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Visualization of Tritri using exponentiation

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.[1][2] 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.[3] 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 \(n > 0\).

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.[4]

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".[5]


Tritri can be computed in the following process:

  • \(a_1 = 3\)
  • \(a_2 = 3^3 = 27\)
  • \(a_3 = 3^{3^3} = 7,625,597,484,987\)
  • \(a_4 = 3^{3^{3^3}} \approx 1.258014290627 \times 10^{3,638,334,640,024}\)
  • etc.
  • 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 111^^^^^111 using Turing machine code.

Tritri is using arrow notation.


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 \(\Sigma(10)\).[6]

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?"[7]

Approximations in other notations

Notation Approximation
BEAF \(\{3,3,3\}\), \(\{3,2,4\}\), \(\{3,3(1)2\}\), \(\{3,2/ 2\}\) (all are exact)
Bird's array notation \(\{3,3,3\}\), \(\{3,2,4\}\), \(\{3,3[2]2\}\) (all are exact)
Hyper-E notation \(\textrm E3638334640023\#7625597484984\)
Chained arrow notation \(3 \rightarrow 3 \rightarrow 3\) or \(3 \rightarrow 2 \rightarrow 4\) (both are exact)
Hyperfactorial array notation \(3!4\) (exact)
Fast-growing hierarchy \(f_4 (3) = f_3 (7,625,597,484,986)\)
Hardy hierarchy \(H_{\omega^{4}}(3)\)
Slow-growing hierarchy \(g_{\zeta_0}(3)\)


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


  1. Bowers, JonathanInfinity Scrapers. Retrieved January 2013.
  2. Bowers, Jonathan. Array Notation up to Three Entries. Retrieved June 2014.
  3. Ackermann Number. Wolfram MathWorld. Retrieved June 2014.
  4. Numbers from linear array notation | Steps Toward Infinity!
  5. From 1,000,000 to Graham's Number - Wait Buy Why
  6. Green, Milton. A lower bound RADO's sigma function for binary turing machines. Retrieved 2013-05-07.
  7. Eliezer, Yudkowsky. Torture vs. Dust SpecksLessWrong. Retrieved 2014-06-07.

Tritri series: tritri · tritet · tripent · trisept · tridecal · grand tridecal
Tetratri series: tetratri · supertet · general(plex)
Pentatri series: pentatri · superpent · pentadecal(plex)
Hexatri series: hexatri · superhex · hexadecal(plex)
Heptatri series: heptatri · supersept · heptadecal
Iteral series: superoct · octadecal · superenn · ennadecal · iteral · ultatri
Dupertri series: dupertri · duperdecal · truperdecal · quadruperdecal
Latri series: latri · emperal(plex) · hyperal(plex) · admiral
Dutritri series: dutritri · dutridecal
Dimentri series: dimentri · dulatri · trilatri · trimentri
Triakulus series: triakulus · tridecatrix
Big boowa series: big boowa · great big boowa · grand boowa
Tiaokhiao's extensions: trihex · trioct · triennet · triundecal · tridodecal · tritriplex · tritriplexian · grand tritri · tetrapent · tetrahex · grand tetratri · octatri · enneatri · decatri · undecatri · dodecatri
RedBandServerYT's extensions: tridecalplex