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the 42 CMPs or Composite Mulanept Patterns
are a small selection of about 140,000 patterns
that can fit comfortably on your laptop screen


they could be composite mulanept patterns
or more generally, functional type shifting patterns


are really fascinated by these things ...but
I was never able to understand how they work and what they are...

I read your first two paper two times! but I cant get them.
I just have the feeling that is a way to "draw" very big numbers , right?
Can you maybe link me your easier explaination about them or explian your work in very easy words for "slow" people like me?
sure thing MphLee
"I just have the feeling that is a way to "draw" very big numbers , right?"
yes that's a good summary of the idea
They are strange things because they are about information pathways
that exist in big and complicated numbers
These numbers have a main starting position
and ending position, and to notice these positions
you need first to recognise the folding pattern being used.
The "default" folding pattern I use is start at bottom right
then Fold to the Left, then Fold in the Up direction
This is Fold_LU and there are 8 possible folding patterns.
Noticing the way the folding pattern actually works
has a subtle Blush aspect, the first fold up looks as though
it is folding down but this is only to obtain the so called
seed value to start the upwards direction.
Likewise, the next Fold Left requires a brief shift
to the Right to pick up the next seed value
before pushing to the Left direction, and so on and so forth.
You can imagine the process as being one of "transference".
You study the symbol representation of the formulae
and notice the symbols or glyphs in the formulae
have a regular pattern, again relative to a folding pattern.


Another thing to notice is about "ordertype" ...
ordertype4 corresponds with hyper4, and ordertype5 with hyper5
and so on. But the thing is there are hyperoperations and the between stages of iterations of a hyperoperation before reaching the successive hyperoperation that "plucks up" a new seedvalue.

The non standard tiling patterns are representations for pure hyperoperations (pure noptiles), iterations of pure hyperoperations (compositions of pure noptiles), and also any composition of hyperoperations using any well defined bracketing pattern and any ordering of hyperoperations as can be seen from the examples.

The other useful thing to recognise is to observe the Unique Initial seed value and the Unique Outer seed value of each of the pure noptiles, no matter what the ordertype is. Assuming the folding pattern used is Fold_LU, we notice The Unique Initial Seed Value, ISV, is always at the bottom right position of a pure noptile and is starting an ordertype4 or ordertype5 subcomponent of the noptile.

However, the Unique Outer Seed Value, OSV, is always alternating between the Right mid region position and the Bottom mid region position, so with hyper4, the OSV is on the right; hyper5 the OSV is at the bottom; hyper6 the OSV is at the right; hyper7 the OSV is at the bottom... And the areas of the "minimal enclosing rectangles" for the pure noptiles grow approximately exponentially in area.

The Initial seed goes with the hyperbase and the Outer seed goes with the hyperexponent. This is a regular feature, and allows the compositions to be well defined, although when pipelines are needed, they are used to join component noptiles together from one of the Non-Unique pure noptile attachment squares and either is attached to, or pipelined to, the ISV or OSV of another noptile, as appropriate according to the subexpression of the formulae expression.
hope this makes it clearer.

this is according to interpreting the patterns as composite mulanept patterns (see the other papers for more explanations and examples)
they can also be seen (a bit more) generally as functional type shifting patterns with other interpretations, such as plus or times instead of exponentiation as the constituent operation, or standard positional notation with fixed arbitrary base(n), but I think only a proper subset of the patterns are valid with the SPN assumption... to avoid the yucky issue of redefining the finite base to nonsensical values.


Thanks!, I'll think more about it. I'll ask you if I don't get something. Smile
by the way

March 14 is international pi day ,.,.,. Cool

don't forget tau though ,.,.,. Big Grin

from The Tau Manifesto

"The Tau Manifesto first launched on Tau Day:
June 28 ( 6/28 ), 2010. Tau Day is a time to celebrate
and rejoice in all things mathematical."


for those unfamiliar with these Zen pattern numbers of the third realm or region, floating beyond the large GIMPS Mersenne prime numbers, I thought I'd mention some of my inspiration and influences; or ideas that seem vaguely relevant
it's easy to find weblinks for these people and websites

an interesting visualisation of one billion is available
Visualisation of powers of ten from one to 1 billion

Robert Munafo
classes of natural numbers aka counting numbers, positive integers

Reuben Goodstein
names for hyperoperations, Goodstein's theorem

John Horton Conway
Conway chained arrow recurrence relation, the fourth realm,
and heaps of wide ranging math ideas

Donald E Knuth
simple but good notation for hyperoperations
concrete math and art of computer programming books

Ronald Graham
Graham's number

Wolfram Research
cellular automata, NKS, Demonstrations project and Wolframalpha

Piet Hein
M C Escher
amazing patterns and art
Edward R Tufte
visual display of quantitative information

Andrzej Grzegorczyk
another perspective on the third realm or region
the Fast Growing Hierarchy and Grzegorczyk Hierarchy

Georges Perec
whose contributions to Oulipo were inspiring
I had to follow the ideas from Oulipo
to think creatively enough about the natural numbers, when in math mode, I was usually trying to think creatively about basic things

Patrick Gunkel
whose contributions to innovative thinking were inspiring

other inspirational people

Benoit Mandelbrot
Mandelbrot set (1979)

David Madore
tree pictures of transfinite ordinals, interesting visualisations but complicated to understand

Jonathan Bowers
polyhedra, polychora
and fourth realm ingenuity ( Bowers arrays )

gmalivuk, deedlit, wardaft and heaps of others .........
xkcd forum, My number is bigger game

another approach to the name big numbers game

I used to be interested in reading papers on these websites
Calresco and CCSR and Principia Cybernetica
complexity studies, artificial life ...

Tim Berners Lee
the worldwide web is 25 years old or young

Quickfur on Eretrandre
who sort of seemed to be thinking along a similar path to me

and of course the wonderful Eretrandre people
Henryk Trappman, Gottfried Helms, Dmitrii Kouznetsov, and others
who invented or discovered the beautiful tetration fractal pictures
and the many deep mathematics ideas on Eretrandre

and of course many others I forget but who also influenced me
or others who I remember, but if I mentioned them all
there would still be others I missed but should have included
Lots of numerical examples related to these patterns
can be found in the Googology Wiki the large number encyclopedia*
there are many examples in the Photos section of the Wiki
*maintained by Jonathan Bowers and Sbiis Saibian


they are pretty and entertaining to look at ...