by Sjaak Adriaanse
Edited by Sara Adams
math
geometric

Paper hoarders will appreciate this nifty tool for cutting leftover pieces of paper into common size ratios like 4×3, Golden and Silver rectangles, or the ratio of the dollar bill.

Now we all want to construct the right paper sizes by folding alone, preferably starting with a square, don't we? But sometimes you just want a piece of paper with the right dimensions and start folding from there. That's where the pdf you can download here comes in. It shows a so-called nomogram.


Click on the image to download the nomogram.

Nomograms consist of a number of rule-like scales with numbers and special markings and they have a very curious property. If you draw a straight line through the scales, the numbers on the scales where the line crosses them ALWAYS satisfy some specific relationship or formula.

Take a look at the sheet that is available for download here. It consists of three lines.

  1. The top line has a number of diamonds on it, each one indicating some common paper size ratio.
  2. The middle line has two scales, marked 'long side (mm)' and 'long side (inch)'.
  3. The bottom line also has two scales, marked 'short side (mm)' and 'short side (inch)'.
Both the middle line and the bottom line have some positions marked with lines, indicating the exact long side and short side of that particular paper size.

The magical property of this nomogram is: if you draw a straight line through a diamond on the top line, the numbers on the other two scales always have the ratio indicated by the diamond.

Let's illustrate this with some examples of how to use the nomogram. The best way to understand the instructions is to print the sheet and draw the straight lines yourself. So grab a ruler and a pencil and go ahead!

Basic Use: Determining Paper Sizes

Simple Example

This example uses sizes in millimeters, but the method is exactly the same when using inches.

Suppose you have a long strip of paper that is 76mm wide and you want to cut some Silver Rectangles for practicing a letterfold. Let's assume that you want the largest possible sheets from your strip of paper, so we'll use the 76mm width as the shorter side of your rectangles. Where do you have to cut the strip to get the right ratio?


Where do you need to cut the strip to get a silver rectangle?

In order to determine this, draw a straight line through the diamond at the top with the text A-size / Silver rectangle and the number 76 on the upper scale of the bottom line (marked on the left short side (mm)). The line you drew cuts the upper scale of the middle line (marked long side (mm)) at slightly less than 108.


Draw a straight line through the diamond and the width of the strip (short side).

So you have to cut off pieces of about 108mm length, and those pieces measuring 76mm×108mm will have the Silver Rectangle ratio.


Cut the strip at 108mm to get a silver rectangle.

Advanced Example


We want a dollar bill ratio.

This example is slightly more elaborate, demonstrating the added value of drawing multiple lines. This example uses inches, but the method is exactly the same when using millimeters.

Suppose you have an A4 sheet of paper, from which you want to cut some rectangular pieces with the same ratio is a dollar bill to practice some dollar bill folds. And suppose you want the short side of the A4 sheet to be the long side of your "dollar bills". In order to determine what the short side of your dollar bill ratio sheets should be, you have to transfer the length of your original sheet, that is the short side of your A4, to the long side scale (middle scale). For this you can use the Square diamond:

  1. Draw a line from the diamond marked Square (1 x 1) at the top left, to the mark A4 on the bottom scale, at the right end at about 8.27 inch. Note that this line crosses the middle line at the same number 8.27 on the inch scale, and this is of course exactly what we want. Let's call this point on the middle scale point A.


    Transfer the short side of an A4 sheet to the scale for the long side.

  2. To determine the short side of your dollar bills, draw a line from the diamond marked dollar bill at the top, through point A. Extend this line and you will see that it cuts the inch scale of the bottom line at almost exactly 3.5. So, the short side of your dollar bills should be 3.5inch.


    Draw a line between the intersection and the dollar bill diamond. Then read the desired value at the intersection with the scale for the short side.

  3. From one A4 sheet you can cut two pieces with the dollar bill size ratio. A small strip will be left over.

Adding your own reference point


Add your own reference points by drawing a line between the dimensions of an example size.

If you have developed your own model with a rectangle in a nonstandard dimension ratio, you can establish a ratio point on the top scale yourself. All you need is a piece of paper with the proper ratio. You can then draw a line through the dimensions of this paper and let it cross the top scale. Mark the crossing point. From now on, you can use this point like all the other points.

If you have mathematically calculated the ratio, you can also use the diamond mark at 100mm in the bottom scale. For instance, for a ratio of 1.23, draw a line through this diamond and 123 on the mm scale on the middle line. This trick also works when you normally would use inches for measuring!

Final Remarks

Here are a few tips for thrifty folders who want a printed nomogram to last as long as possible.

  • You can do the easy one-off readings, like in example 1, with a thin thread, preferably black. Hold it in both hands and stretch it tight while pressing it to the paper with your thumbs.
  • Use a soft pencil when drawing any lines and erase them afterwards.
  • If you have a transparent ruler, draw a thin black line on the underside that runs along its length.
  • If you really like this tool, you can laminate it and draw the lines with a thin non-permanent marker. A laminated nomogram plus a marker make a fine Christmas present for that friend who already has everything origami!

All those fantastic models that need non-square sheets are now within easy reach. To list but a few:

What are you waiting for?

Comments

November 22, 2013 - 1:11pm markmorden

This is a terrific tool. Thanks for sharing. A good way to use this file is to find a drawing program that will open the pdf. If you use the line drawing function of the program, you can anchor one end of the line on a diamond at the top of the nomogram and then extend the line through whatever point you want. This will be more accurate than drawing with a ruler. When you are done, erase the line and save the file for another use. Thanks again for the article.