Maryknoll Fathers’ School
F.1, F.2 and F.3 Mathematics
Mathematics Reading Scheme
Title: Origami Mathematics
Name of teacher-in-charge: Mr. Kwok Yu Fai
Student Name: Chan Tai Man
Class: F.3C
Class No.: 43
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Origami Mathematics
Introduction
Origami may not seem like it involves very much mathematics. Yes, origami involves
symmetry. If we build a polyhedron then, sure, we encounter a shape from geometry. Is that as far
as it goes? Do any interesting mathematical questions arise from the process of folding paper?
Can origami help someone learn arithmetic, algebra, geometry, and other standard math topics? Is
there any deep mathematics in origami? Is the mathematics behind origami useful for anything
other than making pretty decorations?
Mathematics in Origami
People who spend time folding paper often ask themselves questions that are ultimately
mathematical in nature. Is there a simpler procedure for folding a certain figure? Where on the
original square paper do the wings of a crane come from? Why do so many origami figures start
with square paper? What size paper should I use to make a chair to sit at the origami table I
already made? What words should I use to teach people to make a jumping frog? Is it possible to
make an origami beetle that has six legs and two antennae from a single square sheet of paper? Is
there a precise procedure for folding a paper into 5 rectangular strips? Which polyhedra can be
constructed using Sonobe modules and what do they have in common?
Anyone who has practiced origami has probably, at one time or another, unfolded an
origami model and marveled at the intricate crease pattern which forms the "blueprint" of the fold.
Clearly there are some rules at play in these collections of creases. Clearly there is an origami
geometry at work when paper is folded.
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Such geometry, this mathematics of origami, has been studied extensively by origamists,
mathematicians, scientists and artists. The Italian-Japanese mathematician Humiaki Huzita has
formulated a list of axioms to define origami geometrically. Physicist Jun Maekawa has
discovered some fundamental theorems about origami and used them to design origami models of
surprising elegance. Mathematician Toshikazu Kawasaki has a number of origami theorems to his
name and has even generalized some of them to describe paper folding in higher dimensions.
(Origami in the fourth dimension!) Robert Lang of California has developed an ingenious way to
algorithmatize the origami design process, using a computer to help him invent models of
amazing complexity. Educator Shuzo Fujimoto and artist Chris Palmer have discovered amazing
parallels between origami and tessellations. And an enormous number of teachers have been
developing ways to use origami to teach concepts in math, chemistry, physics and architecture
The Challenge Problems section gives some more mathematical origami questions that
you might like to think about. This section contains challenge questions about mathematical
origami and links to other sources of questions. I know the answers to some of these questions.
Some of the questions are open, however. This means that you could be the first person to ever
work on them!
Most people don't realize how much information there is on the subject of mathematics
and origami. There are books and papers published on the subject. In fact, many of the origami
creators and authors are mathematicians, physicists, and other scientists. I am thinking of people
like Robert Lang, Jun Maekawa, Toshikazu Kawasaki, and Thomas Hull. Another origami math
figure with no formal mathematical training is Chris Palmer, the creator of the flower tower.
I hope this report gives you a good overall view of the topic of origami and mathematics
Try it! This time, fold a model, unfold it, and color the crease pattern regions with red and
blue. Now, refold the model and see for yourself!
Reference:
1. Alton T. Olson; Mathematics through Paper Folding; NCTM
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