the 2 possible Q's on the Lside? i have a concept of basic algebra. so not sure i understand. @_@
DJBigz1988 1 year ago
@DJBigz1988 replypart1: You are on the right track on how the equation works, each new Q is feed back into it. So if Q=0 and C=1 the sequence is Q->0*0+1->1*1+1->2*2+1->...->infinity (it can be shown that Q always goes towards infinity if the magnitude is larger than 2). The interesting thing is how the equation behaves depending on the start values for Q and C, there are broadly speaking three behaviours; goes towards infinity, enters a cycle (try Q=0 and C=-2) and goes towards a fixed number.
ImpoliteFruit 1 year ago
@DJBigz1988 Part 2: If Q-> infinity it is outside the fractal, Q->fixed number or cycle it is either on the edge or inside the fractal. The point were it gets really interesting is when you move from normal numbers (1 dimension) to (hyper)complex numbers which are 2 dimensional or higher.
A complex number (2D) is given by Q=a+i*b, if b is zero it behaves just like the normal 1D numbers you are familiar with, the "magic" happens when b isn't zero. i is the square root of -1. See the Mandelbrot.
ImpoliteFruit 1 year ago
@DJBigz1988 Part3:
You can add and subtract complex numbers as normal Q1=a1+i*b1, Q2=a2+i*b2, Q1+Q2=(a1+a2)+i*(b1+b2). However when multiplying it becomes Q1*Q2=(a1*a2-b1*b2)+i*(a1*b2+a2*b1).
The hypercomplex (4D+) numbers have more complex parts and there are different kinds, whose behavior depends on the complex i,j,k.
If you look at the Julia fractal, it shows how the fractal behaves when start conditions are; Q is value from -2-i2 to 2+i2 and C is a fixed (complex) number. Inside if |Q|<4
ImpoliteFruit 1 year ago
@DJBigz1988 Part 4:
A hypercomplex is done the same way, but instead of a plane (2D) one have to trace lines through a volume and since we can't perceive more than 3D, one of the dimensions in a 4D fractal is kept at a fixed value or depend on one of the other 3. I tend to set the 4th at a fixed value. What you see here is a 3D cut of a 4D space (think of a very thin slice of cheese as a 2D slice of a 3D space ;)).
Sorry for the wall of text, I tried to be as brief as possible.
ImpoliteFruit 1 year ago
@ImpoliteFruit Has to be the most intelligent conversation I have ever seen on youtube
Roulden 7 months ago
@Roulden Thank you kindly. If you have any questions feel free to ask and I'll try and answer the best I can. :D
ImpoliteFruit 7 months ago
The lighting/texture looks very realistic.
andurilfromnarsil 1 year ago
very, very complex!! This is astonishing. And i thought some Fractals were in fractional dimensions like the 1/2 dimension.
CurvesOfFractals 2 years ago
Some fractals are, others are not. :p
Simply put; it is a fractal, if the topological dimension is less than the Hausdorff Besicovitch (hope I spelled that right) dimension.
The dimension of the number(s) used does not always translate directly to the fractal dimension.
I hope that made some sense. :)
ImpoliteFruit 2 years ago
This is interesting...any details available?
DeepZoomNet 3 years ago
Strange, the reply I made a while ago have gone AWOL.
The multiplication table is (if I remember correctly);
ii = j
jj =-1
kk =-j
ij = ji = k
ik = ki =-1
kj = jk =-i
ImpoliteFruit 3 years ago
Sort of variation on quaternions? My skill with this kind of thing has faded due to non-use over the decades -- is the algebra formed by these rules isomorphic to quaternions?
I was wondering if you can provide details about the software, the formula used, etc.
DeepZoomNet 3 years ago
They are both hypercomplex numbers. For comparison, a quaternion is given by:
ii=jj=kk=ijk=-1
ij=-ji=k
jk=-kj=i
ki=-ik=j
The software is written by me, it is capable of raytracing any hypercomplex number, either as a Julia or Mandelbrot fractal.
Basically it is ray traced the same way you would a quaternion or a 3D object; Checking if the ray "hits" the set. If it does, find the surface it hits and the vector normal, which is used for light and surface texture (not used in this video).
ImpoliteFruit 3 years ago