# Phyllotaxis By Shiny

Antonio, you don’t know what empathy is! (Cecilia, my beautiful wife)

Spirals are nice. In the wake of my previous post I have done a Shiny app to explore patterns generated by changing angle, shape and number of points of Fermat’s spiral equation. You can obtain an almost infinite number of images. This is just an example:

I like thinking in imaginary flowers. This is why I called this experiment Phyllotaxis.
More examples:

Just one comment about code: I did the Shiny in just one R file as this guy suggested me some time ago because of this post.

This is the code. Do your own imaginary flowers:

```library(shiny)
library(ggplot2)
CreatePlot = function (ang=pi*(3-sqrt(5)), nob=150, siz=15, alp=0.8, sha=16, col="black", bac="white") {
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=col, alpha=alp, size=siz, shape=sha)+
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=bac),
panel.grid=element_blank(),
axis.ticks=element_blank(),
axis.title=element_blank(),
axis.text=element_blank())}
shinyApp(
ui = fluidPage(
titlePanel("Phyllotaxis by Shiny"),
fluidRow(
column(3,
wellPanel(
selectInput("col", label = "Colour of points:", choices = colors(), selected = "black"),
selectInput("bac", label = "Background colour:", choices = colors(), selected = "white"),
selectInput("sha", label = "Shape of points:",
choices = list("Empty squares" = 0, "Empty circles" = 1, "Empty triangles"=2,
"Crosses" = 3, "Blades"=4, "Empty diamonds"=5,
"Asterisks"=8, "Crosed diamonds"=9, "Crossed circles"=10,
"Filled squares" = 15, "Filled circles" = 16, "Filled triangles"=17,
"Filled diamonds"=18), selected = 16),
sliderInput("ang", label = "Angle (degrees):", min = 0, max = 360, value = 180*(3-sqrt(5)), step = .05),
sliderInput("nob", label = "Number of points:", min = 1, max = 1500, value = 60, step = 1),
sliderInput("siz", label = "Size of points:", min = 1, max = 60, value = 10, step = 1),
sliderInput("alp", label = "Transparency:", min = 0, max = 1, value = .5, step = .01)
)
),
mainPanel(
plotOutput("Phyllotaxis")
)
)
),
server = function(input, output) {
output\$Phyllotaxis=renderPlot({
CreatePlot(ang=input\$ang, nob=input\$nob, siz=input\$siz, alp=input\$alp, sha=as.numeric(input\$sha), col=input\$col, bac=input\$bac)
}, height = 650, width = 650 )}
)
```

# Hypnotical Fermat

Se le nota en la voz, por dentro es de colores (Si te vas, Extremoduro)

This is a gif generated with 25 plots of the Fermat’s spiral, a parabolic curve generated through the next expression:

$r^{^2}= a^{2}\Theta$

where $r$ is the radius, $\Theta$ is the polar angle and $a$ is simply a compress constant.

Fermat showed this nice spiral in 1636 in a manuscript called Ad locos planos et solidos Isagoge (I love the title). Instead using paths, I use a polygon geometry to obtain bullseye style plots:

Playing with this spiral is quite addictive. Try to change colors, rotate, change geometry … You can easily discover cool images like this without any effort:

Enjoy!

```library(ggplot2)
library(magrittr)
setwd("YOUR-WORKING-DIRECTORY-HERE")
opt=theme(legend.position="none",
panel.background = element_rect(fill="white"),
panel.grid=element_blank(),
axis.ticks=element_blank(),
axis.title=element_blank(),
axis.text=element_blank())
for (n in 1:25){
t=seq(from=0, to=n*pi, length.out=500*n)
data.frame(x= t^(1/2)*cos(t), y= t^(1/2)*sin(t)) %>% rbind(-.) -> df
p=ggplot(df, aes(x, y))+geom_polygon()+
scale_x_continuous(expand=c(0,0), limits=c(-9, 9))+
scale_y_continuous(expand=c(0,0), limits=c(-9, 9))+opt
ggsave(filename=paste0("Fermat",sprintf("%03d", n),".jpg"), plot=p, width=3, height=3)}
```