There's an igloo-shaped auditorium I've seen which has an extraordinary property: when you're sitting within it, the ceiling appears to float more than 100 feet above your head — but viewed from the outside the whole hemispherical structure seems less than 30 feet tall. Somehow it's bigger on the inside than on the outside!

Maybe that illusion is related to the lack of scale when looking up at a remote surface, as opposed to the situation outdoors when one can compare with trees and other nearby buildings? Perhaps it's loosely connected to the lunar illusion wherein the rising Moon seems huge when it's close to the horizon? I don't know. Maybe there's just some magic going on, as in the old British sf TV series Dr. Who, which featured a time machine ("The Tardis") that somehow had more room inside than it did from the exterior.

And that, in turn, recently reminded me of the (in)famous Banach-Tarski Paradox in mathematics, which states that it's possible to cut up a solid ball into five pieces and then reassemble those pieces into two complete copies of the original ball — or equivalently, cut a marble into pieces that can be reassembled into a ball the side of the planet Earth. Admittedly, as math genius Arthur Rubin tried to explain to me at Caltech many years ago, some of those pieces have to be pretty weird: they're dusty, fractal, perverse things which require an infinitely sharp knife to carve. The name of that knife is the Axiom of Choice.

And reading about that recently in http://www.kuro5hin.org/story/2003/5/23/134430/275 led me to a cute meta-joke-riddle :

 Q: What's an anagram of "Banach-Tarski" ?
 A: "Banach-Tarski Banach-Tarski" !

(see also HearingShapes (28 Mar 2000), ArsMagna (27 Sep 2002), FractalFeynman (30 Jan 2003), PyramidBuilding (21 Feb 2004), ... )

TopicScience - TopicPersonalHistory - TopicHumor - 2004-05-11

(correlates: CircleSquaring, Worst and Bad, SelfStorage, ...)