@GerardWestendorp And a million dollar rich.

[PART3]

Next, you slightly decrease or increase the guess of the variable, depending on the error.

If you do this gently enough it I thought it might be fun to try this with a Sudoku puzzle. For each non-given cell, I demand the difference between any other cell in the same row, column or sub-square is greater than 1 and less than -1, If not, I slightly change the variable away from the other number. (Working modulo 9, so 1 is next to 9).

[PART2]

Next, you slightly decrease or increase the guess of the variable, depending on the error.

If you do this gently enough it will converge to a minimum error. This will produce a local optimum: you cannot decrease the error by slightly wobbling any of the variable. In physics this is often the solution you want.

*Relaxation versus NP*

In engineering, many problems, mechanical, thermal, fluid dynamics, electro magnetic, can be solved by a relaxation method. First you guess all variables, for example by a random number. Then for each variable, you compute the error: You take all equations that the variable should satisfy, and sum over the errors. You take care that for each equation the derivative of the error to your variable is positive.

[PART1]

Inigo Quilez writes about his latest autumnal creation:

"mathematical expressions define the shapes, placement, color (and light), shadows, movement, camera lens simulation, ..."

http://www.youtube.com/watch?v=q1OBrqtl7Yo

#generative autumnal ambient animation made of pure #maths

@GerardWestendorp \( n^2 = 1 + 3 + 5 + \ldots{} + (2n-1) \)

I was reading about Kolmogorov. When he was 5 years old, he noticed that

n^2 = 1 + 2 + 3 +… +(2n-1).

You can interpret this geometrically.

Any odd square can be the last term of such a sequence, and any even square the sum of the last 2 terms. Using this, you can construct Pythagorean triples for every number:

1^2 = 1^2 - 0^2

2^2 = 2^2 - 0^2

3^2 = 5^2 - 4^2

4^2 = 5^2 - 3^2

5^2 = 13^2 - 12^2

6^2 = 10^2 - 8^2

7^2 = 25^2 - 24^2

…

Looping animation made by sequencing key frames in the bifurcation diagram of the logistic map x := a x (1 - x), which is conjugate to the Mandelbrot set's quadratic polynomial x := x^2 + c.

The left hand edge of each keyframe is the parabolic root of a hyperbolic component (using jargon related to the Mandelbrot set). This can be found by Newton's method in 2 variables, starting from a nucleus of the relevant period (the animation starts 1, 2, 4, 8). The nucleus can be found by Newton's method in 1 variable, starting from a guess coordinate found by tracing external rays.

The right hand edge of each keyframe is a tuned / renormalized copy of the tip of the Mandelbrot antenna (c = -2 or a = 4). I found these coordinates by tracing external rays in the Mandelbrot set and then mapping c to a.

The top and bottom edges of the keyframes were found by iterating at the right hand edge and finding the two x that are closest (but not equal to) the starting point (the critical point x = 0.5). I think (not sure) these are at iteration numbers P and 2P, where P is the period of the "owning" hyperbolic component.

Interpolation between keyframes was done using Poincaré half-plane geodesics:

https://mathr.co.uk/blog/2011-12-26_poincare_half-plane_metric_for_zoom_animation.html

Greg Egan, Shrinking the Superpermutations: https://plus.google.com/113086553300459368002/posts/4VB8Xi3i2Gt

Greg suggests a formula for their length, and constructs superpermutations of length only one more than the formula.

They should not be confused with superpatterns (https://en.wikipedia.org/wiki/Superpattern) which contain all permutations of a given length for a different definition of containment, and are a lot shorter, but also somewhat mysterious in length.

#introduction Hi, I've been using Google+ for having interesting discussions on mathematics and physics, particularly visualistations using 3D prints, animations. Since Google+ might be shutting down, I'm trying Mastodon.

I just salvaged all my old posts, and put them on a website:

https://westy31.home.xs4all.nl/SalvagedGooglePlus/SalvagedGooglePlus_Part1.html

Here is an example of graphics I do:

Joined Oct 2018