You are watching: How to fold paper into 3
Fold twice to acquire quarter markings at the file bottom.Fold along the line through the optimal corner and also the third of these marks.The upright lines v the very first two marks intersect this inclined line at thirds, which enables the final foldings.
(Photo by Ross Millikan listed below - if the image aided you, you deserve to up-vote his too...)
Here is a picture to go through Hagen von Eitzen"s answer. The horizontal lines room the an outcome of the an initial two folds. The diagonal heat is the 3rd fold. The heavy lines room the points in ~ thirds for folding into the envelope.
This is both practical (no extra creases) and an exact (no guessing or estimating).
Roll the paper into a 3-ply tube, with the end aligned:
Pinch the file (crease the edge) wherein I"ve drawn the red line
Use the pinch mark to present where the folds need to be
This equipment works just with a sheet of document having facet ratio the sqrt(2) (as A4 has).Only two extra wrinkle required.
Roll in to a cylinder until both edges room opposite to each other.Fold the points wherein the edge touches the paper. (Squish the cylinder indigenous left and right)
Approximation method:Assume a 120 level angle and fold as presented below.For accuracy, enhance edge-side to any of the various other two sides.
Beyond the more geometric methods described so far, there is an iterative algorithm (in practice, as an accurate as any exact method) because of Shuzo Fujimoto that i think no one mentioned. In fact, the following method can be generalised to any shape, size and number of foldings.
Let me signify $d_l$ the distance from the left side of the file to the very first mark on the left and also $d_r$, the distance from the ideal side to the best mark. Come simplify, assume the the lenght of the side you desire to divide is 1.
Make a first approximation because that $d_l$. You want $1/3$, however imagine you take $1/3+varepsilon$ ($varepsilon$ being part error to the right or to the left). Thus, ~ above the appropriate you now have actually $2/3-varepsilon$.
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Next, divide the right part into 2 for the first approximation the $d_r$ (by taking the best side that the paper to your first pinch; again, just a pinch). This offers you $d_r=1/3-varepsilon/2$, thus, a far better approximation!
Now, repeat this procedure top top the left. On the left part you now have $2/3+varepsilon/2$. Take the left side of the file to the second pinch to acquire a 2nd approximation of $d_l = 1/3+varepsilon/4$. Note that, after two pinches, friend have lessened your early error to a quarter!
If your initial guess: v was accurate enough, you will certainly not need to proceed more. But, if you need much more precission, girlfriend just need to repeat the process a pair of times more. Because that instance, an initial error the $varepsilon=1$cm to reduce to much less than 1mm (0.0625mm) after ~ repeating the iteration twice.