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Hyperoperation transitional sequences
#1
Hyperoperation transitional sequence with Base3 indexing
From ordertype=4 ( hyper 4 ) to ordertype=8 ( hyper 8 )
In this PDF animation the answer position power tower is always in the top left corner square.
They are CMPs (composite mulanept patterns) with top down bracketing. Each of the 81 CMPs corresponds with a different COH expression. And they form an "increasing sequence" under an appropriate interpretation for the seedvalues and the formal power towers. Terminology, examples, definitions in the other papers.
Using Base4 index or higher and the composite noptile geometry becomes too cumbersome. Base3 shows the stages involved quite nicely. It is good to view, scroll down, the animation slowly, and think about the information pathways.

.pdf   HTS_B3.pdf (Size: 940.94 KB / Downloads: 199)



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#2
The Base3 transition sequence shows clearly there are huge differences between hyper6, hyper7 and hyper8. The Base3 index sequence is a detailed enough demonstration. The question about a less thorough index that preserves the key stages, was answered by the Stem2Bud3 indexing sequence. A subset of Base3 or superset of Base2 known as Stem2Bud3 has smooth dynamic transitions.
From tetration to pentation, hexation, heptation and octation instead of requiring 81 CMPs there are now only 23 CMPs in the
"increasing" sequence. Showing the same transition sequence using standard formulae expressions for the COH expressions is not possible as it pushes the limit of what type setting formulae are about and if it were, would look ugly and messy for the larger examples. The pattern based approach of composite noptiles is the only clear way to understand, and reveals more of the dynamics, combinatorics and information pathways than is possible from the LaTex or mathtype style formulae expressions. It is the only example in maths where it is of great benefit to transcend complicated formulae expressions into non standard tiling patterns. And in so doing, with formulae expression animations the emergent consideration of requiring a fixed origin and coordinate system relative to the origin arises. Other complicated math expressions such as the Fast Growing Hierarchy can use the pattern method and in the larger examples improves the visibility.
The Stem2Bud3 index sequence animation is a proper subset of the Base3 index sequence animation, using the same colour ensemble and the coordinate system used is such that the origin coincides with the APPT and is always in the top left corner position, because the standard FLU folding pattern is used. There are only four transitional sequences worth considering, the pure noptile transition sequence (see the PDF called introduction) and the Base2, Stem2Bud3 and Base3 transition sequences, that are the in-between stages and involve COH expressions.

.pdf   HTS_S2B3.pdf (Size: 316.91 KB / Downloads: 189)

"A proposition is completely logically analyzed if its grammar is made completely clear: no matter what idiom it may be written or expressed in..."
Philosophical Remarks (1930), Part I (1)
Ludwig Wittgenstein

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