Category Archives: Drawings

A Little Present For Coldplay

November 4, 2014DrawingsButterfly Curve, Coldplay, ggplot, ggplot2, polar coordinates, R@aschinchon

Gravity, release me, and don’t ever hold me down, now my feet won’t touch the ground (Coldplay, Life In Technicolor II)

Inspired by this nice post and by this cover of a Coldplay’s album:

I have dared to do this using ggplot, polar coordinates and Google Fonts:

Coldplay: feel free to use it for some future album.

library(ggplot2)
library(extrafont)
windowsFonts(Monoton=windowsFont("Monoton"))
butterfly=function(x) 8-sin(x)+2*sin(3*x)+2*sin(5*x)-sin(7*x)+3*cos(2*x)-2*cos(4*x)
opt=theme(legend.position="none",
          panel.background = element_rect(fill="black"),
          panel.grid = element_blank(),
          axis.ticks=element_blank(),
          axis.title=element_blank(),
          axis.text =element_blank())
ggplot(data.frame(x = c(0, 2*pi)), aes(x)) +
  stat_function(fun=butterfly, geom="density", fill="#FC0C54", colour="#FC0C54") +
  coord_polar(start=-pi)+
  geom_text(x=.5, y=-14, colour="turquoise2", family="Monoton", label="COLDPLAY", size=12)+
  geom_text(x=1.5, y=14, colour="turquoise2", family="Monoton", angle=90, label="Up Down Up Down Up Down", size=6)+
  opt

Beautiful Curves: The Harmonograph

October 13, 2014Drawingsbeautiful, curves, Harmonograph, R@aschinchon

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:

First harmonographs were built in 1857 by Scottish mathematician Hugh Blackburn, based on the previous work of French mathematician Jean Antoine Lissajous. There is not an unique way to describe mathematically the motion of the pencil. I have implemented the one I found in this sensational blog, where motion in both x and y axis depending on time is defined by:

$<br /> x(t)=e^{-d_{1}t}sin(f_{1}t+p_{1})+e^{-d_{2}t}sin(f_{2}t+p_{2})\\<br /> y(t)=e^{-d_{3}t}sin(f_{3}t+p_{3})+e^{-d_{4}t}sin(f_{4}t+p_{4})<br />$

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'))

Complex Domain Coloring

October 1, 2014Drawingscomplex numbers, ggplot2, hsv, R@aschinchon

Why don’t you stop doodling and start writing serious posts in your blog? (Cecilia, my beautiful wife)

Choose a function, apply it to a set of complex numbers, paint the result using the HSV technique and be ready to be impressed because images can be absolutely amazing. You only need ggplot2 package and your imagination. This is what happens when function is f(x)=(1+i)log(sin((x³-1)/x)):

To learn more about complex domain coloring, you can go here. If you want to try your own functions, you can find the code below. I will try to write a serious post next time but meanwhile, long live doodles!

require(ggplot2)
f = function(x) (1+1i)*log(sin((x^3-1)/x))
z=as.vector(outer(seq(-5, 5, by =.01),1i*seq(-5, 5, by =.01),'+'))
z=data.frame(x=Re(z),
y=Im(z),
h=(Arg(f(z))<0)*1+Arg(f(z))/(2*pi),
s=(1+sin(2*pi*log(1+Mod(f(z)))))/2,
v=(1+cos(2*pi*log(1+Mod(f(z)))))/2)
z=z[is.finite(apply(z,1,sum)),]
opt=theme(legend.position="none",
panel.background = element_blank(),
panel.grid = element_blank(),
axis.ticks=element_blank(),
axis.title=element_blank(),
axis.text =element_blank())
ggplot(data=z, aes(x=x, y=y)) + geom_tile(fill=hsv(z$h,z$s,z$v))+ opt

Space Invaders

September 18, 2014Drawingsaliens, ggplot, ggplot2, invaders, R, reshape@aschinchon

I burned through all of my extra lives in a matter of minutes, and my two least-favorite words appeared on the screen: GAME OVER (Ernest Cline, Ready Player One)

Inspired by the book I read this summer and by this previous post, I decided to draw these aliens:

Do not miss to check this indispensable document to choose your favorite colors:

require("ggplot2")
require("reshape")
mars1=matrix(c(0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,1,0,0,0,0,1,0,0,0,
0,0,0,0,1,0,0,1,0,0,0,0,
0,0,0,1,1,1,1,1,1,0,0,0,
0,0,1,1,0,1,1,0,1,1,0,0,
0,1,1,1,1,1,1,1,1,1,1,0,
0,1,1,1,1,1,1,1,1,1,1,0,
0,1,1,1,1,1,1,1,1,1,1,0,
0,1,1,1,1,1,1,1,1,1,1,0,
0,1,0,1,0,0,0,0,1,0,1,0,
0,1,0,0,1,0,0,1,0,0,1,0,
0,0,0,0,0,0,0,0,0,0,0,0), nrow=12, byrow = TRUE)
mars2=matrix(c(0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,1,1,0,0,0,0,0,
0,0,0,0,1,1,1,1,0,0,0,0,
0,0,0,1,1,1,1,1,1,0,0,0,
0,0,1,1,0,1,1,0,1,1,0,0,
0,1,1,1,1,1,1,1,1,1,1,0,
0,1,1,1,1,1,1,1,1,1,1,0,
0,0,0,0,1,0,0,1,0,0,0,0,
0,0,0,1,0,1,1,0,1,0,0,0,
0,0,1,0,1,0,0,1,0,1,0,0,
0,1,0,1,0,0,0,0,1,0,1,0,
0,0,0,0,0,0,0,0,0,0,0,0), nrow=12, byrow = TRUE)
mars3=matrix(c(0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,1,0,0,1,0,0,0,0,
0,0,0,1,0,0,0,0,1,0,0,0,
0,0,0,1,1,1,1,1,1,0,0,0,
0,0,1,1,0,1,1,0,1,1,0,0,
0,1,1,1,1,1,1,1,1,1,1,0,
0,1,1,1,1,1,1,1,1,1,1,0,
0,1,1,1,1,1,1,1,1,1,1,0,
0,1,0,1,0,0,0,0,1,0,1,0,
0,1,0,0,1,1,1,1,0,0,1,0,
0,0,0,0,1,0,0,1,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0), nrow=12, byrow = TRUE)
mars4=matrix(c(0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,1,1,0,0,0,0,0,
0,0,0,0,1,1,1,1,0,0,0,0,
0,0,0,1,1,1,1,1,1,0,0,0,
0,0,1,1,0,1,1,0,1,1,0,0,
0,1,1,1,1,1,1,1,1,1,1,0,
0,1,1,1,1,1,1,1,1,1,1,0,
0,1,0,0,1,0,0,1,0,0,1,0,
0,0,0,1,0,0,0,0,1,0,0,0,
0,0,0,0,1,0,0,1,0,0,0,0,
0,0,0,1,0,0,0,0,1,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0), nrow=12, byrow = TRUE)
opt=theme(legend.position="none",
panel.background = element_blank(),
panel.grid = element_blank(),
axis.ticks = element_blank(),
axis.title = element_blank(),
axis.text = element_blank())
p1=ggplot(melt(mars1), aes(x=X2, y=X1))+geom_tile(aes(fill=jitter(value, amount=.1)), colour="gray65", lwd=.025)+
scale_fill_gradientn(colours = c("chartreuse", "navy"))+scale_y_reverse()+opt
p2=ggplot(melt(mars2), aes(x=X2, y=X1))+geom_tile(aes(fill=jitter(value, amount=.1)), colour="gray65", lwd=.025)+
scale_fill_gradientn(colours = c("olivedrab1", "magenta4"))+scale_y_reverse()+opt
p3=ggplot(melt(mars3), aes(x=X2, y=X1))+geom_tile(aes(fill=jitter(value, amount=.1)), colour="gray65", lwd=.025)+
scale_fill_gradientn(colours = c("violetred4", "yellow"))+scale_y_reverse()+opt
p4=ggplot(melt(mars4), aes(x=X2, y=X1))+geom_tile(aes(fill=jitter(value, amount=.1)), colour="gray65", lwd=.025)+
scale_fill_gradientn(colours = c("tomato4", "lawngreen"))+scale_y_reverse()+opt

The Zebra Of Riemann

July 13, 2014Curiosities, Drawings, Image Processingggplot2. ggplot, pracma, R, Riemann, tile, zeta@aschinchon

Mathematics is the art of giving the same name to different things (Henri Poincare)

Many surveys among experts point that demonstration of the Riemann Hypothesis is the most important pending mathematical issue in this world. This hypothesis is related to Riemann zeta function, which is supossed to be zero only for those complex whose real part is equal to 1/2 (this is the conjecture of Riemann itself). Confirming the conjecture would imply deep consequences in prime numbers teory and also in our knowledge of their properties. Next plot represents the argument of zeta function over complex with real part between -75 and 5 (x axis) and imaginary part between -40 and 40 (y axis). An explanation of this kind of graph can be found here. Does not it remind you of something?

Solving the hypothesis of Riemann is one of the seven Clay Mathematics Institute Millennium Prize Problems. Maybe zebras keep the secret to solve it in their skins.

This is the code to plot the graph:

require(pracma)
z=outer(seq(-75, 5, by =.1),1i*seq(-40, 40, by =.1),'+')
z=apply(z, c(1,2), function(x) Arg(zeta(x)))
opt=theme(legend.position="none",
panel.background = element_blank(),
panel.grid = element_blank(),
axis.ticks=element_blank(),
axis.title=element_blank(),
axis.text =element_blank())
z=data.frame(expand.grid(x=seq(ncol(z)), y=seq(nrow(z))), z=as.vector(z))
require(ggplot2)
ggplot(z, aes(x=x, y=y, color=z)) + geom_tile() + scale_colour_gradientn(colours=c("black","white", "gray80", "gray40")) + opt

Four Simple Turtle Graphs To Play With Kids

July 7, 2014Drawingskids, R, Turtle Graphics@aschinchon

Technology is just a tool: in terms of getting the kids working together and motivating them, the teacher is the most important (Bill Gates)

Some days ago I read this article in R-bloggers and I discovered the TurtleGraphics package. I knew about turtle graphics long time ago and I was thinking of writing a post about them, doing some draw on my own using this technique. In fact, some of my previous drawings could have been done using this package.

Turtle graphics are a good tool to teach kids some very important mathematical and computational concepts such as length, coordinates, location, angles, iterations … Small changes in some of this parameters (especially in angles) can produce very different drawings. Those who want to know more about turtle graphics can read the Wikipedia article.

Following you can find four simple graphics preceded by the code to generate them. Please, send me your creations if you want:

library(TurtleGraphics)
turtle_init()
turtle_col("gray25")
for (i in 1:150) {
  turtle_forward(dist=1+0.5*i)
  turtle_right(angle=89.5)}
turtle_hide()

library(TurtleGraphics)
turtle_init()
turtle_col("gray25")
turtle_right(angle=234)
for (i in 1:100) {
  turtle_forward(dist=0.9*i)
  turtle_right(angle=144.3)}
turtle_hide()

library(TurtleGraphics)
turtle_init()
turtle_col("gray25")
turtle_setpos(48,36)
d=50
for (i in 1:300) {
  turtle_forward(dist=d)
  if (i%%4==0) {
    turtle_right(angle=75)
    d=d*.95}
else turtle_right(angle=90)}
turtle_hide()

library(TurtleGraphics)
turtle_init()
turtle_col("gray25")
turtle_setpos(50,35)
turtle_right(angle=30)
d=25
turtle_setpos(50-d/2,50-d/2*tan(pi/6))
for (i in 1:100) {
  turtle_forward(dist=d)
  d=d+.5
  turtle_right(angle=120+1)}
turtle_hide()

Fronkonstin

Experiments in R