Tetratri

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This page is about Bowers's definition. For the alternative definition, see Conway's Tetratri.

The tetratri is equal to {3,3,3,3} = 3{{{3}}}3 (3 powerexploded to 3) in BEAF. It can also be written 4 & 3, using the array of operator. The term was coined by Jonathan Bowers. It is larger than Graham's number and comparable to Sbiis Saibian's greegolthra-threxitris.

ARsygo coined great tritri for this number.

Tetratri can be computed in 2-bracket operator notation using the following process:

Visual representation of a tetratri

• \(t_1 = 3\)
• \(t_2 = 3 \{\{ 3 \{\{ 3 \}\} 3 \}\} 3\)
• \(t_3 = 3 \{\{ 3 \{\{ 3 \{\{ \cdots \{\{ 3 \}\} \cdots \}\} 3 \}\} 3 \}\} 3\) with \(t_2\) 3's from center out.
• \(t_4 = 3 \{\{ 3 \{\{ 3 \{\{ \cdots \{\{ 3 \}\} \cdots \}\} 3 \}\} 3 \}\} 3\) with \(t_3\) 3's from center out.
• etc.
• Tetratri is \(t_{t_3}\).

Etymology

The name of this number is based on tetra- (four) and tri- (three), meaning that there are four 3's in the array.

Approximations

Notation	Approximation
Bird's array notation	\(\{3,3,3,3\}\) (exact)
Chained arrow notation	\(3 \rightarrow 3 \rightarrow 3 \rightarrow 3 \rightarrow 4\)
Hyper-E notation	\(E(3)3\#\#3\#\#3\#3\#3\#2\)
Taro's multivariable Ackermann function	\(A(1,0,0,3)\)
X-Sequence Hyper-Exponential Notation	\(3\{X^2\}3\)
Fast-growing hierarchy	\(f_{\omega^2}(3)\)
Hardy hierarchy	\(H_{\omega^{\omega^2}}(3)\)
Slow-growing hierarchy	\(g_{\varphi(2,2,0)}(3)\)

See also

Jonathan Bowers' tritri series

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) · diteral(plex) · admiral
Dutritri series: dutritri · dutridecal
Dimentri series: dimentri · dimendecal · 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
Douglas Shamlin Jr.'s extensions: grand tridecalplex · quintuperdecal · triteral · dutetradecal · quadriteral · dupentadecal · duladecal · triladecal
MultiArrest91's extensions: tritredecal · tritetradecal · sextuperdecal · septuperdecal · octuperdecal · nonuperdecal · decuperdecal · centuperdecal · milluperdecal · quadritetradecal · quintitridecal
SeveralLegend9998's extensions: corboral
XRQ CORPORATION's extensions: grand tridecal · grand emperal · grand grand emperal · quintiteral · quadrilatri · quintilatri · sextilatri · septilatri · octilatri · nonilatri · decilatri
Exeen Management World's extensions: dutritet · dutetratet · dupentapent · duheptasept · duheptadecal · duoctadecal · duennadecal · grand dutritri · demperal · tritridecal · tritetradecal · quadritridecal
BAT4423's extension: duhexadecal
ARsygo's extensions: tritridex · tritetdex · grand tritet · grand tripent · grand dozenal · grand trihectal · grand tritriplex · great tridecal · gigantic tridecal · gorged tridecal · gulp tridecal · gasp tridecal · tetrahept · tetraoct · tetraenn · grand tetradecal · pentatet · hexatet · hexapent
TheInfinitySeeker's extensions: dutritritri