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The tritet is equal to $$\lbrace 4,4,4 \rbrace = 4 \lbrace4\rbrace 4 = 4\uparrow\uparrow\uparrow\uparrow4$$ (4 hexated to 4) in BEAF. The term was coined by Jonathan Bowers. This is the fourth Ackermann number.

Unlike with Tritet Jr., it's impossible to simplify the tritet using the laws of algebra since it's simply way too large.

Googology Wiki user Hyp cos calls this number a trientet, and it's equal to s(4,4,4), s(4,2,5), s(4,4,1,2), s(4,1,1,3), or s(4,1,1,1,2) in strong array notation. 

Computation

Tritet can be computed in the following process:

• $$a_1 = 4$$
• $$a_2 = 4 \uparrow\uparrow 4 = 4 \uparrow\uparrow\uparrow 2$$ (this is tritet Jr.)
• $$a_3 = 4 \uparrow\uparrow a_2 = 4 \uparrow\uparrow\uparrow 3$$
• $$a_4 = 4 \uparrow\uparrow a_3 = 4 \uparrow\uparrow\uparrow 4$$
• etc.
• Tritet is equal to $$a_{a_{a_4}}$$.

Etymology

The name of this number is based on tri- (three) and tetra- (four), meaning that there are 3 4's in the array.

Approximations

Notation Approximation
Up-arrow notation $$4\uparrow\uparrow\uparrow\uparrow4$$ (exact)
BEAF $$\{4,4,4\}$$ (exact)
Hyper-E notation $$E(4)1\#1\#1\#4$$ (exact)
Chained arrow notation $$4\rightarrow4\rightarrow4$$ (exact)
Hyperfactorial array notation $$5!3$$
Fast-growing hierarchy $$f_5(3)$$
Hardy hierarchy $$H_{\omega^5}(3)$$
Slow-growing hierarchy $$g_{\varphi(3,0)}(4)$$

Sources

1. Bowers, JonathanInfinity Scrapers. Retrieved January 2013.
2. Numbers from linear array notation | Steps Toward Infinity!

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

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