# Infinite Staircase

September 24, 2010I’ve been thinking about infinity a lot lately, and how it relates to something called the Droste Effect. It was named for the Droste cocoa box. Here’s the idea: the woman on the package is carrying a Droste box which itself contains a woman holding a tray and on the tray is a box of Droste cocoa which bears the image of a woman holding a tray with a box of cocoa, and so on forever.

So I might define the Droste Effect as: the infinite descent (or ascent) that occurs when an image contains a smaller version of the entire image inside itself.

It’s not hard to see how this connects to the infinite (as this video shows, for example). In fact, one of Cantor’s definitions for an infinite set was a set that contained a subset that was equal in cardinality. In other words, a set is infinite if a part of it is as big as the whole.

What’s cool to consider, though is that to make a video of a Droste effect on your computer only requires a finite amount of information: a finite number of frames, and then a loop. This is very similar to repeating decimals. If I write 1/7 as a decimal, I can do the long division to get 0.142857142857142857142857142857142857142857142857…

I could have represented this information as the six digits: .142857 along with the information that they repeat forever. I could think of this as 7 pieces of information (hey, and I was doing 1/7! Do you think it will always take n pieces of information to represent 1/n?). This is exactly like a Droste effect, just numerical instead of visual.

So what we have is a relationship between this cycling or circular behavior of repetition, parts containing the whole, and the infinite.

Two fun other places to look. The first is my old favorite children’s book *Arm in Arm*, which I finally was able to track down online (it’s out of print, shockingly). The author, Remy Charlip, was a dancer and costume designer and teacher and choreographer (he worked with John Cage) and did all sorts of fun things, and he also wrote books for children. This particular one he describes as

“A collection of connections, endless tales, reiterations, and echolalia.”

And indeed, it’s full of lovely infinite stories, such as:

“It was a dark and stormy night. We were standing on the deck. The ship was sinking. The captain said to me, “Tell me a story my son.” And so I began, “It was a dark and stormy night. We were standing on the deck. The ship was sinking. The captain said to me, ‘Tell me a story my son.’ And so I began, ‘It was a dark and stormy night. We were standing on the deck…”

What we have here is a narrative Droste Effect! It’s a marvelous book. I remember loving it as a kid.

The second thing I wanted to mention was this very cool video: another infinite loop. And when you’re done with it, consider the question: is it possible that our universe is just such an infinite loop? And in fact, is it reasonable to say that our Earth is at all like the loop in the video, given that if you set out in a straight line you’ll come back to where you started?