While most of us are comfortable creating a paper crane, the prospect of solving a complex mathematical equation may seem daunting to many.
Yet, these two activities are more similar than they appear, both requiring precision, algorithmic thinking, a sense for shape, and an eye for patterns and symmetry.
As a mathematician with a passion for origami, I relish the opportunity to introduce mathematical concepts through the art of paper folding. Origami is steeped in mathematical principles, offering a creative and engaging way to explore these ideas.
Modular Origami: The Building Blocks of Creation
My personal favorite within the origami world is modular origami, a technique where multiple folded paper “units” are combined to form larger, often symmetrical, structures.
These units are usually simple to fold, but the real mathematical challenge lies in assembling them into a cohesive pattern, revealing the underlying structure and beauty.
Many modular origami patterns follow a similar approach to unit assembly, making it easy to explore a wide variety of models with relatively little effort.
My website, Maths Craft Australia, offers a range of modular origami patterns alongside other crafts, all of which can take you on intriguing mathematical journeys without any prior mathematical knowledge.
Constructing 3D Shapes from 2D Units
In the realm of mathematics, the most symmetrical shapes are known as the Platonic solids. These shapes, named after the Greek philosopher Plato, are 3D forms composed of identical regular 2D shapes, such as equilateral triangles, squares, and pentagons.
Despite the infinite variety of regular polygons, there are only five Platonic solids:
- The tetrahedron (four triangles)
- The cube (six squares)
- The octahedron (eight triangles)
- The dodecahedron (12 pentagons)
- The icosahedron (20 triangles)
To create these Platonic solids in origami, it’s best to start by learning the “sonobe unit.”
The Sonobe Unit: A Key to Mathematical Origami
The sonobe unit, a simple yet versatile origami shape, resembles a parallelogram with two flaps folded behind it.
Instructions for creating a sonobe unit can be found on my website, and numerous online videos provide visual guidance.
Six sonobe units are needed to construct a cube, 12 for an octahedron, and 30 for an icosahedron. (Note: It’s not possible to create a tetrahedron or dodecahedron using sonobe units.)
Detailed instructions for building a cube are also available on my website, and a quick online search will yield instructions for more complex models.
Diving Deep into Mathematical Origami
Once you’ve grasped the basic structure of each 3D shape, you might find yourself, like others, delving into deeper mathematical inquiries.
Can you arrange sonobe units so that no two units of the same color touch if you only have three colors?
Are there possibilities for even larger symmetric shapes?
Do relationships exist between different 3D shapes?
A single innocent question can open the door to a world of mathematical exploration.
Inquiries about coloring lead to graph theory and network mathematics, while questions about larger models introduce you to the Archimedean solids and Johnson solids, which possess high symmetry but not as much as the Platonic solids.
For a truly mind-expanding experience, you might encounter higher-dimensional symmetric shapes.
Conversely, some researchers are using mathematical frameworks to explore new origami possibilities, turning the traditional approach on its head.
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