Origami, the Japanese art of paper folding, is much more than a decorative or creative activity. Behind each figure lies a world of shapes, symmetries, angles and geometric transformations. Each fold involves a mathematical decision, and each model, however simple or complex, responds to regular patterns that can be analysed from the perspective of plane geometry and spatial geometry.
Many of us have tried to fold a crane, a paper aeroplane or a flower. Although it may seem simple, we soon discover that following the instructions requires precision and spatial understanding. A poorly made fold can prevent the final figure from forming correctly. In schools, workshops and cultural centres, origami is used not only as an art form, but also as an educational tool to teach geometry in a tangible and fun way.
Origami is based on a series of basic geometric operations: folding diagonally, perpendicularly, symmetrically or radially. These folds create lines that divide the surface of the paper into triangles, quadrilaterals and other shapes. As more folds are made, complex angular relationships appear and new shapes emerge.
Key concepts present in origami include:
- Lines of symmetry: many shapes are constructed from bilateral or rotational symmetry.
- Geometric transformations: translations, rotations, reflections, and dilations are performed through each fold.
- Interior angles: adjusting each fold requires understanding how angles add up or distribute at a point.
- Tessellations: some advanced techniques create repetitive patterns that cover surfaces, such as the tessellations of Robert Lang or Eric Gjerde.
There are even formal axioms of origami, known as the Huzita-Hatori Axioms, which define with mathematical precision what can be achieved with folds.
In addition to its artistic and educational value, origami has found surprising applications in science and engineering. Researchers have used folding principles to design:
- Foldable solar panel systems for satellites.
- Medical implants that unfold inside the human body.
- Flexible electronic devices and portable architectural structures.
Origami demonstrates that geometry is not just a discipline for solving abstract problems, but a source of practical and innovative solutions. Its beauty lies in the transformation of a flat sheet of paper into a complex three-dimensional figure, using only the power of folding.
Folding paper, in this context, is a way of thinking, a way of creating, and a way of understanding space. And all from a simple square sheet of paper.
To see how origami relates to mathematics and science, and how to do differente geometric figures I recommend these videos:
https://youtube.com/playlist?list=PLm3p4rsV0UlqD_suy5KtBs8G_RB2mjtwm&si=j3ecZh40VcZXt_Iv
References:
Lang, R. J. (2017). Origami Design Secrets: Mathematical Methods for an Ancient Art. A K Peters/CRC Press
Hull, T. C. (2002). Origami Mathematics. In MAA Notes: Using Origami to Teach Geometry (Vol. 55, pp. 75–87). Mathematical Association of America
Hatori, K. (2001). Mathematical Origami and the Huzita–Hatori Axioms. Journal of Origami Science and Technology, 4(1), 29–33.
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