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Geometric Origami

A book about the geometry of origami, with folding instructions for fifteen regular polygon models.

Geometric Origami

How do you fold regular polygons?

Can more be achieved by folding paper than by traditional ‘ruler and compass’ constructions?

Geometric Origami reveals some of the rich mathematics inherent in paper folding, and will help origamists, mathematicians and the interested general reader appreciate the value of studying the geometrical properties of folding a square of paper.

Part I of the book covers the mathematical theory of origami constructions.

Part II includes step-by-step folding instructions for fifteen regular polygons, from the equilateral triangle to the 19-gon.

Written by the mathematician and origamist Robert Geretschläger, Geometric Origami will appeal to:

• the interested general reader
• origamists interested in folding regular polygons
• mathematicians interested in geometric constructions

A sample page from each chapter can be seen in the Book Gallery.

Reviews of Geometric Origami appear in

• MAA Online (July 2009)
• Crux Mathematicorum Vol 35 No 6 (October 2009)

• MAA Online review
• Crux Mathematicorum review (PDF)
• Crux Mathematicorum
• Book Gallery

The author

Robert Geretschläger teaches mathematics at school and university in Graz, Austria.

His interest in origami goes back to childhood — fond memories of struggling to fold a frog without its legs turning to mush are still very present. A number of his published mathematical papers are somehow related to origami.

List of models

1. Maximum Equilateral Triangle
2. Regular Triangular Grid
3. Maximum Regular Octagon
4. Maximum Regular 16-gon
5. Maximum Regular Hexagon
6. Maximum Regular Dodecagon
7. Easy Regular Pentagon
8. Maximum Regular Pentagon
9. Maximum Regular Decagon
10. Easy Regular Heptagon
11. Maximum Regular Heptagon
12. Regular Nonagon
13. Regular Triskaidekagon
14. Regular 17-Gon
15. Regular 19-Gon

Contents

Part I    Some Theory

1. Euclidean and Origami Procedures
   1. Geometric constructions and where they come from
   2. Elementary Euclidean procedures
   3. Elementary geometric procedures of origami
   4. Reducing Euclidean procedures to origami
   5. Reducing origami procedures to Euclidean constructions
2. Origami Constructions and Algebra
   6. Linear equations
   7. Quadratic equations
   8. Folding cube roots
   9. Solving general cubic equations
   10. Trisecting angles
   11. Solving quartic equations
3. Properties of Origami Constructions
   12. Dividing line segments into sections of equal length
   13. Six problems from one fold
   14. Determining common folding methods for given creases
   15. Origami constructions on a parabola
4. The Maximum Polygon Problem

Part II    Some Practice

5. Triangles, Squares and More
   16. Triangles
   17. The regular octagon and n-gons with n = 2^k
   18. The regular hexagon and n-gons with n = 3 · 2^k
6. The Regular Pentagon and its Cousins
   19. Regular pentagons and the golden section
   20. Some precise folding methods
   21. The regular n-gon with n = 5 · 2^k · 3^l
7. The Regular Heptagon Family
   22. The cubic equation
   23. The “easy” regular heptagon and the regular 14-gon
   24. The maximum regular heptagon
8. A Few More Polygons
   25. The regular nonagon
   26. The regular triskaidekagon
   27. The regular 17-gon
   28. The regular 19-gon

© A K Jobbings 2004-2009