Googology Wiki
Googology Wiki

Hexation or sextation refers to the 6th hyperoperation if addition is to be regarded as the first. It is equal to the binary function \(\uparrow\uparrow\uparrow\uparrow \) in Knuth's up-arrow notation.[1]

"a hexated to b" can be written in array notation as {a,b,4}, in chained arrow notation as \(a\rightarrow b\rightarrow 4\) and in Hyper-E notation as E[a]1#1#1#b. Sunir Shah uses the notation a**b to indicate this.[2] Jonathan Bowers calls it 'a to the b'th layer'.[3] Sbiis Saibian proposes the notation a→b in analogy to ba for tetration, but he usually uses up-arrows.[1]

Hexational growth rate is approximately \(f_5(n)\) in the fast-growing hierarchy.

Tim Urban calls hexation a "power tower feeding frenzy psycho festival".[4]


Here are some examples of hexational-level calculations.




2↑↑↑↑3 =2↑↑65,536 (a power tower of 2's 65,536 terms high)

3↑↑↑↑2 = 3↑↑↑3 = 3↑↑7,625,597,484,987 = tritri (a power tower of 3's 7,625,597,484,987 terms high)

3↑↑↑↑3 = 3↑↑↑tritri = g1 = grahal

4↑↑↑↑2 = 4↑↑↑4 = \(^{^{^{4}4}4}\! 4\) where \({^b} a\) denotes tetration

3 ↑↑↑↑ 8 = 3↑↑↑3↑↑↑3↑↑↑3↑↑↑3↑↑↑3↑↑↑3↑↑↑3

Appearance in other googolisms

Hexation is used in \(G_1\), the first value of the Graham function, which is equal to 3 hexated to 3. This number is also known as grahal.


See also