We use ratios like "2:2+sqrt2" for simplicity/exactness so we can avoid using long decimals.

Some simple examples:

If we didn't know how to fold this Preliminary Base, we would want to know the circled point as a ref. point.

With geometry(or simply intuition), the circled point is the center, so the ref. point can be written as above. We can say "1:1" or "0.5:0.5". We would need to use the latter if we were to use RefFinder.

Now for a Kite Base, the circled point's reference point is shown above. It can be written as "sqrt2:1" or "0.58...:0.42...".

Now the question is,

'why don't we use '1' as the length of the side and compute the 'exact' length?' To do this, it is shown with

Exact*. Notice that the fractions get really big, even for a simple ratio like "sqrt2:1"

Another question might be,

'what is up with the 'sqrt2'? Why do those appear often?' That has to do with using 22.5 degrees. As shown below, Crease Patterns with 22.5 degree folds have many of these triangles which contain 'sqrt2' lengths.

(As you can see, you can easily calculate the length of the long side be imagining the red line. The red line is the hypotenuse of the right triangle with sides (1-1-?) so the hypotenuse must be sqrt2. And by symmetry, the length between the '22.5deg' and the red line is also sqrt2. So the long side is 1+sqrt2)

PleatBox wrote:So now how do the ratios come into play? For example, the ratios that Baltorigamist posted here, 2:2+sqrt2, how do i turn it into a number that I can put into the paper?

So to answer your 2nd question, you need to find the 'total length' of the ratio(which can be done by replacing the colon(:) with plus signs(+) ) then divide each part of the ratio with the 'total length'

For my first example: "1:1"

total length = "1+1" (replace : with +) = 2

therefore: decimal ratio is 1/2 : 1/2 => 0.5 : 0.5

For "2:2+sqrt2":

total length = (2+2+sqrt2) = (4 + sqrt2)

therefore: decimal ratio is 2/(4+sqrt2) : (2+sqrt2)/(4+sqrt2) => 0.369 : 0.631

P.S. Sorry for the long post. I hope I didn't explain it in a confusing manner...Or if I explained something you already understood, sorry!

If you have more question feel free to ask