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I've been toying around with fractals again. My latest investigation is the
Gosper Island. I had been thinking of using the fractal as a tile, but I'm
having trouble locating the center.
I'm using my logo macros to generate the fractal. The file is included below.
#macro triline2(len, epsilon)
#if(len < epsilon)
fd(len)
#else
lt(acos(5 * sqrt(7) / 14))
triline2(len/sqrt(7), epsilon)
rt(60)
triline2(len/sqrt(7), epsilon)
lt(60)
triline2(len/sqrt(7), epsilon)
rt(acos(5 * sqrt(7) / 14))
#end
#end
// Gosper fractal can tile the plane
#macro Gosper(len, epsilon)
#local a = 0;
#while(a < 6)
triline2(len, epsilon)
lt(60)
#local a = a + 1;
#end
#end
init()
Gosper(2, 0.01)
The problem is that as epsilon gets smaller, the fractal seems to rotate about
the origin, and I'm not sure why. I don't have the time to dig through the
calculations, but I suspect that it has to do with where the recursion stops.
If anyone can help, I'd appreciate it.
Regards,
A.D.B.
P.S: as a side note, it might be neat to have some of these fractals like the
Koch flake and the Gosper island available as patterns in a later version...
/* ! ----- Begin koch.pov ----- ! */
#include "kolors.inc"
#include "math.inc"
light_source { -50.0*z, color rgb 1 }
camera {
orthographic
location <0.0, 0.0, -10.0>
look_at 0.0
}
#macro init()
#declare _cP_ = <0.0, 0.0, 0.0>;
#declare _cD_ = 0;
#declare _DRAW_ = true;
#declare _INIT_ = 1;
#end
// Pen Up
#macro pu()
#declare _DRAW_ = false;
#end
// Pen Down
#macro pd()
#declare _DRAW_ = true;
#end
// left turn (degrees)
#macro lt(d)
#ifdef(_INIT_)
#declare _cD_ = _cD_ + d;
#if(_cD_ > 360)
#declare _cD_ = _cD_ - 360;
#end
#end
#end
// right turn (degrees)
#macro rt(d)
#ifdef(_INIT_)
#declare _cD_ = _cD_ - d;
#if(_cD_ < -360)
#declare _cD_ = _cD_ + 360;
#end
#end
#end
// foreward (length)
#macro fd(l)
#ifdef(_INIT_)
#local _pL_ = _cP_;
#declare _cP_ = _cP_ + l*<cosd(_cD_), sind(_cD_), 0>;
#if(_DRAW_)
cylinder { _pL_, _cP_, 0.015625 pigment {Orange}
finish { ambient 0.6 diffuse 0.3 } }
#end
#end
#end
#macro home()
#debug concat("_cD_ = " str(_cD_,6,6) "\n")
#debug concat("_cP_ = <" str(_cP_.x,6,6) ", " str(_cP_.y,6,6) ", "
str(_cP_.z,6,6)">\n")
#local rV = <0,0,0> - _cP_;
#local mag_rV = sqrt(pow(rV.x,2) + pow(rV.y,2));
#local rV = rV / mag_rV;
#debug concat("rV = <" str(rV.x,6,6) ", " str(rV.y,6,6) ", "
str(rV.z,6,6)">\n")
#local cV = _cP_ + <cosd(_cD_),sind(_cD_),0>;
#local cV = cV / sqrt(pow(cV.x,2) + pow(cV.y,2));
#debug concat("cV = <" str(cV.x,6,6) ", " str(cV.y,6,6) ", "
str(cV.z,6,6)">\n")
#local theta = degrees(acos(vdot(rV,cV)));
#debug concat(str(theta,6,6) "\n")
#debug concat(str(mag_rV,6,6) "\n")
#local hV = <0,0,0>;
/*
lt(theta)
fd(mag_rV)
lt(acos(vdot(rV,hV)))
*/
#end
#macro triline(len)
#if(len < 0.01)
fd(len)
#else
triline(len/3)
lt(60)
triline(len/3)
rt(120)
triline(len/3)
lt(60)
triline(len/3)
#end
#end
#macro triline2(len, epsilon)
#if(len < epsilon)
fd(len)
#else
lt(acos(5 * sqrt(7) / 14))
triline2(len/sqrt(7), epsilon)
rt(60)
triline2(len/sqrt(7), epsilon)
lt(60)
triline2(len/sqrt(7), epsilon)
rt(acos(5 * sqrt(7) / 14))
#end
#end
#macro koch(len)
pu()
lt(90)
fd(len*sind(30) - (len/3)*sind(30))
lt(90)
fd(len/2)
rt(180)
pd()
#local a = 0;
#while(a < 3)
triline(len)
rt(120)
#local a = a + 1;
#end
home()
#end
#macro gosper(len, epsilon)
#local a = 0;
#while(a < 6)
triline2(len, epsilon)
lt(60)
#local a = a + 1;
#end
#end
sphere { 0.0, 0.1 pigment { rgb 1 } }
init()
/*
pu()
lt(180)
fd(1.25)
rt(90)
fd(3.5)
rt(90)
pd()
*/
//triline2(2, 1)
gosper(2, 2/pow(sqrt(7),7))
//rt(30)
//fd(2)
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