Hi Ruby,

I am not sure why you want to call your 9/14 enlargement 'linear'.

...

The braiding process is messing up the perceived 'linearity'.

...

hi,

you are right, that something is messing up.

9/14, ( and 13/20, 17/26 ...) is linear enlargement of 5/8,

but obviously 5/8 is not linear enlargement of any THK.

my point is that, for a speific THK, like 5/8 , or any THK you already knows how to tie it,

you can apply a linear enlargement to it.

and it is clear and easy as an instruction.

Schaake's theory is not that easy to follow.

though it actually can guide me to reach any THK.

and

I think this linear enlargement is part of Schaake's theory. they do not conflict.

a tiny small part of it.

by walking at that so called RKT tree,

or actually a Stern-Brocot Tree,

at any THK you have infinite ways to enlarge it to any THK belongs to current branch,

by changing direction randomly left and right.

and among all these ways, the simplest ways are the 2 linear enlargements:

RightSplit and LeftSplit, like from (3,2) to (5,4), or from (3,2) to (7,4),

by not changing direction, and walk only left 2 steps, or only right 2 steps.

---

so, I am not going to say that I can tie a 13/21， or 21/34 , with a linear enlargement method,

but,

if I can already tie a specified THK, say 13/21， 21/34， or any p/b THK,

or if somebody else just give to me a finished THK like 13/21， 21/34， or any p/b THK,

then, without knowing how he tied it beforewards,

I can apply a linear enlargement method to it, and build a series of enlarged THK.

(also infinite, actually the same size of all THKs , mathematically)