Erik Demaine's Folding and Unfolding:

Fold-and-Cut Examples

The crease patterns below all have the property that, when folded up, a single complete straight cut can simultaneously cut all the bold lines and no more, producing the desired shape or shapes. For more information, refer to the page on the fold-and-cut problem and its mathematics. These examples were designed by Erik Demaine and Martin Demaine.

How To Print

There are three main ways to print these figures, ranging from easiest to highest quality:
  1. You can use your webbrowser to print the images at 72 dpi. The lines will be grainy, but printing this way is the easiest.
  2. To get higher quality, save one of the high-resolution images (150 dpi or 300 dpi) to a file, open it with an image viewer, and print it scaled down to fit to the page.
  3. To get prefect quality, use the PDF or Encapsulated PostScript versions. Print to fill the entire page. If you don't know how to print these files with your printer, see my information on file formats.

How To Fold

The examples below all follow the following notation: To make one of these examples, here is one recommendation procedure, depending on your expertise in folding crease patterns:
  1. Print the crease pattern as large as possible, and possibly also photo-enlarge using a photocopier onto larger paper.
  2. Precrease all the creases (dashed and dot-dashed) by pinching the paper and making sure that you follow along the printed lines. This may get your hands a little dirty with toner.
  3. Reverse the valley (dashed) creases, so that all crease have their proper orientation.
  4. Now the hard part: collapse all the creases simultaneously. This can take some practice, especially on the harder examples, but with some effort you should be able to get it.

Straight Skeleton Method

These examples are all based on the straight-skeleton solution to fold-and-cut, because this method typically produces practical foldings. The foldings can be made even more practical (as in several of the examples below) by slightly perturbing the graph of desired cuts so that multiple edges of the straight skeleton align to produce high-degree straight-skeleton vertices and fewer straight-skeleton edges and creases overall.

MIT Logo


[PDF] [Encapsulated PostScript] [36 dpi image (shown)] [72 dpi image] [150 dpi image] [300 dpi image]

Thanks to Nina Strohminger for finding a good folding of this model.


Swan


[PDF] [Encapsulated PostScript] [72 dpi image (shown)] [150 dpi image] [300 dpi image]

Angelfish


[PDF] [Encapsulated PostScript] [72 dpi image (shown)] [150 dpi image] [300 dpi image]

Butterfly


[PDF] [Encapsulated PostScript] [72 dpi image (shown)] [150 dpi image] [300 dpi image]

Fancy Star


[PDF] [Encapsulated PostScript] [72 dpi image (shown)] [150 dpi image] [300 dpi image]

G4G5, prepared for the 5th Gathering for Martin Gardner


[PDF] [Encapsulated PostScript] [72 dpi image (shown)] [150 dpi image] [300 dpi image]

Jackolantern (make out of orange paper!)


[PDF] [Encapsulated PostScript] [72 dpi image (shown)] [150 dpi image] [300 dpi image]

Entire Tangram Set from a Square


[PDF] [Encapsulated PostScript] [72 dpi image (shown)] [150 dpi image] [300 dpi image]
Last updated July 19, 2007 by Erik Demaine.