Trimentri is equal to \(\{3,3 ((1) 1) 2\} = \{3,3 (0,0,0,1) 2\} = \{3,3 (0,0,3) 2\} = \{3,3 (0,3,2) 2\}\) in BEAF.[1] The term was coined by Jonathan Bowers. It can be thought as:
- 3-superdimensional array of 3's
- 1-trimensional array of 3's
- "3 tetrated to 3" array of 3's
- "3 pentated to 2" array of 3's.
- This signifies that trimentri is the smallest pentational array that is not degenerate, as trimentri is an X^^X-sized array of 3's, which decomposes into this tetrational array. This is because an X^^X-sized array of 2's degenerate into 4. However, you'll see that most pentational arrays cannot be decomposed into simple tetrational arrays, as the pentational arrays grow so fast so that:
- Consider the array {3, 3, 3([1]1)2}:
- Let an be a function so that:
- a1 = 3
- a2 = {3, 3((1)1)2}
- a3 = {3, trimentri((1)1)2}
- ...
- This array will be equal to aa₃.
- Let an be a function so that:
- Consider the array {3, 3, 3([1]1)2}:
- Therefore, the true limit of well-defined BEAF is slightly higher than the a^^b & a-sized arrays.
- This signifies that trimentri is the smallest pentational array that is not degenerate, as trimentri is an X^^X-sized array of 3's, which decomposes into this tetrational array. This is because an X^^X-sized array of 2's degenerate into 4. However, you'll see that most pentational arrays cannot be decomposed into simple tetrational arrays, as the pentational arrays grow so fast so that:
The array of trimentri is the array which defines trilatri, but with all entries replaced with the entire array twice. So, it has (39)3 = 327 = 7,625,597,484,987 entries.
Approximations[]
Notation | Approximation |
---|---|
Bird's Array Notation | \(\{3,3[1,1,1,2]2\}\) |
Cascading-E notation | \(\textrm E[3]3\#\text{^}\#\text{^}\#\text{^}\#3\) |
Fast-growing hierarchy (using CNF's fundamental sequences) | \(f_{\omega^{\omega^{\omega^{\omega}}}}(3)\) |
Hardy hierarchy | \(H_{\omega^{\omega^{\omega^{\omega^{\omega}}}}}(3)\) |
Slow-growing hierarchy | \(g_{\vartheta(\Omega^{\Omega^{\Omega^2\omega}})}(3)\) |
Sources[]
- ↑ Bowers, Jonathan. Infinity Scrapers. Retrieved January 2013.
See also[]
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 · 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: 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: quintiteral · quadrilatri · quintilatri · sextilatri · septilatri · octilatri · nonilatri · decilatri
Exeen Management World's extensions: dutritet · dutetratet · dupentapent · duheptasept · duheptadecal · duoctadecal · duennadecal · grand dutritri · tritridecal · tritetradecal · quadritridecal
BAT4423's extensions: duhexadecal