I just found out David Metzler exists, here's his very gentle 47 part intro into differential forms:


Or this cool series, about the Hodge connjecture:


#maths #lecture #differential #algebra

My snowdecahedron is being struck by global warming.
Interestingly, it seems to melt slower on the edges, so that they stay sharp, as the shape shrinks.

Made a snowdecahedron. Not as nice as some people on the web, but still, I deserve a beer.

Fitting In

Stereographic projection of putatively optimal packing of 124 points on the unit sphere.

Source code and explanation: community.wolfram.com/groups/-

Doing a relativity project, I came accros a talk by Einstein, in which he argues for not rejecting ether theory!
He says that although special relativity suggests that the ether is a redundant concept, in general relativity it might be usefull, since space-time, according the general relativity, has local physical properties.

Following up on my previous post on Newton’s theorem that you can have circular orbits that pass infinitely fast through the origin, if and only if the gravity law were r^-5.
I did an animation with more frames, but fewer graphics.

Amazing, that Newton figured this out around 1680! Newton had to first invent calculus himself, and come up with the revolutionary new way of looking at the physical world using mathematics.

Newton is of course famous for discovering the inverse square law of gravity. But he also found a number of interesting theorems about central force fields. One of them is illustrated here: If the force law is 1/r^5, then there are solutions that are circles which pass through centre! The velocity is infinite as the ‘planet’ shoots through the central ‘black hole.
Wikipedia on central force theorems:

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

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