## Beautiful math formulas,series,sequences

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### Beautiful math formulas,series,sequences

1+e^i(pi) =0 i is imaginary number, pi is a known number close to value 3.14.

only math.
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cavarka9

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### Re: Beautiful math formulas,series,sequences

A zoomed out image of the mandelbrot set, the set of points in the complex plane for which zn+1 = zn2+c.

Thommo

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### Re: Beautiful math formulas,series,sequences

Thommo wrote:

A zoomed out image of the mandelbrot set, the set of points in the complex plane for which zn+1 = zn2+c
...is bounded with z0 = 0 (and that just covers the black parts, which on its own, isn't quite so pretty).

Alternatively: the set of points c such that the keep set of zn+1 = zn2+c is connected (and these sequences classify the long term behaviour of any quadratic iteration sequence).
Here we go again. First, we discover recursion.
VazScep

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### Re: Beautiful math formulas,series,sequences

Always had a passion for akermann's function. Something about this level of recursion that floats my boat. Writing down the values and watching it wind down, then wind all the way up again. Goosebumps.
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colubridae

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### Re: Beautiful math formulas,series,sequences

VazScep wrote:...is bounded with z0 = 0 (and that just covers the black parts, which on its own, isn't quite so pretty).

Alternatively: the set of points c such that the keep set of zn+1 = zn2+c is connected (and these sequences classify the long term behaviour of any quadratic iteration sequence).

Thanks, I'm not quite sure what happened to the rest of that sentence, I intended to post the full definition (of the base set).

Thommo

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### Re: Beautiful math formulas,series,sequences

Conway's Game of Life

.. that such a childishly simple algorithm can produce such complexity makes me feel dizzy in a good way.
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BlackBart

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### Re: Beautiful math formulas,series,sequences

BlackBart wrote:Conway's Game of Life

.. that such a childishly simple algorithm can produce such complexity makes me feel dizzy in a good way.

Every moment is a choice.Choices you make now determine your destiny.free yourself of old choices made. Success is a journey,not a destination.

cavarka9

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### Re: Beautiful math formulas,series,sequences

cavarka9 wrote:
BlackBart wrote:Conway's Game of Life

.. that such a childishly simple algorithm can produce such complexity makes me feel dizzy in a good way.

No idea if it can be expressed as a formula. The algorithm is described here; http://mathworld.wolfram.com/GameofLife.html
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BlackBart

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### Re: Beautiful math formulas,series,sequences

BlackBart wrote:No idea if it can be expressed as a formula. The algorithm is described here; http://mathworld.wolfram.com/GameofLife.html
Every program can be given by a term in the untyped lambda calculus, which stands for an object in a Scott domain. Not sure it would be very elegant though.

Relatively recently, functional programmers have noticed that cellular automata are elegantly described by the very general and important notion of a comonad of a very general and important data structure called a zipper. Expressed in this language, you can code up Conway's Game of Life in half a dozen lines of code without ever having to muck around with recursion or loops.
Here we go again. First, we discover recursion.
VazScep

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### Re: Beautiful math formulas,series,sequences

Thommo wrote:

Nice way to explore the Mandelbrot and Julia sets @ FalcoSoft.
"No-one is exempt from speaking nonsense – the only misfortune is to do it solemnly."
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THWOTH
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### Re: Beautiful math formulas,series,sequences

One particularly elegant result I like, is that for any arbitrary tensor, you can pick two indices thereof, subject to certain constraints (usually that both indices are contravariant or covariant), and rearrange that tensor as a sum of two new tensors, one symmetric and one antisymmetric.

In the case of rank 2 tensors, if Tij is an arbitrary tensor, then this result says that you can write Tij as the sum of two other tensors, Sij and Aij, where Sij and Aij obey:

[1] Sij = Sji

[2] Aij = -Aji

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### Re: Beautiful math formulas,series,sequences

THWOTH wrote:
Thommo wrote:

Nice way to explore the Mandelbrot and Julia sets @ FalcoSoft.

That's cool.

Thommo

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### Re: Beautiful math formulas,series,sequences

VazScep wrote:
BlackBart wrote:No idea if it can be expressed as a formula. The algorithm is described here; http://mathworld.wolfram.com/GameofLife.html
Every program can be given by a term in the untyped lambda calculus, which stands for an object in a Scott domain. Not sure it would be very elegant though.

Relatively recently, functional programmers have noticed that cellular automata are elegantly described by the very general and important notion of a comonad of a very general and important data structure called a zipper. Expressed in this language, you can code up Conway's Game of Life in half a dozen lines of code without ever having to muck around with recursion or loops.

Interesting! I'll see if I can speed up my clunky life program
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BlackBart

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### Re: Beautiful math formulas,series,sequences

BlackBart wrote:
VazScep wrote:
BlackBart wrote:No idea if it can be expressed as a formula. The algorithm is described here; http://mathworld.wolfram.com/GameofLife.html
Every program can be given by a term in the untyped lambda calculus, which stands for an object in a Scott domain. Not sure it would be very elegant though.

Relatively recently, functional programmers have noticed that cellular automata are elegantly described by the very general and important notion of a comonad of a very general and important data structure called a zipper. Expressed in this language, you can code up Conway's Game of Life in half a dozen lines of code without ever having to muck around with recursion or loops.

Interesting! I'll see if I can speed up my clunky life program
I finally got around to implementing this:

Code: Select all
`type GridCursor a = CursorT (CursorT Identity) aeast :: GridCursor a -> Maybe (GridCursor a)east = fmap extract . leftwest :: GridCursor a -> Maybe (GridCursor a)west = fmap extract . rightnorth :: GridCursor a -> Maybe (GridCursor a)north x = fmap (const \$ cohoist (runIdentity . fromJust . left) x) (left . lower \$ x)south :: GridCursor a -> Maybe (GridCursor a)south x = fmap (const \$ cohoist (runIdentity . fromJust . right) x) (right . lower \$ x)neighboursC :: [GridCursor a -> Maybe (GridCursor a)]neighboursC = [north, north >=> east, east, east >=> south, south, south >=> west,west, west >=> north]neighboursAlive :: GridCursor Bool -> IntneighboursAlive = length . filter id . f  where f = fmap catMaybes . sequence . (fmap.fmap.fmap) extract \$ neighboursCconway =  extend (\w ->           if neighboursAlive w < 2 || neighboursAlive w > 3 then False              else if neighboursAlive w == 3 then True else extract w)`

I assume it reads as line-noise, but pretty much everything there is backed up by an absolutely watertight mathematical abstraction. This is the closest I know how to get a working implementation of Conway to elegant maths. 10 lines of code. The rest might as well be documentation.
Here we go again. First, we discover recursion.
VazScep

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### Re: Beautiful math formulas,series,sequences

For fractals I still say that
Code: Select all
`IF (real(z) < 0)   z = (z+1)*cELSE    z = (z-1)*conj(c)`

Is superior to the Mandlebrot set

halucigenia

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### Re: Beautiful math formulas,series,sequences

Re Conway's Game of Life.

Am I right in thinking that; for any cell in the grid, if the sum of it's 8 adjacent cells is 2 or 3 the cell is turned on (value, 1) if it's off or stays on if it's on, otherwise the cell is turned off (value, 0) if it's on or stays off if it's off. I.E:

if (sum < 2 OR sum > 3) cell = 0
else cell = 1
?
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THWOTH
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### Re: Beautiful math formulas,series,sequences

Re

Thought I'd have a go at this using HTML5 Canvas API and JavaScript.

http://thwoth.net/gameoflife/

(Create your own seeds at the bottom of the page)
"No-one is exempt from speaking nonsense – the only misfortune is to do it solemnly."
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THWOTH
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### Re: Beautiful math formulas,series,sequences

http://en.wikipedia.org/wiki/Cyclic_cellular_automaton

Also the Dragon Curve which I ploted out once when I was in my teens when I first leared of fractals.

http://en.wikipedia.org/wiki/Dragon_curve
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CdesignProponentsist

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### Re: Beautiful math formulas,series,sequences

That's interesting.
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THWOTH
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### Re: Beautiful math formulas,series,sequences

BlackBart wrote:Conway's Game of Life

.. that such a childishly simple algorithm can produce such complexity makes me feel dizzy in a good way.

epepke

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