descartes :descartes: duchatelet :duchatelet: euler :euler: fibonacci :fibonacci: gauss :gauss: germain :germain: hilbert :hilbert: hopper :hopper: kovalevskaya :kovalevskaya: leibniz :leibniz: lovelace :lovelace: newton :newton: noether :noether: pythagoras :pythagoras: ramanujan :ramanujan: turing :turing:

[PART 4]
Sudoku puzzles are NP complete, if you increase the size sufficiently beyond 9*9. This means that if you find an efficient method for solving them, you will have proved that P=NP, and you will be famous.

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).

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.

Inigo Quilez writes about his latest autumnal creation:

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

#generative autumnal ambient animation made of pure #maths

Practice Latex:
\(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\\

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:

#fractals #chaos #mandelbrot #maths

Greg Egan, Shrinking the Superpermutations:
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 ( 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.

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:
Here is an example of graphics I do:

Scholar Social

A Mastodon instance for academics

Scholar Social is a microblogging platform for researchers, grad students, librarians, archivists, undergrads, academically inclined high schoolers, educators of all levels, journal editors, research assistants, professors, administrators—anyone involved in academia who is willing to engage with others respectfully.

We strive to be a safe space for queer people and other minorities, recognizing that there can only be academic freedom where the existence and validity of interlocutors' identities is taken as axiomatic.

"A Mastodon profile you can be proud to put on the last slide of a presentation at a conference"

"Official" monthly journal club!

(Participation is, of course, optional)

Scholar Social features a monthly "official" journal club, in which we try to read and comment on a paper of interest.

Any user of Scholar Social can suggest an article by sending the DOI by direct message to and one will be chosen by random lottery on the last day of the month. We ask that you only submit articles that are from *outside* your own field of study to try to ensure that the papers we read are accessible and interesting to non-experts.

Read more ...