# Fractal Art

R day 1:

#### Fractal Background

I am reading Godel, Escher, Bach, and recursion is one of the main topics the book discussed. Recursion is a pattern and never ends, or goes to infinite. ( EX. Bach’s Fugue, Escher’s stairs, Godel incompleteness theorem).

I remember there was a Fractal Artbook in my home, i used to flip through when i didnt want to study. I think Fractal Art is one of the most interesting recursion topics! geralt / Pixabay

Fractal Art is a never-ending self -similar pattern across different scales, like the snowflakes (Koch Snowflake).

Today we will be using GGPLOT2 to draw a fractal!

#### Fractal Basic

we use complex plain to express fractals.

Z=(a+b*i), i= sqrt(-1)

General Recursive formula algorithm

the initial value is C, then we repeating the calculations!

Z_{N}=Z_{n-1}^{2} +C, C=a+b*i

####  ```library(ggplot2)

max_iter=25
cl=colours()
step=seq(-2,0.8,by=0.005)
points=array(0,dim=c(length(step)^2,3))
t=0

for(a in step)
{
for(b in step+0.6)
{
x=0;y=0;n=0;dist=0
while(n<max_iter & dist<4)
{
n=n+1
newx=a+x^2-y^2
newy=b+2*x*y
dist=newx^2+newy^2
x=newx;y=newy
}

if(dist<4)
{
color=24 # black
}
else
{
color=n*floor(length(cl)/max_iter)
}

t=t+1
points[t,]=c(a,b,color)
}
}

df=as.data.frame(points)

ggplot(data=df, aes(V1, V2, color=cl[V3]))+
geom_point() + theme(legend.position = 'none',
axis.text.x=element_blank(),
axis.text.y=element_blank(),
panel.background=element_blank(),
axis.title.x=element_blank(),
axis.title.y=element_blank(),
panel.grid.major=element_blank(),
panel.grid.minor=element_blank(),
axis.ticks=element_blank()
)

```

#### Summary

Besides fun to look at the fractal artwork, it has practical applications. Such as fractal dimensions provide a solution to measuring the coastline (coastline Paradox).

Reference:

http://www.r-chart.com/2010/08/fractals-in-r.html