NewTwitter design based on a Golden Spiral

I finally got the new version of Twitter yesterday, and it looks great. And that's no accident: according to the designer, the layout of the new Twitter interface is based on the Golden Spiral:

You can describe the Golden Spiral by laying consecutive squares in a spiral fashion, each square being smaller than the last by a factor of the Golden Ratio, (1+sqrt(5))/2 or approximately 1.618. As a ratio of a length to a width, the Golden Ratio has been known by artists for millenia as an aesthetically pleasing aspect ratio for rectangular features in works of art (such as the dimensions of a painting). It's also known to mathematicians and statisticians as the number to which ratios of successive members of the Fibonacci sequence converge.

In a classic case of using a sledgehammer to crack a walnut, I created a custom iterator in R to represent the Fibonacci sequence (code after the fold) and verified the ratio converges to the Golden Ratio at the R command line:

> fib1 <- iFib(); nextElem(fib1)
[1] 1
> fib2 <- iFib()
> nextElem(fib1)/nextElem(fib2)
[1] 1
> nextElem(fib1)/nextElem(fib2)
[1] 2
> nextElem(fib1)/nextElem(fib2)
[1] 1.5
> nextElem(fib1)/nextElem(fib2)
[1] 1.666667
> nextElem(fib1)/nextElem(fib2)
[1] 1.6
> nextElem(fib1)/nextElem(fib2)
[1] 1.625
> nextElem(fib1)/nextElem(fib2)
[1] 1.615385
> nextElem(fib1)/nextElem(fib2)
[1] 1.619048
> nextElem(fib1)/nextElem(fib2)
[1] 1.617647
> nextElem(fib1)/nextElem(fib2)
[1] 1.618182
> nextElem(fib1)/nextElem(fib2)
[1] 1.617978
> nextElem(fib1)/nextElem(fib2)
[1] 1.618056
> nextElem(fib1)/nextElem(fib2)
[1] 1.618026

See? Golden ratio.

Update: See this followup post on calculating the Fibonacci sequence in R.

Twitter: @stop

require(iterators)
## Generator for Fibonacci sequence
iFib <- function() {
	lastFib <- 0
	nextFib <- 1
	nextEl <- function() {
		newLast <<- nextFib
		nextFib <<- lastFib + nextFib
  lastFib <<- newLast
  lastFib
	}
	it <- list(nextElem = nextEl)
	class(it) <- c('abstractiter','iter')
	it
}

Comments

[mondo print]

Very cool.

[jebyrnes]

Here's an easy 3-liner that I'm sure could simply be wrapped into a function.

fv<-c(0,1)
sapply(3:20, function(i) fv[i]<<-fv[i-1]+fv[i-2])
fv[-1]/fv[-length(fv)

Which, oh, that could be compressed into 2 lines (you have to initialize the vector)

fv<-c()
sapply(1:20, function(i) if(i<3){fv[i] <<- 1}else{fv[i]<<-fv[i-1]+fv[i-2]})[-1]/fv[-length(fv)]

[David Smith]

Via John Myles White, here's an amazingly elegant way of deriving the Golden Mean from its continued fraction representation using R's Reduce function:

> cfrac <- function(x) Reduce(function(u, v) u + 1 / v, x, right = TRUE)

> cfrac(rep(1,10))

[1] 1.618182

cfrac(rep(1,n)) better approximates the Golden Ratio as n gets larger.

[Rick Wicklin]

It turns out that you can understand why the ratio is the golden mean if you cast the sequence in terms of iteration of a certain matrix:
http://blogs.sas.com/iml/index.php?/archives/22-Matrices,-Eigenvalues,-Fibonacci,-and-the-Golden-Ratio.html

[Thomas]

This is very exciting and I like your template as well. Keep it up.