Thursday, April 15, 2010

Folding Paper in Thirds without Guessing or using Ruler

Update: New method added: "Folding any rectangular (or square) sheet of paper in thirds"

I've been thinking about this lately. How to fold a paper in thirds? I found two separate methods. One works for the A series of papers (most popular of which is the A4 size paper). The other works for any size.

Folding an A series paper (A0, A4, etc.)
  1. Fold the sheet of paper diagonally using two opposing corners (Let us call these A and B).
  2. Orient the paper in such a way that the two other corners (Call these C and D) are at the bottom and a W shape is formed.
  3. Fold the left flap of the W shape over such that the previous fold line is perfectly aligned on itself, and the new fold passes through one of the corners, C or D, whichever is nearer to the flap.
  4. Unfold, and the do the same procedure with the right flap. The new fold must pass through the other one of the corners C and D.
  5. Unfold this new fold, and also the diagonal. Now you must notice that there are two intersection points on the paper. Fold the paper into thirds by folding it through these two points.
Tips and notes:
  • You may want to avoid performing this procedure on your original paper as it leaves unwanted fold lines or creases on the paper. Instead you may do it on a rough sheet of paper of the exact same size, then transfer the final folds (the thirds) onto your original. Or you may follow procedure below.
  • A series paper has the unique property that two sheets of paper of a particular number in the series (e.g. A4) joined together would create a paper that is the same size as the previous number (A3). A sheet of paper of a particular number (e.g. A4) when folded or cut in equal halves across the width would give two sheets of paper of the next number (A5). Mathematically, the ratio of length to breadth is 1:√2.
  • The ability to fold the paper in thirds by the above procedure is a side effect of this property.
  • Theoretically this should perfectly fold the paper in thirds, however due to practical limitations in ensuring accurate paper size, the folds may not be perfect.
  • I figured this out on my own. I didn't find anything similar on the web.
Folding any sheet of paper (with straight edge on the side that should be folded in thirds, such as Letter or Legal)
  1. Take another sheet of paper where the width (breadth) is at least 0.662 times the length. In the case of an A series paper, another sheet of paper of the same number in the series is sufficient.
  2. Fold this other sheet in fourths. (First fold in half, then fold the two halves each in half.)
  3. Unfold the paper. You will find three crease lines on the paper.
  4. Now take the original sheet of paper and place it on top of the folded paper, while following the next step.
  5. Place the end corners of the side you want to fold such that one corner coincides exactly with a corner of the folded sheet of paper.
  6. The other corner of the original sheet's side must coincide exactly with a point on the third fold line away from the corner that coincided in step 5. When the papers are flat on a flat surface, there is only one such point.
  7. You will notice that the two other folds of the folded paper divide the side of the original paper in thirds. Simply use these intersections to fold the paper in thirds.
Tips and notes:
  • The folded sheet of paper is simply a tool to fold the original in thirds. You can use other tools that maybe more readily available such as four parallel lines in which the distance between any two consecutive lines is always equal. You can place the corners of the side such that they coincide with line 1 and line 4.
  • I found this procedure on the internet, but the website I found only talked about square paper. However it is true for all sizes of paper as long as you can make a tool long enough (and short enough) to accommodate the length of the side to be folded.
Folding any rectangular (or square) sheet of paper in thirds
  1. Fold (and crease) the paper into two equal halves. Unfold it.
  2. Fold a diagonal between two opposing corners. You may optionally fold back to equal halves after this.
  3. Fold a diagonal between two opposing corners of one half of the paper such that this diagonal intersects the original diagonal from step 2. (Only one of the two possible diagonals can intersect the original diagonal).
  4. Unfold everything. The intersection point divides the length and the breadth of the paper into one-thirds and two-thirds.
  5. You can continue to fold a similar diagonal on the other half of the paper, or you can simply fold through the intersection point to divide the sheet into one-thirds and two-thirds. The two-thirds part can then be folded in half, and the paper is folded in thirds.
Tips and notes
  • As with the first method, you may want to fold a rough paper of same size and transfer the final folds to your target paper.
  • This method gives better accuracy than the first, and also helps you avoid the hassle (in the second method) of aligning and coinciding corners and fold lines of two sheets of paper.
  • This method is universal for rectangular sheets of paper (including squares). However the previous method is universal for sheets of paper of any shape, as long as a straight edge of full length can be created, perpendicular to the desired folds.
  • This too, I found online. However the web page was only talking about square paper. But when I checked, the method works for any size of paper as long as it is rectangular. I checked with A4 and a random sized rectangular paper.
    A visual interpretation may be posted when I have the time.