Beautiful math formulas,series,sequences

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Beautiful math formulas,series,sequences

#1  Postby cavarka9 » Jan 18, 2015 7:33 am

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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Re: Beautiful math formulas,series,sequences

#2  Postby Thommo » Jan 18, 2015 9:04 am

Image

A zoomed out image of the mandelbrot set, the set of points in the complex plane for which zn+1 = zn2+c.
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Re: Beautiful math formulas,series,sequences

#3  Postby VazScep » Jan 18, 2015 11:05 am

Thommo wrote:Image

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).
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Re: Beautiful math formulas,series,sequences

#4  Postby colubridae » Jan 18, 2015 11:33 am

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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Re: Beautiful math formulas,series,sequences

#5  Postby Thommo » Jan 18, 2015 11:50 am

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). :scratch:
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Re: Beautiful math formulas,series,sequences

#6  Postby BlackBart » Jan 18, 2015 11:55 am

Conway's Game of Life
Image

.. that such a childishly simple algorithm can produce such complexity makes me feel dizzy in a good way.
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Re: Beautiful math formulas,series,sequences

#7  Postby cavarka9 » Jan 18, 2015 12:02 pm

BlackBart wrote:Conway's Game of Life
Image

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

with equations please
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Re: Beautiful math formulas,series,sequences

#8  Postby BlackBart » Jan 18, 2015 12:48 pm

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

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

with equations please

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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Re: Beautiful math formulas,series,sequences

#9  Postby VazScep » Jan 18, 2015 6:07 pm

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.
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Re: Beautiful math formulas,series,sequences

#10  Postby THWOTH » Jan 18, 2015 6:36 pm

Thommo wrote:Image

Nice way to explore the Mandelbrot and Julia sets @ FalcoSoft.
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Re: Beautiful math formulas,series,sequences

#11  Postby Calilasseia » Jan 18, 2015 7:32 pm

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

There's something deeply satisfying about this. :)
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Re: Beautiful math formulas,series,sequences

#12  Postby Thommo » Jan 18, 2015 7:40 pm

THWOTH wrote:
Thommo wrote:Image

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


That's cool.
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Re: Beautiful math formulas,series,sequences

#13  Postby BlackBart » Jan 18, 2015 7:46 pm

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

#14  Postby VazScep » Mar 14, 2015 8:45 pm

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 :cheers:
I finally got around to implementing this:

Code: Select all
type GridCursor a = CursorT (CursorT Identity) a

east :: GridCursor a -> Maybe (GridCursor a)
east = fmap extract . left

west :: GridCursor a -> Maybe (GridCursor a)
west = fmap extract . right

north :: 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 -> Int
neighboursAlive = length . filter id . f
  where f = fmap catMaybes . sequence . (fmap.fmap.fmap) extract $ neighboursC

conway =
  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.
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Re: Beautiful math formulas,series,sequences

#15  Postby halucigenia » Mar 30, 2015 3:04 pm

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

Is superior to the Mandlebrot set
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Re: Beautiful math formulas,series,sequences

#16  Postby THWOTH » Apr 05, 2015 12:29 pm

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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Re: Beautiful math formulas,series,sequences

#17  Postby THWOTH » Apr 06, 2015 10:39 pm

Re :this:

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)
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Re: Beautiful math formulas,series,sequences

#18  Postby CdesignProponentsist » Apr 07, 2015 12:35 am

Image

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.

Image

http://en.wikipedia.org/wiki/Dragon_curve
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Re: Beautiful math formulas,series,sequences

#19  Postby THWOTH » Apr 07, 2015 1:17 am

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

#20  Postby epepke » Apr 07, 2015 2:49 am

BlackBart wrote:Conway's Game of Life
Image

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


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