Wednesday, September 08, 2021
A map to a four dimensional space
In addition to expanding the output of an old program, I have also been expanding the mapping of the chaotic system (and like the last post, this post will have a very limited audience but again, I don't care). So this image:
is generated by iterating these two equations:
xi+1 = (Ayi + B) xi (1 - xi)
yi+1 = (Cxi + D) yi (1 - yi)
with the following values:
- A = +2.4376
- B = +1.5624
- C = -0.8659
- D = +4.0000
is a type of map where one can find chaotic attractors. It's a 2-dimensional slice of a 4-dimension space, where the horizontal axis is A, which runs from -4 (on the left) to 4, and the vertical axis is B, which also runs from -4 (on the bottom) to 4. If you look closely, you can make out a small red cross in the upper right hand area that marks the spot where the chaotic attractor above can be found in said space. Along the bottom, you'll see the four values listed above, with “Ax” marking the attractor location along the horizontal axis, and “By” marking the location along the vertical axis. The brighter the area, the more … um … expansive the chaotic attractor becomes.
But unlike thirty years ago, I decided to slice this a few different ways. If you imaging the above image as a slice through a 3-dimentional cube, this is the image you'd see if you were looking straight down on the cube at a horizontal slice through the cube. Shifting your view to the front, where A still runs left-to-right, C now runs up-down, and we get a vertical slice through the cube:
Looking at the cube from the left side—B values now run along the horizontal, C is still up-down and we get another vertical slice:
In each of those images, you should find a small red cross that marks the location of the attractor.
There are still three more planes we can cut through, although in this case, I can't quite make out the front, side or top. One is the A-D slice (and in each of the following images, you can make out the cross along the top edge):
The second is the B-D slice:
And the final one is the C-D slice:
Yeah … D ends up being vertical in all three … and that … kind of … makes sense … to me. Or am I going crazy? This is 4-dimensional space we're talking about.