Circulation is a hyper-operator denoted as \(a \uparrow^\infty b\), defined as \(\lim_{n \rightarrow \infty} a \uparrow^n b\).[1] In most cases, \(a \uparrow^n b\) must be defined for real a and b for circulation to be of any use. The term was most likely coined by Daniel Geisler, who names this operator without any attribution.

Using only nonnegative integers, there are only four cases where circulation is defined:

  • \(1 \uparrow^\infty n = 1\)
  • \(2 \uparrow^\infty 2 = 4\)
  • \(n \uparrow^\infty 0 = 1\) (provided \(n\neq 0\))
  • \(n \uparrow^\infty 1 = n\)

All of the above cases are caused from degenerate expressions: for example, \(2 \uparrow^a 2 = 4\) for all positive integers a.


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