# Neighborhoods: Experimenting with Cyclic Cellular Automata

On candy stripe legs the Spiderman comes, softly through the shadow of the evening sun (Lullaby, The Cure)

Cellular automata are an inmense source of artistical images. Today, I experimented with Cyclic automata, which are ruled with these simple rules:

• Create a grid of cells.
• Give a state to each cell randomly; states a numbers between 0 and M-1 (you choose the value of M previously).
• For each cell, count how many of its neighbouring cells have their state value exactly 1 unit greater than the cell’s state.
• If the resulting number is greater than a certain thresold (that you also choose previously) increment the state of the cell by 1; if cell state reaches value of M, then you have to put 0 (in other words, you have to add 1 modulus M).
• Repeat two previous steps a number of times.

A key concept in this algorithm is defining which is the neighborhood of a cell. There are two of them quite famous: Moore and Von Neumann neighborhoods, but you can define your own ones. Once you decide to stop iterating, you can give color to each cell according its final state, and you will obtain images like this one:

If you have a look to the code, you will see the next parameters:

• `neighborhood`: the pattern of neighbouring cells
• `range`: the depth of neighborhood
• `states`: maximum number of states allowed (the M of the algorithm)
• `thresold`
• `iter`: number of iterations
• `width` and `height` of the grid

Apart from Moore and Von Neumann, I implemented some other neighborhoods. This chart shows some of them. In columns you can find the folowing: `M` (Moore), `N` (Von Neumann), `Mr` (Moore remote), `Nr` (Von Neumann remote), `Cr` (Cross), `S1` (S-Shape #1), `Bl` (Blade), `C2` (Corners #2) and `TM` (Tick Mark). In rows, you can find different ranges for each neighborhood, from 1 to 4:

You will find more neighborhood in the code of the experiment, which is here. There are infinite combinations of the previous parameters, and each one results in a different image:

I used again COLOULovers palettes As always, I encourage you to experiment with the code, create your own neighborhoods, and see how they work. Happy New Year 2021!

# Abstractions

Spinning on that dizzy edge (Just Like Heaven, The Cure)

This post talks about a generative system called Physarum model, which simulates the evolution of a colony of extremely simple organisms that, under certain environmental conditions, result into complex behaviors. Apart from the scientific interest of the topic, this model produce impressive images like this one, that I call The Death of a Red Dwarf:

You can find a clear explanation of how a physarum model works in this post, by Sage Jenson. A much deeper explanation can be found in this paper by Jeff Jones, from the University of the West of England. Briefly, a physarum model evolves a set of particles (agents), making them move over a surface. Agents turn towards locations with higher concentrations of a pheromone trail. Once they move, they make a deposition of pheromone as well. These are the steps of a single iteration of the model:

• Sensor stage: Each agent looks to three positions of the trail map (left, front and right) according a certain sensor angle
• Motor stage: then it moves to the place with the higher concentration with some rules to deal with ties
• Deposition stage: once in the new location, the agent deposites a certain amount of pheromone.
• Diffuse stage: the pheromones diffusing over the surface to blur the trail array.
• Decay stage: this make to decay the concentration of pheromone on the surface.

Sage Jenson explains the process with this illustrative diagram:

My implementation is a bit different from Jones’ one. The main difference is that I do not apply the diffuse stage after deposition: I prefer a high defined picture instead blurriyng it. I also play with the initial arrangement of agents (location and heading angle) as well with the initial configuration of environment. For example, In The Death of a Red Dwarf, agents start from a circle and the environment is initialized in a dense disc. You can find the details in the code. There you will see that the system is governed by the following parameters:

• Front and left sensor angles from forward position
• Agent rotation angle
• Sensor offset distance
• Step size (how far agent moves per step)
• Chemoattractant deposition per step
• Trail-map chemoattractant diffusion decay factor

In adition to them (specific of the physarum model), you also can change others like colors, noise of angles (parameter `amount` of `jitter` function), number of agents and iterations as well as the initial arrangement of the environment and the location of agents. I invite you to do it. You will discover many abstractions: butterfly wings, planets, nets, explosions, supernovas … I have spent may hours playing with it. Some examples:

You can find the code here. Please, let me know if you do something interesting with it. Share your artworks with me in Twitter or drop me an email (you can find my address here).

# Watercolors

MoÃ§a do corpo dourado
Do Sol de Ipanema
O seu balanÃ§ado
Ã‰ mais que um poema
(Garota de Ipanema, JoÃ£o Gilberto)

Sometimes I think about the reasons why I spend so many time doing experiments and writing my discoveries in a blog. Even although the main reason to start this blog was some kind of vanity, today I have pretty clear why I still keep writing it: to keep my mind tuned. I really enjoy looking for ideas, learning new algorithms, figuring out the way to translate them into code and trying to discover new territories going a step further. I cannot imagine my life without coding. Many good times in the last years have been in front of my laptop listening music and drinking a beer. In these strange times, confined at house, coding has became in something more important. It keeps me ahead from the sad news and moves my mind to places where everything is quiet, friendly and perfect. Blogging is my therapy, my mindfulness.

This post is inspired in this post from Softology, an amazing blog I recommend you to read. In it, you can find a description of the stepping stone cellular automaton as well as a appealing collection of images generated using this technique. I modified the original algorithm described in the post to create images like these, which remind me a watercolor painting:

I begin with a 400 x 400 null matrix. After that, I choose a number of random pixels that will act as centers of circles. Around them I substitute the initial zeros by numbers drawned from a normal distribution which mean depends on the distance of pixels to the center. The next step is to apply the stepping stone algorithm. For each pixel, I substitute its value by a weighted average of itself and the value of some of its neighbors, choosen randomly. I always mix values of the pixels. The original algorithm, as described in the Softology’s blog, performs these mixings randomly. Another difference is that I mix values intead interchanging them, as the original algorithm does. Once I repeat this process a number of times, I pick a nice palette from COLOURLovers and turn values of pixels into colors with `ggplot`:

The code is here. Let me know if you do something interesting with it. Turning numbers into bright colors: I cannot imagine a better way to spend some hours in these shadowy times.

# Coloring Sudokus

Someday you will find me
caught beneath the landslide
(Champagne Supernova, Oasis)

I recently read a book calledÂ Snowflake Seashell Star: Colouring Adventures in Numberland by Alex Bellos and Edmund Harris which is full of mathematical patterns to be coloured. All images are truly appealing and cause attraction to anyone who look at them, independently of their age, gender, education or political orientation. This book demonstrates how maths are an astonishing way to reach beauty.

One of my favourite patterns are tridokus, a sophisticated colored version of sudokus. Coloring a sudoku is simple: once that is solved it is enough to assign a color to each number (from 1 to 9).Â  If you superimpose three colored sudokus with no cells at the same position sharing the same color, and using again nine colors, the resulting image is a tridoku:

There is something attractive in a tridoku due to the balance of colors but also they seem a quite messy: they are a charmingly unbalanced.Â  I wrote a script to generalize the concept to n-dokus. The idea is the same: superimpose n sudokus without cells sharing color and position (I call them disjoint sudokus) using just nine different colors. I did’n’t prove it, but I think the maximum amount of sudokus can be overimposed with these constrains is 9. This is a complete series from 1-doku to 9-doku (click on any image to enlarge):

I am a big fan of `colourlovers` package. These tridokus are colored with some of my favourite palettes from there:

Just two technical things to highlight:

• There is a package called sudoku that generates sudokus (of course!). I use it to obtain the first solved sudoku which forms the base.
• Subsequent sudokus are obtained from this one doing two operations: interchanging groups of columns first (there are three groups: columns 1 to 3, 4 to 6 and 7 to 9) and interchanging columns within each group then.

You can find the code here: do you own colored n-dokus!

# Mandalas Colored

ApriÃ©tame bien la mano, que un lucero se me escapa entre los dedos (Coda Flamenca, Extremoduro)

I have the privilege of being teacher at ESTALMAT, a project run by Spanish Royal Academy of Sciences that tries to detect, guide and stimulate in a continuous way, along two courses, the exceptional mathematical talent of students of 12-13 years old. Some weeks ago I gave a class there about the importance of programming. I tried to convince them that learning R or Python is a good investment that always pays off; It will make them enjoy more of mathematics as well as to see things with their own eyes. The main part of my classÂ was a workshop about Voronoi tesselations in R. We started drawing points on a circle and we finished drawing mandalas like these ones. You can find the details of the workshop here (in Spanish). It was a wonderful experience to see the faces of the students while generating their own mandalas.

In that case all mandalas were empty, ready to be printed and coloured as my 7 years old daughter does. In this experiment I colour them. These are the changes I have done to myÂ  previous code:

• Remove external segments whichÂ intersects the boundary of the enclosing
rectangle
• Convert the tesselation into a list of polygons with `tile.list` function
• Use `colourlovers` package to fill the polygons with beautiful colour palettes

This is an example of the result:

Changing three simple parameters (`iter`, `points` and `radius`) you can obtain completely different images (clicking on any image you can see its full size version):

You can find details of these parameters in my previous post. I cannot resist to place more examples:

You can find the code here. Enjoy.

# 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 )}
```