# Office Origami

Some time in 2020, I had to mail a letter to register to vote. One step of the process was folding an 8.5 by 11 sheet of paper into thirds so it’d fit into an envelope. However, I didn’t want to freehand it. I looked up online methods of folding paper exactly in thirds, but I didn’t find anything satisfactory. Every method I found either needed an external tool, left extra creases across the paper, had too much of a “freehand” aspect, or worked only with square paper. So, I wrote a program to find a method for me.

The program was pretty rudimentary. I equipped it with ony one origami operation: given two points $P$ and $Q$, mark all the points on the edge of the paper equidistant from $P$ and $Q$. That is, fold $P$ onto $Q$, but don’t make the entire crease. Just crease near the edges. The program was not able to find an exact solution, and I’m not sure if one exists with only this operation. If one did, it would be too complex to be practical. It did find some approximate solutions, which I’ll describe below:

Conventions:

Orient the paper so that the longer edges are vertical.

All paper considered will have aspect ratio $y:x$ with $y>x$.

Method 1:

- Crease the midpoint of the bottom edge.
- Fold the top left corner onto the bottom midpoint.
- Place the your finger on the point where the top and right edge intersect, and while keeping the intersection point fixed, slide the paper so that the top edge is horizontal, and crease.
- Fold the larger section in half.
The paper is divided into three sections with ratio $4y^2-3x^2:4y^2-3x^2:4x^2$.

For $11:8.5$ paper, this is results in $1069:1069:1156$. The largest third is 0.19 inches too large, a 5.28 % error.

For $\sqrt{7}:2$ paper, this process is perfect.

Method 2:

- Crease the midpoint of the bottom edge, and then the point one quarter from the bottom left corner to the bottom right.
- Fold the top right corner onto the quarterpoint and crease the part of the fold that intersects the right edge.
- Fold the top edge to meet the previous creasepoint.
- Fold the larger section in half.
The paper is divided into three sections with ratio $16y^2+9x^2:16y^2+9x^2:32y^2-18x^2$.

For $11:8.5$ paper, this is results in $10345:10345:10286$. The smallest third is 0.014 inches too small, a 0.38 % error.

For $3\sqrt{3}:4$ paper, this process is perfect.

It would be interesting to one day make a more general version of this program, i.e. a program that synthesizes a geometric construction given a list of allowed operations.