# Penultimate Modular Origami

James S. Plank

plank@cs.utk.edu

March, 1996

## Introduction

This is a description of how to make polyhedra out of the ``penultimate'' module. This module is originally described in Jay Ansill's book Lifestyle Origami and he attributes the module to Robert Neale. I have omitted how to put the modules together -- buy the book, or figure it out for yourself. It's pretty obvious. The pentagon module is lifted straight from the book (although I've found 3x4 paper easier to work with than 4x4 paper), but the others are my own tweaks.

A note about cutting and glue The triangle and square modules as pictured have cuts. These are not necessary --- you may use inside folds to achieve the same purpose (i.e. the tabs that you are inserting would be too long or wide otherwise). When you do use the inside folds, the tabs become thick, and it takes more patience to get the modules together. Also, the resulting polyhedron is often less stable. However, the choice is yours. If you care more about the purity of the art form than the stability of the polyhedron, then that is achievable. I'd recommend the dodecahedron and truncated icosahedron as excellent models that are very stable without cuts or glue.

This method of making modules lends itself to many variations besides the ones shown here. All you need is a calculater with trigonometric functions and you can figure them out for yourself. Besides the Platonic and Archimedian solids, I have made various others: rhombic dodecahedron, rhombic triacontahedron, numerous prisms and antiprisms, stella octangula, great and lesser stellated dodecahdra, compound of 5 tetrahedra, compound of 5 octahedra, etc. If you're interested, I can give descriptions of the modules, although perhaps not quickly. Pictures of most of these are available at http://www.cs.utk.edu/~plank/plank/origami/origami.html.

The polyhedron numbers referenced below are from the pictures of the Archimedean solids in Fuse's book Unit Origami. Kasahara/Takahama's Origami for the Connoisseur also has pictures of these polyhedra with a different numbering.

I have not included modules for octagons or decagons. I've made octagonal ones, but they're pretty flimsy, meaning that the resulting polyhedra cannot exist in the same house as cats without the aid of glue or a gun. If you can't figure out how to make octagonal or decagonal modules, send me email, and I'll make the diagrams.

If you are interested in polyhedrons, I'd recommend reading Wenninger's Polyhedron Models, Holden's Shapes, Space and Symmetry and for a more mathematical treatment, Coxeter's Regular Polytopes. There is a web page with beautiful renderings of the uniform polyhedra at http://www.mathconsult.ch/showroom/unipoly/index.html.

Modular origami is found in many origami books. Notable in these are the Fuse and Kasahara books mentioned above, as well as Gurkewitz's 3-D Geometric Origami, and Yamaguchi's Kusudama. Jeannine Mosely has invented a brilliantly simple module for the greater and lesser stellated dodecahedrons. If you are interested in that module, let me know and I'll dig it up for you.

Introduction -- Making the Modules -- Making the Polyhedra -- Jim's Origami Page
Jim Plank --- Jim's Origami Page