Gurkewitz Rona - Modular Origami Polyhedra

This book is dedicated to the memories of Karan Hall and Lewis Simon.

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Clockwise :

12-Module Decoration Box from 2x1s adapted from dollar bills

30-Module Dodecahedron from 108-degree module from 4x3s

12-Module Decoration Box from 2x1s

15-Module Pentagonal Dipyramid from 60-degree half-square Plank modules

Part 1

Introduction

Clockwise from upper right:

24-Module Cuboctahedron from 90- to 60-degree half-square Plank modules

6-Module Tetrahedron from 60-degree half-square Plank modules

12-Module Octahedron from 60-degree half-square Plank modules

30-Module Icosahedron from 60-degree half-square Plank modules

As the title indicates, this book is a revised and enlarged edition of the original Modular Origami Polyhedra , by Lewis Simon and Bennett Arnstein, which was published by Bennett Arnstein in 1989. The models included here have been developed by the authors during the course of four decades. Lewis Simon was a pioneer in the field of modular origami in the 1960s. Since that time, many people have learned to enjoy the branch of origami called modular or unit origami. This type of origami consists of making models from several pieces of paper folded in the same way. Folding the modules can be simple. It is assembly that takes some practice if you are new to this type of folding. Even assembly may not be difficult until the last piece or two must be added to the model. With practice, the assembly process becomes very clear and dexterity develops. The results make the effort worthwhile.

This book contains diagrams for constructing modules that can be used to build polyhedral shapes. It is divided into sections for three main systems and a section for miscellaneous modules. As in the earlier book 3-D Geometric Origami: Modular Polyhedra (by Gurkewitz and Arnstein), we define a system of models as a collection of models that can be folded from different numbers of a given module or from modules that have related folding sequences. Another possibility is to vary the starting polygonal shape or the first few folds of the paper used for a module. A third possibility is to systematically vary an angle or angles on a module to produce a new module that makes a different polyhedron.

The Sonobe system, named after Mitsunobu Sonobe of Japan, is comprised of models made from different numbers of the basic module, creased in different ways, as well as a module with properties related to those of the basic module. The basic module can be viewed as a chain of four right triangles. The Sonobe polyhedral shapes made from the basic module are distinctive because their faces are all 45-45-90 right triangles or a combination of several such triangles. This book includes some foldings of the basic module designed by Lewis Simon, namely the 12-module cube and the 24-module cube. New in this book is the integration of the Sonobe shapes into the Decoration Box system. The generalization of the basic Sonobe module from a chain of four 45-45-90 right triangles to a chain of four equilateral triangles is different from the two similarly functioning modules in 3-D Geometric Origami. Other foldings of the basic module are possible and have been explored by Tomoko Fuse and Michael Naughton.

We think that Lewis Simon was one of the first to fold a stellated octahedron from Sonobe modules. We have included the stellated octahedron and the stellated icosahedron as models in this book. It should be noted that the stellation process has been generalized by Kasahara and others to building polyhedra with 3-, 4-, 5-, 6-, or 8-sided faces. To do this one builds a compound of pyramids for each polygon that is a face of the polyhedron. The number of pyramids in the compound will be based on the number of sides of the polygon. This book illustrates how to join three modules together to make a cube-corner pyramid. For 3-, 4-, 5-, and 6-sided faces of the polyhedron, the same number of pyramids are joined at a vertex to make a compound. For an 8-sided face, an octagonal compound is made up of four 3-sided compounds and five 4-sided compounds. There is a central 4-sided compound surrounded by alternating 3- and 4-sided compounds.