Tritri

Tritri is equal to \(3 \uparrow\uparrow\uparrow 3 = 3 \uparrow\uparrow 7,625,597,484,987 = 3 \uparrow\uparrow\uparrow\uparrow 2\) or 3 pentated to 3 in up-arrow notation. In BEAF it can be written as {3,3,3}, {3,3(1)2}, or {3,2/2}. Using array of operator, it can be written as 3&3. It is the third Ackermann Number.
Jonathan Bowers, who coined the name, has created many other googologisms based on the number 3 (such as ultatri and triakulus). This is because 3 is the smallest positive integer that does not create degenerate arrays like 1 and 2, since \(\{2,2,n\} = 4\) for all \(n\).
The last 10 digits of tritri are ...2,464,195,387.
Googology Wiki user Hyp cos calls this number a trientri, and it's equal to s(3,3,3), s(3,2,4), s(3,3,1,2), s(3,1,1,3), or s(3,1,1,1,2) in strong array notation.Numbers from linear array notation | Steps Toward Infinity!
Tim Urban calls the 7,625,597,484,987-term power tower evaluating to this number the "sun tower", and the number itself "INSANITY" in his Wait Buy Why article "From 1,000,000 to Graham's Number".From 1,000,000 to Graham's Number - Wait Buy Why
BlankEntity calls this number kolmas kenttä tyhjä, and it's the 43rd number of Celtchar of New World.BlankEntity's Googology - Celtchar of New World (Retrieved 31 July, 2022)
 coined coral for this number, and it's equal to \(3A1\) in Greek Letter Notation, as well as \(3 \rightarrow 3 \rightarrow 3\) in Chained arrow notation.
== Computation ==
Tritri can be computed in the following process:
* \(a_1 = 3\)
* \(a_2 = 3^3 = 27\)
* \(a_3 = 3^{3^3} = 7,625,597,484,987\)
* \(a_4 = 3^{3^{3^3}} \approx 1.258014290627 \times 10^{3,638,334,640,024}\)
* \(a_5 = 3^{3^{3^{3^3}}} \approx 10^{6.0 \times 10^{3,638,334,640,023}}\)
* \(a_6 = 3^{3^{3^{3^{3^3}}}} \approx 10^{10^{6.0 \times 10^{3,638,334,640,023}}}\)
* repeat this process
* Tritri is equal to \(a_{a_3} = a_{7,625,597,484,987}\).
Tritri is \(f(f(3)\)) or \((f^2(3))\) when: \(\left\{ \begin{array}{ll} f(n)=3^{f(n-1)} \\ f(1)=3 \end{array} \right.\)
Tritri is \(\underbrace{ 3\uparrow3\uparrow...\uparrow3\uparrow3 }_{\underbrace{ 3\uparrow3\uparrow3 }_{3}}\) using arrow notation.
== Size ==
The number is equal to a power tower of 3's, 7,625,597,484,987 levels high. In contrast, a power tower of only 4 3's is between googol and googolplex, and \(3 \uparrow\uparrow 7\) is larger than the Poincare recurrence time of a Linde-type super-inflationary universe, often touted as the largest number to appear in physics. It is of course much smaller than Graham's number, on the other hand.
Milton Green proved that tritri is much less than .
On the group blog LessWrong, the following question was proposed as a moral thought experiment: "Would you prefer that one person be horribly tortured for fifty years without hope or rest, or that \(3 \uparrow\uparrow\uparrow 3\) people get dust specks in their eyes?"
== Simpifying using the rules of BEAF. ==
This number is excellent for amateur googologists when they are first trying to understand BEAF linear arays.
Simplifying it to {3,327,2}=\(3 \uparrow\uparrow\uparrow 3\): {3,3,3}={3,{3,2,3},2}={3,{3,{3,1,3},2},2}={3,{3,3,2},2}={3,{3,{3,2,2},1},2}={3,{3,{3,2,2}},2}={3,{3,{3,{3,1,2},1}},2}={3,{3,{3,{3,1,2}}},2}={3,{3,{3,3}},2}= {3,{3, 33},2}={3,{3,27},2}={3,327,2}
==Approximations in other notations==
Calculation
\(\begin{array}{rl}
3 \uparrow \uparrow \uparrow 3 &= 3 \uparrow \uparrow 7625597484987 \\
&\approx 10 \uparrow \uparrow 7625597484986.041 \approx 10\uparrow \uparrow (7.6256 \times 10^{12}) \\
&\approx 10 \uparrow \uparrow \uparrow 2.31076 \\
E1.11\#2\#2 &= E1.11\#(E1.11\#2) \\
&\approx 10\uparrow\uparrow(10\uparrow10\uparrow1.11) \\
&\approx 10\uparrow \uparrow (7.6295 \times 10^{12})
\end{array}\)
==Etymology==
The name of this number is based on the word "tri-" (Greek for three), meaning a size-3 linear array of 3's.
== See also ==
*Tritet
*Tripent
*Trisept
*Tridecal
*Grahal
*Ackermann number
*7,625,597,484,987
*grand tridecal
*Arrow notation
== Sources ==
ja:トリトリ
zh:特利特利
Category:Numbers
Category:Numbers by Jonathan Bowers
Category:3 entry linear
Category:Tetration level
Category:Greek-based name
Category:Powers of 3
Category:Pentation