''Geometric Origami'' is a book on the mathematics of paper folding, focusing on the ability to simulate and extend classical straightedge and compass constructions using origami. It was written by Austrian mathematician and published by Arbelos Publishing (Shipley, UK) in 2008. The Basic Library List Committee of the

Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...

has suggested its inclusion in undergraduate mathematics libraries.

Topics

The book is divided into two main parts. The first part is more theoretical. It outlines the

Huzita–Hatori axioms The Huzita–Justin axioms or Huzita–Hatori axioms are a set of rules related to the Mathematics of paper folding, mathematical principles of origami, describing the operations that can be made when folding a piece of paper. The Axiom, axioms assu ...

for mathematical origami, and proves that they are capable of simulating any straightedge and compass construction. It goes on to show that, in this mathematical model, origami is strictly more powerful than straightedge and compass: with origami, it is possible to solve any cubic equation or quartic equation. In particular, origami methods can be used to trisect angles, and for doubling the cube, two problems that have been proven to have no exact solution using only straightedge and compass. The second part of the book focuses on folding instructions for constructing regular polygons using origami, and on finding the largest copy of a given regular polygon that can be constructed within a given square sheet of origami paper. With straightedge and compass, it is only possible to exactly construct regular for which

n

is a product of a

power of two A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent. In a context where only integers are considered, is restricted to non-negative ...

with distinct Fermat primes (powers of two plus one): this allows

n

to be 3, 5, 6, 8, 10, 12, etc. These are called the constructible polygons. With a construction system that can trisect angles, such as mathematical origami, more numbers of sides are possible, using

Pierpont prime In number theory, a Pierpont prime is a prime number of the form 2^u\cdot 3^v + 1\, for some nonnegative integers and . That is, they are the prime numbers for which is 3-smooth. They are named after the mathematician James Pierpont, who us ...

s in place of Fermat primes, including for

n

equal to 7, 13, 14, 17, 19, etc. ''Geometric Origami'' provides explicit folding instructions for 15 different regular polygons, including those with 3, 5, 6, 7, 8, 9, 10, 12, 13, 17, and 19 sides. Additionally, it discusses approximate constructions for polygons that cannot be constructed exactly in this way.

Audience and reception

This book is quite technical, aimed more at mathematicians than at amateur origami enthusiasts looking for folding instructions for origami artworks. However, it may be of interest to origami designers, looking for methods to incorporate folding patterns for regular polygons into their designs. Origamist David Raynor suggests that its methods could also be useful in constructing templates from which to cut out clean unfolded pieces of paper in the shape of the regular polygons that it discusses, for use in origami models that use these polygons as a starting shape instead of the traditional square paper. ''Geometric Origami'' may also be useful as teaching material for university-level geometry and

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...

, or for undergraduate research projects extending those subjects, although reviewer Mary Fortune cautions that "there is much preliminary material to be covered" before a student would be ready for such a project. Reviewer Georg Gunther summarizes the book as "a delightful addition to a wonderful corner of mathematics where art and geometry meet", recommending it as a reference for "anyone with a working knowledge of elementary geometry, algebra, and the geometry of complex numbers".

References

Mathematics books 2008 non-fiction books Paper folding {{Mathematics of paper folding