mathematics

Geometric Origami

Geometric Origami is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Geometric Origami rather than just read about it. In short: 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 Robert Geretschläger and published by Arbelos Publishing (Shipley, UK) in 2008.

Key takeaways

Geometric Origami belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Geometric Origami to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Geometric Origami from memory before moving on to harder problems.

Reference excerpt

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 Robert Geretschläger and published by Arbelos Publishing (Shipley, UK) in 2008. The Basic Library List Committee of the Mathematical Association of America 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 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 n {\displaystyle n} -gons for which n {\displaystyle n} is a product of a power of two with distinct Fermat primes (powers of two plus one): this allows n {\displaystyle 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 primes in place of Fermat primes, including n {\displaystyle n} -gons for n {\displaystyle 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, 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

Worked examples

Example 1 — a first encounter with Geometric Origami

Start with the simplest possible case. Write down what Geometric Origami claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Geometric Origami before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Geometric Origami ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Geometric Origami

In research: Geometric Origami appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Geometric Origami in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Geometric Origami is common in secondary-school and first-year university syllabi. It links to neighbouring topics 2008 non-fiction books, Mathematics books, Paper folding, so understanding it makes those chapters shorter.
In everyday life: Look for Geometric Origami outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Geometric Origami in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Geometric Origami means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Geometric Origami out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Geometric Origami in simple terms?

Why does Geometric Origami matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Geometric Origami?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Geometric Origami.