03/13/2014, 06:23 AM
MphLee
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.
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.