01/21/2014, 06:32 AM
Technical Note:
the idea about other shapes, regular hexagons, it seems was a little silly, color squares are the way to go for composite mulanept patterns.
the " four number regions " ...
1) up to about 10^19
2) up to about 10^10^7
3) hyper(4) to about hyper(m) with 9 <= m <= 27
4) Conway Chain Arrows with lengths 3, 4 and 5
the number region2 has an interesting demo project by Ed Pegg Jr
http://demonstrations.wolfram.com/HowTri...ginAndEnd/
I feel some people have observed about composite noptiles the following:
There are composite ordertype 4s but not composite ordertype 3s
Why is this?
answer:
hyper4 corresponds with ordertype4
iterated hyper4 corresponds with composite ordertype4
hyper5 corresponds with ordertype5
this matches the wikipedia information for hyper4, hyper5 and hyper6
although without direct linguistic reference to " mulanept "
so then what about composite ordertype 3s ?
the noptile geometry doesn't refer to this
and therefore doesn't refer clearly to
iterated exponentiation, before hyper4
is this a problem?
I feel it isn't for several reasons
iterated exponentiation is the domain of several things
* iterations of exponential function and Lambert's W
* the demo project above
* the next FGH function coming after (2^n)*n
* typical problems described as "laddered exponents"
* etindao numbers
( ie triplepowers, quadruplepowers etc
indexed with appended operations )
* the generalised formula for the First Value of ordertype(5)
with arbitrary finite base(n) in type dimensional PVN
so the stage between PVN and hyper4
is served well by the standard notations
if there were a desire to express iterated exponentiation
using composite noptiles it is still possible,
but looks unnecessary & silly
the ordertype3 seedvalue becomes the topmost exponent
the 3coloured squares of ordertype4
correspond to either seedvalue^seedvalue or n^seedvalue
2 iterates of ordertype4 (5coloured squares)
correspond to n^(n^seedvalue)
and so on
then ordertype5 corresponds with hyper4
so this is a little cumbersome
and why I think I did it the right way
the conventional symbol formula expressions power tower notations
look after the levels of iterated exponentiation
that lead up to hyper4
The term Composite Mulanept Patterns is the correct term when the Composite Noptile Patterns are given the suitable interpretation relative to Compositions Of Hyperoperations. There are other interpretations for the patterns, coming under the umbrella term Functional Type Shifting Patterns
The following quote comes from “the music of the primes – why an unsolved problem in mathematics matters” by Marcus du Sautoy.
It has relevance to my work as I explain below.
“ While Robinson [Julia Robinson] was putting paid to the tenth of Hilbert’s problems, a friend of hers in Stanford was demolishing Hilbert’s [David Hilbert] belief that in mathematics there is no unknowable. As a student in 1962, Paul Cohen had rather arrogantly asked his professors in Stanford which of Hilbert’s problems would make him famous if he could solve it. They thought for a while, then told him that the first one was the most important. In crude terms, it was asking how many numbers there are. To head this list, Hilbert had chosen Cantor’s question about different infinities. Is there an infinite collection of numbers which is bigger in size than the set of all fractional numbers, yet small enough that they can’t be paired with all the real numbers – including all irrational numbers such as pi, sqrt(2) or any number with an infinite decimal expansion?
Hilbert probably turned in his grave when Cohen returned a year later with the solution: both answers are possible! Cohen proved that this most basic of questions was one of Godel’s unprovable statements. Gone, then, was any hope that only obscure questions were undecidable. What Cohen had proved was this: it was impossible to prove, on the basis of the axioms [starting assumptions] we currently use for mathematics, that there is a set of numbers whose size is strictly between the number of fractions and the number of all real numbers; equally, it can’t be proved that there isn’t such a set. Indeed, he had managed to build two different mathematical worlds which satisfied the axioms we are using in mathematics. In one of those worlds the answer to Cantor’s question was ‘yes’; in the other world it was ‘no’. “
I think I was mostly happy with this answer due to Cohen’s work, but also, to some extent, I guess I thought the answer could be improved upon, and I discovered an answer that seems more easy to understand or more of an ideal answer, in the same way as Hilbert must have felt when he famously said:
" We must know — we will know! "
Wir müssen wissen — wir werden wissen!
for more quotes see
http://en.wikiquote.org/wiki/David_Hilbert
http://en.wikipedia.org/wiki/Ignoramus_et_ignorabimus
http://en.wikipedia.org/wiki/David_Hilbert
http://plus.maths.org/content/we-must-know-we-will-know
That was one of the motivations for my work, I wanted the context of infinity, different kinds of infinity to be understood from the more stable mathematics of hyperoperations and compositions of hyperoperations, quoting from the quote above “In crude terms, it was asking how many numbers there are. To head this list, Hilbert had chosen Cantor’s question about different infinities.“
The Composite Mulanept Patterns are the optimal answer about how to visualise in “crude terms” how many numbers there are and thereby much elucidating the “Frivolous Law of Arithmetic” that says almost all numbers are very, very big.
there are four realms of useful math with natural numbers
numbers up to 10^19
numbers up to 10^10^7
numbers from 4^^4 to 4(^m)4 where 9 <= m <= 27
Conway chained arrows of length 3, 4 or 5
the first realm is the realm of the trades and engineering
where confidence about thoroughness assumption
of natural numbers is valid
the second realm is the realm of math tricks that work
in order to verify some very large prime numbers
for all the talk about prime gaps there are many
genuine prime gaps of indescribabilty up to the
largest known prime numbers
the third realm is the realm of generalised recursion
and there is a perfectly good geometry for this
the key hyperoperations that can be described clearly
this way are hyper7, 8 and 9
the "heavy particles" of mathematics
with a particle wave duality
likewise of fundamental importance
are the first 2 non-trivial terms of the FGH
the fourth realm is also important
as it comes from a well defined recurrence relation
and shows the regions of numbers clearly
and ... transfinite arithmetic inconsistencies or glitches
at e1, e2, e3, e4, e5 and so on --- the epsilon numbers
check the arithmetic and see that its true
the workaround uses hyperoperation style of logic
Sierpinski didn't mention about this
... ... ... he got famous for his
Sierpinski gasket ... ... ...
the idea about other shapes, regular hexagons, it seems was a little silly, color squares are the way to go for composite mulanept patterns.
the " four number regions " ...
1) up to about 10^19
2) up to about 10^10^7
3) hyper(4) to about hyper(m) with 9 <= m <= 27
4) Conway Chain Arrows with lengths 3, 4 and 5
the number region2 has an interesting demo project by Ed Pegg Jr
http://demonstrations.wolfram.com/HowTri...ginAndEnd/
I feel some people have observed about composite noptiles the following:
There are composite ordertype 4s but not composite ordertype 3s
Why is this?
answer:
hyper4 corresponds with ordertype4
iterated hyper4 corresponds with composite ordertype4
hyper5 corresponds with ordertype5
this matches the wikipedia information for hyper4, hyper5 and hyper6
although without direct linguistic reference to " mulanept "
so then what about composite ordertype 3s ?
the noptile geometry doesn't refer to this
and therefore doesn't refer clearly to
iterated exponentiation, before hyper4
is this a problem?
I feel it isn't for several reasons
iterated exponentiation is the domain of several things
* iterations of exponential function and Lambert's W
* the demo project above
* the next FGH function coming after (2^n)*n
* typical problems described as "laddered exponents"
* etindao numbers
( ie triplepowers, quadruplepowers etc
indexed with appended operations )
* the generalised formula for the First Value of ordertype(5)
with arbitrary finite base(n) in type dimensional PVN
so the stage between PVN and hyper4
is served well by the standard notations
if there were a desire to express iterated exponentiation
using composite noptiles it is still possible,
but looks unnecessary & silly
the ordertype3 seedvalue becomes the topmost exponent
the 3coloured squares of ordertype4
correspond to either seedvalue^seedvalue or n^seedvalue
2 iterates of ordertype4 (5coloured squares)
correspond to n^(n^seedvalue)
and so on
then ordertype5 corresponds with hyper4
so this is a little cumbersome
and why I think I did it the right way
the conventional symbol formula expressions power tower notations
look after the levels of iterated exponentiation
that lead up to hyper4
The term Composite Mulanept Patterns is the correct term when the Composite Noptile Patterns are given the suitable interpretation relative to Compositions Of Hyperoperations. There are other interpretations for the patterns, coming under the umbrella term Functional Type Shifting Patterns
The following quote comes from “the music of the primes – why an unsolved problem in mathematics matters” by Marcus du Sautoy.
It has relevance to my work as I explain below.
“ While Robinson [Julia Robinson] was putting paid to the tenth of Hilbert’s problems, a friend of hers in Stanford was demolishing Hilbert’s [David Hilbert] belief that in mathematics there is no unknowable. As a student in 1962, Paul Cohen had rather arrogantly asked his professors in Stanford which of Hilbert’s problems would make him famous if he could solve it. They thought for a while, then told him that the first one was the most important. In crude terms, it was asking how many numbers there are. To head this list, Hilbert had chosen Cantor’s question about different infinities. Is there an infinite collection of numbers which is bigger in size than the set of all fractional numbers, yet small enough that they can’t be paired with all the real numbers – including all irrational numbers such as pi, sqrt(2) or any number with an infinite decimal expansion?
Hilbert probably turned in his grave when Cohen returned a year later with the solution: both answers are possible! Cohen proved that this most basic of questions was one of Godel’s unprovable statements. Gone, then, was any hope that only obscure questions were undecidable. What Cohen had proved was this: it was impossible to prove, on the basis of the axioms [starting assumptions] we currently use for mathematics, that there is a set of numbers whose size is strictly between the number of fractions and the number of all real numbers; equally, it can’t be proved that there isn’t such a set. Indeed, he had managed to build two different mathematical worlds which satisfied the axioms we are using in mathematics. In one of those worlds the answer to Cantor’s question was ‘yes’; in the other world it was ‘no’. “
I think I was mostly happy with this answer due to Cohen’s work, but also, to some extent, I guess I thought the answer could be improved upon, and I discovered an answer that seems more easy to understand or more of an ideal answer, in the same way as Hilbert must have felt when he famously said:
" We must know — we will know! "
Wir müssen wissen — wir werden wissen!
for more quotes see
http://en.wikiquote.org/wiki/David_Hilbert
http://en.wikipedia.org/wiki/Ignoramus_et_ignorabimus
http://en.wikipedia.org/wiki/David_Hilbert
http://plus.maths.org/content/we-must-know-we-will-know
That was one of the motivations for my work, I wanted the context of infinity, different kinds of infinity to be understood from the more stable mathematics of hyperoperations and compositions of hyperoperations, quoting from the quote above “In crude terms, it was asking how many numbers there are. To head this list, Hilbert had chosen Cantor’s question about different infinities.“
The Composite Mulanept Patterns are the optimal answer about how to visualise in “crude terms” how many numbers there are and thereby much elucidating the “Frivolous Law of Arithmetic” that says almost all numbers are very, very big.
there are four realms of useful math with natural numbers
numbers up to 10^19
numbers up to 10^10^7
numbers from 4^^4 to 4(^m)4 where 9 <= m <= 27
Conway chained arrows of length 3, 4 or 5
the first realm is the realm of the trades and engineering
where confidence about thoroughness assumption
of natural numbers is valid
the second realm is the realm of math tricks that work
in order to verify some very large prime numbers
for all the talk about prime gaps there are many
genuine prime gaps of indescribabilty up to the
largest known prime numbers
the third realm is the realm of generalised recursion
and there is a perfectly good geometry for this
the key hyperoperations that can be described clearly
this way are hyper7, 8 and 9
the "heavy particles" of mathematics
with a particle wave duality
likewise of fundamental importance
are the first 2 non-trivial terms of the FGH
the fourth realm is also important
as it comes from a well defined recurrence relation
and shows the regions of numbers clearly
and ... transfinite arithmetic inconsistencies or glitches
at e1, e2, e3, e4, e5 and so on --- the epsilon numbers
check the arithmetic and see that its true
the workaround uses hyperoperation style of logic
Sierpinski didn't mention about this
... ... ... he got famous for his
Sierpinski gasket ... ... ...