Consider the
Thomae function (or
modified Dirichlet function):
This function is continuous at all irrational numbers and discontinuous at all rational numbers.
We will define f approximately as a gnuplot function (see f(1,x) below) such that we can plot
the graph of y = f(sin(x)) directly.
1. y = f(sin(x)) with linespoints
(Click on the image for a larger view.)
Script 1:
d = 0.001
m = 250
f(k,x) = k>m ? 0 : (abs(k*x-int(k*x))<d ? 1.0/k : f(k+1,x))
set xrange [-pi:pi]
set yrange [-0.5:1.5]
set samples 100000
set size ratio -1
unset key
set pointsize 0.25
plot f(1,sin(x)) with linespoints ls 7 lw 0.25 lc rgb "red"
2. y = f(sin(x)) with points
(Click on the image for a larger view.)
Script 2:
d = 0.001
m = 250
f(k,x) = k>m ? 0 : (abs(k*x-int(k*x))<d ? 1.0/k : f(k+1,x))
set xrange [-pi:pi]
set yrange [-0.5:1.5]
set samples 100000
set size ratio -1
unset key
set pointsize 0.1
plot f(1,sin(x)) with points pt 7 lc rgb "#00cc00"
3. y = f(sin(x)) with dots
(Click on the image for a larger view.)
Script 3:
d = 0.001
m = 250
f(k,x) = k>m ? 0 : (abs(k*x-int(k*x))<d ? 1.0/k : f(k+1,x))
set xrange [-pi:pi]
set yrange [-0.5:1.5]
set samples 100000
set size ratio -1
unset key
plot f(1,sin(x)) with dots lc rgb "black"
4. Golden Ring (parametric plot)
(Click on the image for a larger view.)
Script 4:
d = 0.001
m = 250
f(k,x) = k>m ? 0 : (abs(k*x-int(k*x))<d ? 1.0/k : f(k+1,x))
set xrange [-1.3:1.3]
set yrange [-1.05:1.55]
set size ratio -1
set samples 100000
set parametric
unset key
unset xtics
unset ytics
unset border
set term svg size 800, 800
set bmargin 1
set lmargin 1
set rmargin 1
set tmargin 1
set output 'C:\golden_ring.svg'
set pointsize 0.125
set multiplot
plot [0:2*pi] -f(1,sin(t))*sin(t), 1+f(1,sin(t))*cos(t) \
w p pt 7 lc rgb "#ffd700"
plot [0:2*pi] -(f(1,sin(t))*sin(t)+sin(2*t)), \
f(1,sin(t))*cos(t)+cos(2*t) w p pt 7 lc rgb "#ffd700"
( Mathematical software used:
gnuplot )
Related posts:
