Continuing on from yesterday… Yann’s sparse numbers inspired me to think about how you might picture a base system that is not an integer – say 1.7, which is between one and two, or 9.3, which is between nine and ten. Doug’s comment the other day points to work that shows how to do this for one particular non-integer – the golden ratio φ, which is (1+√5)/2. But what about all the other non-integers?
Here’s a thought: Imagine Snow White and the Seven Dwarves has been reimagined by a crazed mathematician. In this version, each of the dwarves rides in an elevator – one elevator each for Bashful, Doc, Dopey, Grumpy, Happy, Sleepy and Sneezy. Each elevator represents one digit. When Snow White calls out a seven digit number, each dwarf is supposed to press the elevator button corresponding to his digit.
Snow White makes sure her elevator operators have clean hands
But there’s a problem – the dwarves might not be able to reach all the buttons, because some of the buttons might be too high and therefore out of reach:
Suppose the taller dwarves, Doc, Grumpy and Happy, can reach the ninth floor button, but the four shorter dwarves can only reach up to the eighth floor. Also, in a curious plot twist, our crazed mathematician author decides that Snow White can slip the dwarves a magic potion to make them all taller. As the potion continually increases everyone’s height, at some point each dwarf realizes he can reach one more button. By carefully doling out the magical potion, Snow White can set her button pushing friends to start to look like a base system between base to and three, or between base nine and ten, and so forth.
If we replace the dwarves by brain synapses, we can see that as we get more synapses firing, say if the brain’s dopamine level rises, then fewer neurons will be needed to represent the same variety of brain states. In effect, the brain is shifting further from base one and nearer to base two. Of course the scale is different: Snow White had only seven dwarves, whereas our brain has about 100 trillion synapses.
What would we call this way of representing numbers? Bashful binaries? Dopey decimals? Snow White and the Seven Digits? I’m open to suggestions.