Origami Polyhedra Design: 1st Edition (Paperback) book cover

Origami Polyhedra Design

1st Edition

By John Montroll

A K Peters/CRC Press

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pub: 2009-10-26
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This book unravels the mystery of Geometry in Origami with a unique approach: 64 Polyhedra designs, each made from a single square sheet of paper, no cuts, no glue; each polyhedron the largest possible from the starting size of square and each having an ingenious locking mechanism to hold its shape.

The author covers the five Platonic solids (cube, tetrahedron, octahedron, icosahedron and dodecahedron). There are ample variations with different color patterns and sunken sides. Dipyramids and Dimpled Dipyramids, unexplored before this in Origami, are also covered. There are a total of 64 models in the book. All the designs have an interesting look and a pleasing folding sequence and are based on unique mathematical equations.


… a teacher of geometry because it allows students to fold figures ranging from simple two-dimensional polygons to complex three-dimensional figures … can easily be blended into the regular classroom curriculum … an excellent resource. I recommend it for geometry teachers as a supplement to their regular curriculum.

—Gary S. Luck, University of Wisconsin-Milwaukee, Milwaukee, USA

… It covers a truly wide range of objects, including the Platonic polyhedra, di-pyramids and some stellated or ‘sunken’ counterparts. The staggering thing is that each and every example is made of one single sheet of paper. It is astonishing what is possible in the realm of origami. The procedures of generating such a model are explained step by step, which enables the reader to follow the recipe. … I sincerely recommend this nice and absorbing book for all afficionados of origami. …

—O. Roschel, International Mathematical News (IMN), December 2010

Origami's recent explosive increase in sophistication owes much to the pioneering ingenuity of noted origami artist Montroll, as documented by his more than two dozen books published since 1980. … The present volume collects instructions for folding diverse polyhedra, both convex and concave, including many beautiful variations on Platonic solids, more than 70 models in all. … Notwithstanding the considerable detail the author provides regarding the matematical proportions, miraculously, the folding processes generally involve no explicit numerical measurement. … Highly recommended.

—D. V. Feldman, CHOICE, August 2010

Origami Polyhedra Design is a breakthrough collection of original designs created by the author to make polyhedral shapes from a single sheet of paper through folding. In modular origami, a polyhedral shape is built by interlocking many identical origami pieces, where each piece is constructed from a separate sheet of paper. In this book, Montroll, an origami master who pioneered the folding of origami animals from a single sheet of paper, presents methods for folding various polyhedra from a single sheet of paper. The book is well designed and organized and makes you want to start folding polyhedra. … and the instructions for the individual models are clear and easy to understand…

—Thomas Hagedorn, MAA Reviews, April 2010

Table of Contents

Part I: Designing Origami Polyhedra

Polyhedra Overview

Design Factors and Techniques

Math and Design

Design Method Examples


Equilateral Triangle





Silver Rectangle

Bronze Rectangle

Golden Rectangle

Part II: Platonic and Related Polyhedra

Tetrahedron Design


Duo-Colored Tetrahedron

Striped Tetrahedron

Tetrahedron of Triangles

Stellated Tetrahedron

Dimpled Truncated Tetrahedron

Cube Design


Striped Cube

Triangles on Cube

Cube with Squares

Stellated Octahedron


Dimpled Rhombicuboctahedron

Stacked Cubes

Octahedron Design


Striped Octahedron

Duo-Colored Octahedron

Stellated Cube


Dimpled Truncated Octahedron

More Platonic Solids Design



Sunken Platonic Solids Design

Sunken Octahedron

Sunken Tetrahedron

Sunken Cube

Sunken Dodecahedron

Sunken Icosahedron

Part III: Dipyramids and Dimpled Dipyramids

Dipyramid Design

Triangular Dipyramid 90°

Triangular Dipyramid

Triangular Dipyramid in a Sphere

Tall Triangular Dipyramid

Tall Square Dipyramid

Silver Square Dipyramid

Squat Silver Square Diamond

Pentagonal Dipyramid

Pentagonal Dipyramid 45°

Pentagonal Dipyramid in a Sphere

Golden Pentagonal Dipyramid

Squat Golden Pentagonal Dipyramid

Hexagonal Dipyramid

Silver Hexagonal Dipyramid

Hexagonal Dipyramid 36°

Hexagonal Dipyramid in a Sphere

Squat Silver Hexagonal Dipyramid

Heptagonal Dipyramid

Heptagonal Dipyramid 30°

Heptagonal Dipyramid in a Sphere

Octagonal Dipyramid

Octagonal Dipyramid 26°

Octagonal Dipyramid in a Sphere

Nonagonal Dipyramid

Decagonal Dipyramid

Dimpled Dipyramid Design

Tall Dimpled Square Dipyramid

Dimpled Silver Square Dipyramid


Dimpled Squat Square Dipyramid

Tall Dimpled Hexagonal Dipyramid

Dimpled Silver Hexagonal Dipyramid

Dimpled Hexagonal Dypyramid

Dimpled Hexagonal Dipyramid in a Sphere

Octagonal Flying Saucer

Dimpled Octagonal Dipyramid in a Sphere

Dimpled Octagonal Dipyramid

Dimpled Silver Octagonal Dipyramid

About the Author

John Montroll, St. Anselm’s Abbey School, Washington, D.C., USA

Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Geometry / General
MATHEMATICS / Recreations & Games