G
galathaea
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?
-+-+-
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<double> const 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<double> rootOfUnityGeneralisedPolynomial(unsigned n,
std::complex<double> const& z)
{
std::complex<double> sum(0., 0.);
for (unsigned termIndex(0); termIndex < n; ++termIndex)
{
std::complex<double> term(std::exp(std:ow(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!
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
!! 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?
-+-+-
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<double> const 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<double> rootOfUnityGeneralisedPolynomial(unsigned n,
std::complex<double> const& z)
{
std::complex<double> sum(0., 0.);
for (unsigned termIndex(0); termIndex < n; ++termIndex)
{
std::complex<double> term(std::exp(std:ow(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!
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar