Category Archives: Drawings

Plants

Blue dragonflies dart to and fro
I tie my life to your balloon and let it go
(Warm Foothills, Alt-J)

In my last post I did some drawings based on L-Systems. These drawings are done sequentially. At any step, the state of the drawing can be described by the position (coordinates) and the orientation of the pencil. In that case I only used two kind of operators: drawing a straight line and turning a constant angle. Today I used two more symbols to do stack operations:

  • “[“ Push the current state (position and orientation) of the pencil onto a pushdown
    operations stack
  • “]” Pop a state from the stack and make it the current state of the pencil (no line is drawn)

These operators allow to return to a previous state to continue drawing from there. Using them you can draw plants like these:

Each image corresponds to a different axiom, rules, angle and depth. I described these terms in my previous post. If you want to reproduce them you can find the code below (each image corresponds to a different set of axiom, rules, angle and depth parameters). Change colors, add noise to angles, try your own plants … I am sure you will find nice images:


library(gsubfn)
library(stringr)
library(dplyr)
library(ggplot2)

#Plant 1
axiom="F"
rules=list("F"="FF-[-F+F+F]+[+F-F-F]")
angle=22.5
depth=4

#Plant 2
axiom="X"
rules=list("X"="F[+X][-X]FX", "F"="FF")
angle=25.7
depth=7

#Plant 3
axiom="X"
rules=list("X"="F[+X]F[-X]+X", "F"="FF")
angle=20
depth=7

#Plant 4
axiom="X"
rules=list("X"="F-[[X]+X]+F[+FX]-X", "F"="FF")
angle=22.5
depth=5

#Plant 5
axiom="F"
rules=list("F"="F[+F]F[-F]F")
angle=25.7
depth=5

#Plant 6
axiom="F"
rules=list("F"="F[+F]F[-F][F]")
angle=20
depth=5


for (i in 1:depth) axiom=gsubfn(".", rules, axiom)

actions=str_extract_all(axiom, "\\d*\\+|\\d*\\-|F|L|R|\\[|\\]|\\|") %>% unlist

status=data.frame(x=numeric(0), y=numeric(0), alfa=numeric(0))
points=data.frame(x1 = 0, y1 = 0, x2 = NA, y2 = NA, alfa=90, depth=1)


for (action in actions) 
{
  if (action=="F")
  {
    x=points[1, "x1"]+cos(points[1, "alfa"]*(pi/180))
    y=points[1, "y1"]+sin(points[1, "alfa"]*(pi/180))
    points[1,"x2"]=x
    points[1,"y2"]=y
    data.frame(x1 = x, y1 = y, x2 = NA, y2 = NA, 
               alfa=points[1, "alfa"],
               depth=points[1,"depth"]) %>% rbind(points)->points
  }
  if (action %in% c("+", "-")){
    alfa=points[1, "alfa"]
    points[1, "alfa"]=eval(parse(text=paste0("alfa",action, angle)))
  }
  if(action=="["){ 
    data.frame(x=points[1, "x1"], y=points[1, "y1"], alfa=points[1, "alfa"]) %>% 
      rbind(status) -> status
    points[1, "depth"]=points[1, "depth"]+1
  }
  
  if(action=="]"){ 
    depth=points[1, "depth"]
    points[-1,]->points
    data.frame(x1=status[1, "x"], y1=status[1, "y"], x2=NA, y2=NA, 
               alfa=status[1, "alfa"],
               depth=depth-1) %>% 
      rbind(points) -> points
    status[-1,]->status
  }
}

ggplot() + 
  geom_segment(aes(x = x1, y = y1, xend = x2, yend = y2), 
               lineend = "round", 
               colour="white",
               data=na.omit(points)) + 
  coord_fixed(ratio = 1) +
  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())

A Shiny App to Draw Curves Based on L-System

Don’t worry about a thing ’cause every little thing gonna be alright (Three Little Birds, Bob Marley)

One of my favourite books is The Computational Beauty of Nature by Gary William Flake where there is a fantastic chapter about fractals in which I discovered the L-Systems.

L-Systems were conceived  in 1968 by Aristide Lindenmayer, a Hungarian biologist, as a mathematical description of plant growth. Apart from the Wikipedia, there are many places on the Internet where you can read about them. If you are interested, don’t miss The Algorithmic Beauty of Plants, an awesome book by Przemysław Prusinkiewicz that you can obtain here for free.

Roughly speaking, a L-System is a very efficient way to make drawings. In its simplest way consists in two different actions: draw a straigh line and change the angle. This is just what you need, for example, to draw a square: draw a straigh line of  any length, turn 90 degrees (without drawing), draw another straigh line of the same length, turn 90 degrees in the same direction, draw, turn and draw again. Denoting F as the action of drawing a line of length d and + as turning 90 degrees right, the whole process to draw a square can be represented as F+F+F+F.

L-Systems are quite simple to program in R. You only need to substitute the rules iteratively into the axiom (I use gsubfn function to do it) and split the resulting chain into parts with str_extract_all, for example. The result is a set of very simple actions (draw or turn) that can be visualized with ggplot and its path geometry. There are four important parameters in L-Systems:

  • The seed of the drawing, called axiom
  • The substitutions to be applied iteratively, called rules
  • How many times to apply substitutions, called depth
  • Angle of each turning

For example, let’s define the next L-System:

  • Axiom: F-F-F-F
  • Rule: F → F−F+F+FF−F−F+F

The rule means that every F must be replaced by F−F+F+FF−F−F+F while + means right turning and - left one. After one iteration, the axiom is replaced by F-F+F+FF-F-F+F-F-F+F+FF-F-F+F-F-F+F+FF-F-F+F-F-F+F+FF-F-F+F and iterating again, the new string is F-F+F+FF-F-F+F-F-F+F+FF-F-F+F+F-F+F+FF-F-F+F+F-F+F+FF-F-F+FF-F+F+FF-F-F+F-F-F+F+FF-F-F+F-F-F+F+FF-F-F+F+F-F+F+FF-F-F+F-F-F+F+FF-F-F+F-F-F+F+FF-F-F+F+F-F+F+FF-F-F+F+F-F+F+FF-F-F+FF-F+F+FF-F-F+F-F-F+F+FF-F-F+F-F-F+F+FF-F-F+F+F-F+F+FF-F-F+F-F-F+F+FF-F-F+F-F-F+F+FF-F-F+F+F-F+F+FF-F-F+F+F-F+F+FF-F-F+FF-F+F+FF-F-F+F-F-F+F+FF-F-F+F-F-F+F+FF-F-F+F+F-F+F+FF-F-F+F-F-F+F+FF-F-F+F-F-F+F+FF-F-F+F+F-F+F+FF-F-F+F+F-F+F+FF-F-F+FF-F+F+FF-F-F+F-F-F+F+FF-F-F+F-F-F+F+FF-F-F+F+F-F+F+FF-F-F+F. As you can see, the length of the string grows exponentially. Converting last string into actions, produces this drawing, called Koch Island:

It is funny how different axioms and rules produce very different drawings. I have done a Shiny App to play with L-systems. Although it is quite simple, it has two interesting features I would like to undeline:

  • Delay reactions with eventReactive to allow to set depth and angle values before refreshing the plot
  • Build a dynamic UI that reacts to user input depending on the curve choosen

There are twelve curves in the application: Koch Island (and 6 variations), cuadratic snowflake, Sierpinsky triangle, hexagonal Gosper, quadratic Gosper and Dragon curve. These are their plots:

The definition of all these curves (axiom and rules) can be found in the first chapter of the Prusinkiewicz’s book. The magic comes when you modify angles and colors. These are some examples among the infinite number of possibilities that can be created:

I enjoyed a lot doing and playing with the app. You can try it here. If you do a nice drawing, please let me know in Twitter or dropping me an email. This is the code of the App:

ui.R:

library(shiny)

shinyUI(fluidPage(
  titlePanel("Curves based on L-systems"),
  
  sidebarLayout(
    sidebarPanel(
      selectInput("cur", "Choose a curve:",
                  c("","Koch Island",
                    "Cuadratic Snowflake",
                    "Koch Variation 1",
                    "Koch Variation 2",
                    "Koch Variation 3",
                    "Koch Variation 4",
                    "Koch Variation 5",
                    "Koch Variation 6",
                    "Sierpinsky Triangle",
                    "Dragon Curve",
                    "Hexagonal Gosper Curve",
                    "Quadratic Gosper Curve"),
                  selected = ""),
      
      conditionalPanel(
        condition = "input.cur != \"\"",
        uiOutput("Iterations")),
      
      conditionalPanel(
        condition = "input.cur != \"\"",
        uiOutput("Angle")),
      
      conditionalPanel(
        condition = "input.cur != \"\"",
        selectInput("lic", label = "Line color:", choices = colors(), selected = "black")),
      
      
      conditionalPanel(
        condition = "input.cur != \"\"",
        selectInput("bac", label = "Background color:", choices = colors(), selected = "white")),
      
      conditionalPanel(
        condition = "input.cur != \"\"",
        actionButton(inputId = "go", label = "Go!", 
                     style="color: #fff; background-color: #337ab7; border-color: #2e6da4"))
      
      
      
      
    ), 
    mainPanel(plotOutput("curve", height="550px", width = "100%"))
  )
  
))

server.R:

library(shiny)
library(gsubfn)
library(stringr)
library(dplyr)
library(ggplot2)
library(rlist)

shinyServer(function(input, output) {
   
  curves=list(
    list(name="Koch Island",
         axiom="F-F-F-F",
         rules=list("F"="F-F+F+FF-F-F+F"),
         angle=90,
         n=2,
         alfa0=90),
    list(name="Cuadratic Snowflake",
         axiom="-F",
         rules=list("F"="F+F-F-F+F"),
         angle=90,
         n=4,
         alfa0=90),
    list(name="Koch Variation 1",
         axiom="F-F-F-F",
         rules=list("F"="FF-F-F-F-F-F+F"),
         angle=90,
         n=3,
         alfa0=90),
    list(name="Koch Variation 2",
         axiom="F-F-F-F",
         rules=list("F"="FF-F-F-F-FF"),
         angle=90,
         n=4,
         alfa0=90),
    list(name="Koch Variation 3",
         axiom="F-F-F-F",
         rules=list("F"="FF-F+F-F-FF"),
         angle=90,
         n=3,
         alfa0=90),
    list(name="Koch Variation 4",
         axiom="F-F-F-F",
         rules=list("F"="FF-F--F-F"),
         angle=90,
         n=4,
         alfa0=90),
    list(name="Koch Variation 5",
         axiom="F-F-F-F",
         rules=list("F"="F-FF--F-F"),
         angle=90,
         n=5,
         alfa0=90),
    list(name="Koch Variation 6",
         axiom="F-F-F-F",
         rules=list("F"="F-F+F-F-F"),
         angle=90,
         n=4,
         alfa0=90),
    list(name="Sierpinsky Triangle",
         axiom="R",
         rules=list("L"="R+L+R", "R"="L-R-L"),
         angle=60,
         n=6,
         alfa0=0),
    list(name="Dragon Curve",
         axiom="L",
         rules=list("L"="L+R+", "R"="-L-R"),
         angle=90,
         n=10,
         alfa0=90),
    list(name="Hexagonal Gosper Curve",
         axiom="L",
         rules=list("L"="L+R++R-L--LL-R+", "R"="-L+RR++R+L--L-R"),
         angle=60,
         n=4,
         alfa0=60),
    list(name="Quadratic Gosper Curve",
         axiom="-R",
         rules=list("L"="LL-R-R+L+L-R-RL+R+LLR-L+R+LL+R-LR-R-L+L+RR-", 
                    "R"="+LL-R-R+L+LR+L-RR-L-R+LRR-L-RL+L+R-R-L+L+RR"),
         angle=90,
         n=2,
         alfa0=90))
  
  output$Iterations <- renderUI({ if (input$cur!="") curve=list.filter(curves, name==input$cur) else curve=list.filter(curves, name=="Koch Island") iterations=list.select(curve, n) %>% unlist
    numericInput("ite", "Depth:", iterations, min = 1, max = (iterations+2))
  })
  
  output$Angle <- renderUI({ curve=list.filter(curves, name==input$cur) angle=list.select(curve, angle) %>% unlist
    numericInput("ang", "Angle:", angle, min = 0, max = 360)
  })
  
  data <- eventReactive(input$go, { curve=list.filter(curves, name==input$cur) axiom=list.select(curve, axiom) %>% unlist
    rules=list.select(curve, rules)[[1]]$rules
    alfa0=list.select(curve, alfa0) %>% unlist
    
    for (i in 1:input$ite) axiom=gsubfn(".", rules, axiom)
    actions=str_extract_all(axiom, "\\d*\\+|\\d*\\-|F|L|R|\\[|\\]|\\|") %>% unlist
    
    points=data.frame(x=0, y=0, alfa=alfa0)
    for (i in 1:length(actions)) 
    {
      if (actions[i]=="F"|actions[i]=="L"|actions[i]=="R")
      {
        x=points[nrow(points), "x"]+cos(points[nrow(points), "alfa"]*(pi/180))
        y=points[nrow(points), "y"]+sin(points[nrow(points), "alfa"]*(pi/180))
        alfa=points[nrow(points), "alfa"]
        points %>% rbind(data.frame(x=x, y=y, alfa=alfa)) -> points
      }
      else{
        alfa=points[nrow(points), "alfa"]
        points[nrow(points), "alfa"]=eval(parse(text=paste0("alfa",actions[i], input$ang)))
      }
    }
    return(points)
  })
  
  output$curve <- renderPlot({    
    ggplot(data(), aes(x, y)) + 
      geom_path(color=input$lic) + 
      coord_fixed(ratio = 1) +
      theme(legend.position="none",
            panel.background = element_rect(fill=input$bac),
            panel.grid=element_blank(),
            axis.ticks=element_blank(),
            axis.title=element_blank(),
            axis.text=element_blank())
  })
    
})

Sunflowers for COLOURlovers

Andar, lo que es andar, anduve encima siempre de las nubes (Del tiempo perdido, Robe)

If you give importance to colours, maybe you know already COLOURlovers. As can be read in their website, COLOURlovers is a creative community where people from around the world create and share colors, palettes and patterns, discuss the latest trends and explore colorful articles… All in the spirit of love.

There is a R package called colourlovers which provides access to the COLOURlovers API. It makes very easy to choose nice colours for your graphics. I used clpalettes function to search for the top palettes of the website. Their names are pretty suggestive as well: Giant Goldfish, Thought Provoking, Adrift in Dreams, let them eat cake … Inspired by this post I have done a Shiny app to create colored flowers using that palettes. Seeds are arranged according to the golden angle. One example:

Some others:

You can play with the app here.

If you want to do your own sunflowers, here you have the code. This is the ui.R file:

library(colourlovers)
library(rlist)
top=clpalettes('top')
sapply(1:length(top), function(x) list.extract(top, x)$title)-&gt;titles

fluidPage(
  titlePanel("Sunflowers for COLOURlovers"),
  fluidRow(
    column(3,
           wellPanel(
             selectInput("pal", label = "Palette:", choices = titles),
             sliderInput("nob", label = "Number of points:", min = 200, max = 500, value = 400, step = 50)
           )
    ),
    mainPanel(
      plotOutput("Flower")
    )
  )
  )

And this is the server.R one:

library(shiny)
library(ggplot2)
library(colourlovers)
library(rlist)
library(dplyr)

top=clpalettes('top')
sapply(1:length(top), function(x) list.extract(top, x)$title)->titles

CreatePlot = function (ang=pi*(3-sqrt(5)), nob=150, siz=15, sha=21, pal="LoversInJapan") {
  
  list.extract(top, which(titles==pal))$colors %>% 
    unlist %>% 
    as.vector() %>% 
    paste0("#", .) -> all_colors
  
  colors=data.frame(hex=all_colors, darkness=colSums(col2rgb(all_colors)))
  colors %>% arrange(-darkness)->colors
  
  background=colors[1,"hex"] %>% as.character

  colors %>% filter(hex!=background) %>% .[,1] %>% as.vector()->colors

  ggplot(data.frame(r=sqrt(1:nob), t=(1:nob)*ang*pi/180), aes(x=r*cos(t), y=r*sin(t)))+
    geom_point(colour=sample(colors, nob, replace=TRUE, prob=exp(1:length(colors))), aes(size=(nob-r)), shape=16)+
    scale_x_continuous(expand=c(0,0), limits=c(-sqrt(nob)*1.4, sqrt(nob)*1.4))+
    scale_y_continuous(expand=c(0,0), limits=c(-sqrt(nob)*1.4, sqrt(nob)*1.4))+
    theme(legend.position="none",
          panel.background = element_rect(fill=background),
          panel.grid=element_blank(),
          axis.ticks=element_blank(),
          axis.title=element_blank(),
          axis.text=element_blank())}

function(input, output) {
 output$Flower=renderPlot({
    CreatePlot(ang=180*(3-sqrt(5)), nob=input$nob, siz=input$siz, sha=as.numeric(input$sha), pal=input$pal)
  }, height = 550, width = 550 )}

Frankenstein

Remember me, remember me, but ah! forget my fate (Dido’s Lament, Henry Purcell)

A Voronoi diagram divides a plane based on a set of original points. Each polygon, or Voronoi cell, contains an original point and all that are closer to that point than any other.

This is a nice example of a Voronoi tesselation. You can find good explanations of Voronoi diagrams and Delaunay triangulations here (in English) or here (in Spanish).

A grayscale image is simply a matrix where darkness of pixel located in coordinates (i, j) is represented by the value of its corresponding element of the matrix: a grayscale image is a dataset. This is a Voronoi diagraman of Frankenstein:

To do it I followed the next steps:

  1. Read this image
  2. Convert it to gray scale
  3. Turn it into a pure black and white image
  4. Obtain a random sample of black pixels (previous image corresponds to a sample of 6.000 points)
  5. Computes the Voronoi tesselation

Steps 1 to 3 were done with imager, a very appealing package to proccess and analice images. Step 5 was done with deldir, also a convenient package which computes Delaunay triangulation and the Dirichlet or Voronoi tessellations.

The next grid shows tesselations for sample size from 500 to 12.000 points and step equal to 500:

I gathered all previous images in this gif created with magick, another amazing package of R I discovered recently:

This is the code:

library(imager)
library(dplyr)
library(deldir)
library(ggplot2)
library(scales)

# Download the image
file="http://ereaderbackgrounds.com/movies/bw/Frankenstein.jpg"
download.file(file, destfile = "frankenstein.jpg", mode = 'wb')

# Read and convert to grayscale
load.image("frankenstein.jpg") %>% grayscale() -> x

# This is just to define frame limits
x %>% 
  as.data.frame() %>% 
  group_by() %>% 
  summarize(xmin=min(x), xmax=max(x), ymin=min(y), ymax=max(y)) %>% 
  as.vector()->rw

# Filter image to convert it to bw
x %>%
  threshold("45%") %>% 
  as.cimg() %>% 
  as.data.frame() -> df

# Function to compute and plot Voronoi tesselation depending on sample size
doPlot = function(n)
{
  #Voronoi tesselation
  df %>% 
  sample_n(n, weight=(1-value)) %>% 
  select(x,y) %>% 
  deldir(rw=rw, sort=TRUE) %>% 
  .$dirsgs -> data

  # This is just to add some alpha to lines depending on its longitude
  data %>% 
    mutate(long=sqrt((x1-x2)^2+(y1-y2)^2),
         alpha=findInterval(long, quantile(long, probs = seq(0, 1, length.out = 20)))/21)-> data

  # A little bit of ggplot to plot results
  data %>% 
    ggplot(aes(alpha=(1-alpha))) +
    geom_segment(aes(x = x1, y = y1, xend = x2, yend = y2), color="black", lwd=1) +
    scale_x_continuous(expand=c(0,0))+
    scale_y_continuous(expand=c(0,0), trans=reverse_trans())+
    theme(legend.position  = "none",
            panel.background = element_rect(fill="white"),
            axis.ticks       = element_blank(),
            panel.grid       = element_blank(),
            axis.title       = element_blank(),
            axis.text        = element_blank())->plot

return(plot)
}

# I call the previous function and store resulting plot in jpeg format
i=5000
name=paste0("frankie",i,".jpeg")
jpeg(name, width = 600, height = 800, units = "px", quality = 100)
doPlot(i)
dev.off()

# Once all images are stored I can create gif
library(magick)
frames=c()
images=list.files(pattern="jpeg")

for (i in length(images):1)
{
  x=image_read(images[i])
  x=image_scale(x, "300")
  c(x, frames) -> frames
}
animation=image_animate(frames, fps = 2)
image_write(animation, "Frankenstein.gif")

The Ex Libris Generator

Go ahead stomp your feet on the floorboards
Clap your hands if that’s really what you came here for
(Heaven, The Milk Carton Kids)

Inspired by curves created by the harmonograph, I have done a Shiny app to generate random images that you can personalize and use as an Exlibris.  You can try the App here. For me, an exlibris (also known as bookplates) can be a nice, original and useful present for book-lovers. This is an example:
exlibris8
More examples:

I always put the code at the end of my posts. Since I always have doubts about how many people are interested in what I do, today will be different. I will share the code with those who ask it to me in any of the following ways:

  • Sending me a direct message on Twitter
  • Droping me an email

Cheers!

The Somnambulist and Pi

How wary we are of something warm and unborn. Something calmly by zero will divide (Unbegotten, The Somnambulist)

Some time ago, I assumed the mission to draw a plot for the cover of the new album of The Somnambulist, a music band from Berlin. They wanted a circlization of Pi, which is a graphic where numbers are represented in a circular layout. The idea is connecting each digit of Pi to its successive digit with links to the position of the numerically corresponding external sectors. I used a color palette composed by 10 nuances of the visible spectrum as a tribute for Planck, as Marco (the vocalist) requested me. After a number of attempts:

somnambulist

The album is named Unbegotten, a german word which means archaic. As Marco told me, in theology it also means kind of eternal because of being never born and so never dying. I like how π is integrated into the title to substitute the string “tt” in the middle. Pi is also eternal so the association is genuine.

The music of The Somnambulist is intense, dark and powerful and is waiting for you here to listen it. My favorite song is the one that gives name to the album.

If you want to know more about circlizong numbers, you can visit this post, where you also can see the code I used as starting point to do this plot.

Chaotic Galaxies

Tell me, which side of the earth does this nose come from? Ha! (ALF)

Reading about strange attractors I came across with this book, where I discovered a way to generate two dimensional chaotic maps. The generic equation is pretty simple:

x_{n+1}= a_{1}+a_{2}x_{n}+a_{3}x_{n}^{2}+a_{4}x_{n}y_{n}+a_{5}y_{n}+a_{6}y_{n}^{2}
y_{n+1}= a_{7}+a_{8}x_{n}+a_{9}x_{n}^{2}+a_{10}x_{n}y_{n}+a_{11}y_{n}+a_{12}y_{n}^{2}

I used it to generate these chaotic galaxies:

Changing the vector of parameters you can obtain other galaxies. Do you want to try?

library(ggplot2)
library(dplyr)
#Generic function
attractor = function(x, y, z)
{
  c(z[1]+z[2]*x+z[3]*x^2+ z[4]*x*y+ z[5]*y+ z[6]*y^2, 
    z[7]+z[8]*x+z[9]*x^2+z[10]*x*y+z[11]*y+z[12]*y^2)
}
#Function to iterate the generic function over the initial point c(0,0)
galaxy= function(iter, z)
{
  df=data.frame(x=0,y=0)
  for (i in 2:iter) df[i,]=attractor(df[i-1, 1], df[i-1, 2], z)
  df %>% rbind(data.frame(x=runif(iter/10, min(df$x), max(df$x)), 
                          y=runif(iter/10, min(df$y), max(df$y))))-> df
  return(df)
}
opt=theme(legend.position="none",
          panel.background = element_rect(fill="#00000c"),
          plot.background = element_rect(fill="#00000c"),
          panel.grid=element_blank(),
          axis.ticks=element_blank(),
          axis.title=element_blank(),
          axis.text=element_blank(),
          plot.margin=unit(c(-0.1,-0.1,-0.1,-0.1), "cm"))
#First galaxy
z1=c(1.0, -0.1, -0.2,  1.0,  0.3,  0.6,  0.0,  0.2, -0.6, -0.4, -0.6,  0.6)
galaxy1=galaxy(iter=2400, z=z1) %>% ggplot(aes(x,y))+
  geom_point(shape= 8, size=jitter(12, factor=4), color="#ffff99", alpha=jitter(.05, factor=2))+
  geom_point(shape=16, size= jitter(4, factor=2), color="#ffff99", alpha=jitter(.05, factor=2))+
  geom_point(shape=46, size= 0, color="#ffff00")+opt
#Second galaxy
z2=c(-1.1, -1.0,  0.4, -1.2, -0.7,  0.0, -0.7,  0.9,  0.3,  1.1, -0.2,  0.4)
galaxy2=galaxy(iter=2400, z=z2) %>% ggplot(aes(x,y))+
  geom_point(shape= 8, size=jitter(12, factor=4), color="#ffff99", alpha=jitter(.05, factor=2))+
  geom_point(shape=16, size= jitter(4, factor=2), color="#ffff99", alpha=jitter(.05, factor=2))+
  geom_point(shape=46, size= 0, color="#ffff00")+opt
#Third galaxy
z3=c(-0.3,  0.7,  0.7,  0.6,  0.0, -1.1,  0.2, -0.6, -0.1, -0.1,  0.4, -0.7)
galaxy3=galaxy(iter=2400, z=z3) %>% ggplot(aes(x,y))+
  geom_point(shape= 8, size=jitter(12, factor=4), color="#ffff99", alpha=jitter(.05, factor=2))+
  geom_point(shape=16, size= jitter(4, factor=2), color="#ffff99", alpha=jitter(.05, factor=2))+
  geom_point(shape=46, size= 0, color="#ffff00")+opt
#Fourth galaxy
z4=c(-1.2, -0.6, -0.5,  0.1, -0.7,  0.2, -0.9,  0.9,  0.1, -0.3, -0.9,  0.3)
galaxy4=galaxy(iter=2400, z=z4) %>% ggplot(aes(x,y))+
  geom_point(shape= 8, size=jitter(12, factor=4), color="#ffff99", alpha=jitter(.05, factor=2))+
  geom_point(shape=16, size= jitter(4, factor=2), color="#ffff99", alpha=jitter(.05, factor=2))+
  geom_point(shape=46, size= 0, color="#ffff00")+opt

Gummy Worms

Just keep swimming (Dory in Finding Nemo)

Inspired by this post, I decided to create gummy worms like this:

worm
Or these:

When I was young I used to eat them.

Do you want to try? This is the code:

library(rgl)
library(RColorBrewer)
t=seq(1, 6, by=.04)
f = function(a, b, c, d, e, f, t) exp(-a*t)*sin(t*b+c)+exp(-d*t)*sin(t*e+f)
v1=runif(6,0,1e-02)
v2=runif(6, 2, 3)
v3=runif(6,-pi/2,pi/2)
open3d()
spheres3d(x=f(v1[1], v2[1], v3[1], v1[4], v2[4], v3[4], t),
          y=f(v1[2], v2[2], v3[2], v1[5], v2[5], v3[5], t),
          z=f(v1[3], v2[3], v3[3], v1[6], v2[6], v3[6], t),
          radius=.3, color=sample(brewer.pal(8, "Dark2"),1))

Playing With Julia (Set)

Viento, me pongo en movimiento y hago crecer las olas del mar que tienes dentro (Tercer Movimiento: Lo de Dentro, Extremoduro)

I really enjoy drawing complex numbers: it is a huge source of entertainment for me. In this experiment I play with the Julia Set, another beautiful fractal like this one. This is what I have done:

  • Choosing the function f(z)=exp(z3)-0.621
  • Generating a grid of complex numbers with both real and imaginary parts in [-2, 2]
  • Iterating f(z) over the grid a number of times so zn+1 = f(zn)
  • Drawing the resulting grid as I did here
  • Gathering all plots into a GIF with ImageMagick as I did in my previous post: each frame corresponds to a different number of iterations

This is the result:

julia

I love how easy is doing difficult things in R. You can play with the code changing f(z) as well as color palettes. Be ready to get surprised:

library(ggplot2)
library(dplyr)
library(RColorBrewer)
setwd("YOUR WORKING DIRECTORY HERE")
dir.create("output")
setwd("output")
f = function(z,c) exp(z^3)+c
# Grid of complex
z0 <- outer(seq(-2, 2, length.out = 1200),1i*seq(-2, 2, length.out = 1200),'+') %>% c()
opt <-  theme(legend.position="none",
              panel.background = element_rect(fill="white"),
              plot.margin=grid::unit(c(1,1,0,0), "mm"),
              panel.grid=element_blank(),
              axis.ticks=element_blank(),
              axis.title=element_blank(),
              axis.text=element_blank())
for (i in 1:35)
{
  z=z0
  # i iterations of f(z)
  for (k in 1:i) z <- f(z, c=-0.621) df=data.frame(x=Re(z0), y=Im(z0), z=as.vector(exp(-Mod(z)))) %>% na.omit() 
  p=ggplot(df, aes(x=x, y=y, color=z)) + 
    geom_tile() + 
    scale_x_continuous(expand=c(0,0))+
    scale_y_continuous(expand=c(0,0))+
    scale_colour_gradientn(colours=brewer.pal(8, "Paired")) + opt
  ggsave(plot=p, file=paste0("plot", stringr::str_pad(i, 4, pad = "0"),".png"), width = 1.2, height = 1.2)
}
# Place the exact path where ImageMagick is installed
system('"C:\\Program Files\\ImageMagick-6.9.3-Q16\\convert.exe" -delay 20 -loop 0 *.png julia.gif')
# cleaning up
file.remove(list.files(pattern=".png"))