Triakulus

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Triakulus is equal to \(3\&3\&3\) or \(\{ 3,3 /2 \}\) or \(\{L,1\}_{3,3}\) using the array of operator and BEAF.^[1] Since the array of operator solves from left to right, \(3\&3\&3 = \{ 3,3,3 \} \& 3 \) = \(3 \uparrow\uparrow\uparrow 3 \& 3\), a pentational array. The structure of an array which defines triakulus is supposedly X^^^X and the array is intended to have tritri entries.

It must be remembered that although it has tritri entries, a triakulus is in no way the same thing as a linear array with tritri repetitions of 3. That would be dupertri, which is a much smaller number. This is because you have to group the terms in the arrays, as an array of size X+1 & 10, when the terms are grouped, is equal to {10, 10(1)10}. If the terms are ungrouped, the array will be a linear array of size 10+1, which will be {10, 10+1(1)2}, which is just a linear array with size 11. One of the major challenges behind defining this level of BEAF would be successfully conceive a representation of triakulus that captures all its tritri entries while setting it apart from a dupertri, as a dupertri is an ungrouped 3^^^3 & 3, while triakulus is X^^^X & 3, where the X's are grouped.

If it's well-defined, this would be the smallest number named by Jonathan Bowers which lies in the pentational-array range.^[2] This is the first and the smallest Bowerism which is not well-defined. See Introduction to BEAF#Pentational arrays for more details of that level.

Approximations[]

Since it is not defined by now, the following approximation is just an expected value under climbing method (strong bound):

Notation	Approximation
Bird's array notation	\(\{3,3 [1 [1 [1 \backslash 2] 2 ¬ 2] 2] 2\}\)
Extended Cascading-E notation	E[3]3#^^#^^#3 = E[3]3#^^^#3
Fast-growing hierarchy	\(f_{\varphi(\varepsilon_0,0)}(3)\)
Hardy hierarchy	\(H_{\varphi(\varepsilon_0,0)}(3)\)

Sources[]

↑ Bowers, Jonathan. Infinity Scrapers. Retrieved January 2013.
↑ Sbiis Saibian, Ultimate Finite Numbers List part 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

Jonathan Bowers' array of series

[1] Bowers, Jonathan. Infinity Scrapers. Retrieved January 2013.

[2] Sbiis Saibian, Ultimate Finite Numbers List part 3

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