you will fall in love

!! these are the cutest critters !!

3-ary polynomial:

http://i10.tinypic.com/2hwlir6.png
4-ary polynomial:

full

http://i13.tinypic.com/47u0yfm.png
close

http://i14.tinypic.com/2heajo5.png
5-ary polynomial:

full

http://i18.tinypic.com/2cifxja.png
close

http://i16.tinypic.com/2wcr9js.png
6-ary polynomial:

full

http://i14.tinypic.com/4fy3wd3.png
close

http://i17.tinypic.com/48zmlbl.png
7-ary polynomial:

full

http://i18.tinypic.com/2dalt9l.png
close

http://i10.tinypic.com/2j4rc7o.png
these are plots

of the class of generalised polynomials

i have been playing with

2 pi i / n

w = e

n

2 3 n-1

w_n w_n w_n w_n

y = x + x + x + x + ... + x

n

i coded the poynomials

in my openGL simulations engine

and color mapped the absolute value

over a part of the complex plane centered at the origin

x axis real positive to right

y axis imaginary positive up

there are a lot of immediate conjectures to make

but they are also tantalisingly difficult to prove

the zeroes (black) appear to have

some interesting number theoretic information

but above all they are adorable

the surfaces are fascinating

with the symmetries of their ripples

and the intricacies of their "mouths"

has their tale already been told?

or is this one those stories

that still needs to be written?

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for those interested in the simulation

the behavior is coded as c++ or OCaml

( this particular class of functions in c++ )

first i define omega (w above)

or pull it from my library

something like

std::complex<doubleconst nthRoot(std::exp(2. * pi * i /

std::complex<double>(n, 0.)));

now i was using runtime variation of arity

but i also have a compiletime version as well

to unroll the loops

remove loop counter

and other optimisations

but it didn't make much a difference in time

having to compile so often

std::complex<doublerootOfUnityGeneralisedPolynomia l(unsigned n,

std::complex<doubleconst& z)

{

std::complex<doublesum(0., 0.);

for (unsigned termIndex(0); termIndex < n; ++termIndex)

{

std::complex<doubleterm(std::exp(std::pow(nthRoot, termIndex) *

std::log(z)));

sum += term;

//std::cout << "current term: " << term << std::endl;

//std::cout << "running sum: " << sum << std::endl;

}

return sum;

}

the simulator has visualisers for the complex plane

so i ran these through the engine

and colored it pretty!

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galathaea: prankster, fablist, magician, liar