Golden Rectangles

Exercise Type: Exploration

The following bit of code plots the golden rectangles in golden spiral patterns which is frequently used in art, and is often found in natural phenomena:

# the golden ratio
phi <- (1 + sqrt(5))/2

# starting coordinates
x1 <- 0
x2 <- 1
y1 <- 0
y2 <- 1/phi

# basic plot
plot(0,0, axes=F, xlab="", ylab="", type="n",asp=1, xlim=c(0,1), ylim=c(0,y2))

# base rectange
rect(xleft=x1,xright=x2,ybottom=y1, ytop=y2)

##########################
# Iteration 1
# 1st square
x3 <- x1 + (x2-x1) / phi
rect(xleft=x1, xright=x3, ybottom=y1, ytop=y2)

# 2nd square
y3 <- y1 + (y2-y1) / phi
rect(xleft=x3, xright=x2, ybottom=y1, ytop=y3)

# 3rd square
x4 <- x2 - (x2 - x3) / phi 
rect(xleft=x4, xright=x2, ybottom=y3, ytop=y2)

# 4th square
y4 <- y2 - (y2-y3) / phi 
rect(xleft=x3, xright=x4, ybottom=y4, ytop=y2)

# 5th
x5 <- x3 + (x4-x3) / phi
rect(xleft= x3, xright=x5, ybottom=y3, ytop=y4)

# 6th
y5 <- y3 + (y4-y3) / phi
rect(xleft=x5, xright=x4, ybottom=y3, ytop=y5)

# 7th
x6 <- x4 - (x4-x5) / phi
rect(xleft=x6, xright=x4, ybottom=y5, ytop=y4)

# 8th
y6 <- y4 - (y4-y5) / phi 
rect(xleft=x5, xright=x6, ybottom=y6, ytop=y4)

Instructions

Explore how you can make this plot more beautiful by setting the parameters of the squares (e.g. adding colors, change line widths)! Explore ?rect!

Bonus

This pattern is recursive and plotting code above can actually be iterated relatively easily. Can you make this a proper iteration?