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Each of us has their own mappa mundi (Gala, my indispensable friend)
The harmonograph is a mechanism which, by means of several pendulums, draws trajectories that can be analyzed not only from a mathematical point of view but also from an artistic one. In its double pendulum version, one pendulum moves a pencil and the other one moves a platform with a piece of paper on it. You can see an example here. The harmonograph is easy to use: you only have to put pendulums into motion and wait for them to stop. The result are amazing undulating drawings like this one:
x(t)=e-d1tsin(tf1+p1)+e-d2tsin(tf2+p2)
y(t)=e-d3tsin(tf3+p3)+e-d4tsin(tf4+p4)
I initialize parameters randomly so every time you run the script, you obtain a different output. Here is a mosaic with some of mine:
This is the code to simulate the harmonograph (no extra package is required). If you create some nice work of art, I will be very happy to admire it (you can find my email here):
f1=jitter(sample(c(2,3),1));f2=jitter(sample(c(2,3),1));f3=jitter(sample(c(2,3),1));f4=jitter(sample(c(2,3),1)) d1=runif(1,0,1e-02);d2=runif(1,0,1e-02);d3=runif(1,0,1e-02);d4=runif(1,0,1e-02) p1=runif(1,0,pi);p2=runif(1,0,pi);p3=runif(1,0,pi);p4=runif(1,0,pi) xt = function(t) exp(-d1*t)*sin(t*f1+p1)+exp(-d2*t)*sin(t*f2+p2) yt = function(t) exp(-d3*t)*sin(t*f3+p3)+exp(-d4*t)*sin(t*f4+p4) t=seq(1, 100, by=.001) dat=data.frame(t=t, x=xt(t), y=yt(t)) with(dat, plot(x,y, type="l", xlim =c(-2,2), ylim =c(-2,2), xlab = "", ylab = "", xaxt='n', yaxt='n'))
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