Ever tried to fold up a map and couldn’t? Edit: what’s a paper map? You’re not alone. Maps don’t have clear fold lines that direct you towards an ordered process of folding, resulting in that complete mess by the fourth step or so. It gets much easier if the fold lines are clear.

Recently I’ve been working with shapes and patterns that do have clear fold lines and have been converting 2D sheets into three-dimensional objects by bending, twisting, or folding. A reverse example: if we take the [3D] cube below and unfold it we obtain six squares in the [2D] pattern. I’ve been using these 2D patterns, called nets, to fold up polyhedra back into 3D.

Nerd note: if you unfold a 4D cube (hypercube) you are left with something called a “tesseract” that looks like this:

, not this.

The polyhedra I’m folding are taken from the Platonic and Archimedean set, something I’ve used heavily in my thesis and related work. This time they’re being used in a shape study for a doghouse that will be showcased at Barkitecture Houston next month. The picture below shows one the unfolded net for the Truncated Icosahedron.

After folding and gluing the polyhedra they look like this:

Ok, so maybe you don’t have a laser cutter available to burn through cardboard. Luckily you can get the same effect using paper, Elmer’s, and some scissors. There are a number of resources available on the web where you can download and print out some of these patterns to fold up your own polyhedra. I’ve listed some of them below:

Places to download existing polyhedra patterns (nets) to print out and fold up:

  • Platonic Solids Fold Up Patterns. A well-constructed infographic like image including all the platonic solids and additional geometric info about the polyhedra.
  • Archimedean Polyhedra Folding Patterns. Colored patterns for the Platonic and Archimedean solids. You’ll need to remember to cut out tabs for gluing these together.
  • Paper Models of Polyhedra. Nearly every major polyhedra, compounds, and other available, in color, for download in .pdf form. Includes excellent pictures of all the examples.
  • Map Foldouts. Includes color images of the globe mapped to platonic polyhedra for folding your own pseudoglobe.

Places that allow you to download 3D files of polyhedra for manipulation or 3D printing:

Programs that will allow you to import your own files and export sheets for printing and folding:

  • Pepakura 3D is an easy-to-use, free program that accepts 3D files and exports 2D patterns for folding. Perfect for those of you who are thinking you need a complicated mask for Halloween but don’t know how to start.
  • Javagami is a free, java-based program for designing and printing polyhedra patterns for folding
  • Stella is a polyhedral viewing program that also has hundreds of sample polyhedra patterns that can be downloaded, printed, and folded.
  • Ori-Revo is a free japanese modeling program that creates complex fold patterns based complicated 3D geometry

If you know how to work with vertex files to generate geometry, here are some resources to download or visualize polyhedra:

Hope you enjoy these resources. You now have no excuse for a high-quality Halloween mask. If you print anything out and make it please post a link to it in the comments!

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