Exploding dots machines arise from patterns of dot replacement, which should, in turn, determine a “base”.
For example the replacement pattern gives rise to a base satisfying , whereas the replacement pattern gives rise to a rational base satisfying .
The replacement pattern gives rise to an irrational base satisfying
So for a machine we see that the base is 2 because :
On the other hand a machine also has base 2 because :
So these two machines are representing positive integers as finite sums of “digits” times powers of 2.
What’s different is what counts as a digit.
In the machine the digits are 0 and 1, while in the machine the (allowable) digits are 0, 1, 2, and 3.
This gives rise to correct, but distinct representations, but both “base 2”, (!!) coming from the ways in which the two different machines operate:
|Base 10||Base 2/1||Base 4/2|
The connections between the pattern that defines an exploding dots machine, the digits that are allowable, and the algebraic number that results from the pattern, is not entirely clear.
And what happens if a pattern gives a root of a cubic as the base? Do any of the – possibly complex – roots also work equally well? And what are the allowable digits then?
For that matter, what if take the negative root of as the base for an exploding dots machine with pattern ?