Imagine a space telescope getting ready to take a picture, but the bright light from a nearby star is in the way. Luckily, the telescope has a clever trick up its sleeve: a huge shield that blocks the glare. This shield, called a starshade, is about 35 meters wide but can fold down to just under 2.5 meters, making it small enough to fit on a rocket. The secret to its compact design comes from an ancient art form called origami, which means “folding paper” and has been practiced in Japan since at least the 17th century.
Origami starts with simple ideas that can lead to amazing creations. You can make a paper crane with about 20 steps or a dragon with over 1,000 steps, and even something as complex as a starshade. With just a single square sheet of paper, you can fold it into almost any shape. When you unfold it, you’ll see a pattern of lines, each showing a concave valley fold or a convex mountain fold. Origami artists use these folds to create crease patterns, which are like blueprints for their designs.
Most origami models are three-dimensional, but their crease patterns are designed to fold flat without adding new creases or cutting the paper. The math behind flat-foldable crease patterns is simpler than that for three-dimensional ones, making it easier to create a two-dimensional design and then shape it into a three-dimensional form.
There are four important rules for flat-foldable crease patterns:
The crease pattern must be two-colorable, meaning you can fill the areas between creases with two colors so that areas of the same color don’t touch. Adding another crease can mess up this two-colorability.
The number of mountain and valley folds at any interior vertex must differ by exactly two.
If you number all the angles at an interior vertex in a clockwise or counterclockwise direction, the even-numbered angles must add up to 180 degrees, and the odd-numbered angles must do the same. If a new crease changes this, the model won’t fold flat.
A layer cannot go through a fold.
A two-dimensional, flat-foldable base often represents a final three-dimensional shape. Understanding how crease patterns relate to two-dimensional bases and three-dimensional forms helps origami artists design complex shapes. For example, origami artist Robert J. Lang created a crease pattern that allocates areas for a creature’s legs, tail, and other parts. When folded into a flat base, each area becomes a separate flap. By shaping these flaps, the two-dimensional base turns into a three-dimensional scorpion.
If you wanted to fold seven flowers from the same sheet of paper, you could duplicate the flower’s crease pattern and connect them in a way that follows all four rules, creating a tessellation—a repeating pattern of shapes that covers a plane without gaps or overlaps. The ability to fold a large surface into a compact shape has many uses, from space to the tiny world of our cells.
Using origami principles, medical engineers have redesigned traditional stent grafts, which are tubes used to open and support damaged blood vessels. By using tessellation, the rigid tube can fold into a compact sheet about half its expanded size. Origami ideas have also been used in airbags, solar panels, self-folding robots, and even DNA nanostructures. Who knows what amazing possibilities will come next?
Explore the world of origami by creating your own crease patterns. Start with a simple square sheet of paper and try to design a flat-foldable model. Use the four rules of flat-foldable crease patterns to guide your design. Share your creation with the class and explain how you applied each rule to ensure your model folds flat.
Take a crease pattern and attempt to color it using only two colors, ensuring that no two adjacent areas share the same color. This activity will help you understand the concept of two-colorability. Discuss with your classmates how adding or removing creases affects the two-colorability of your pattern.
Choose a vertex from a crease pattern and measure the angles around it. Verify that the even-numbered angles sum to 180 degrees and the odd-numbered angles do the same. Experiment by adding a new crease and observe how it affects the angle sums. Present your findings to the class.
Design a tessellation using a simple origami flower pattern. Duplicate the pattern and connect them to form a repeating design that covers a plane without gaps or overlaps. Ensure your tessellation follows all four rules of flat-foldable crease patterns. Display your tessellation and explain the process you used to create it.
Research a real-world application of origami, such as its use in medical devices or space technology. Create a presentation that explains how origami principles are applied in your chosen example. Discuss the benefits and challenges of using origami in this context and share your insights with the class.
Here’s a sanitized version of the provided YouTube transcript:
—
As the space telescope prepares to take a photo, the light from a nearby star obstructs its view. However, the telescope has a solution: a large shield designed to block the glare. This starshade has a diameter of approximately 35 meters but can fold down to just under 2.5 meters, making it small enough to be carried on the end of a rocket. Its compact design is inspired by an ancient art form: origami, which translates to “folding paper” and is a Japanese practice that dates back to at least the 17th century.
In origami, simple concepts can lead to a variety of creations, from a paper crane with about 20 steps to a dragon with over 1,000 steps, and even a starshade. A single, traditionally square sheet of paper can be transformed into almost any shape through folding. When unfolded, the sheet reveals a pattern of lines, each representing either a concave valley fold or a convex mountain fold. Origami artists arrange these folds to create crease patterns that serve as blueprints for their designs.
While most origami models are three-dimensional, their crease patterns are typically designed to fold flat without introducing new creases or cutting the paper. The mathematical rules governing flat-foldable crease patterns are simpler than those for three-dimensional patterns, making it easier to create an abstract two-dimensional design and then shape it into a three-dimensional form.
There are four rules that any flat-foldable crease pattern must follow. First, the crease pattern must be two-colorable, meaning the areas between creases can be filled with two colors so that areas of the same color do not touch. Adding another crease can disrupt this two-colorability. Second, the number of mountain and valley folds at any interior vertex must differ by exactly two.
The third rule states that if we number all the angles at an interior vertex in a clockwise or counterclockwise direction, the even-numbered angles must sum to 180 degrees, as must the odd-numbered angles. If a crease is added and the new angles do not meet this requirement, the model will not fold flat. Finally, a layer cannot penetrate a fold.
A two-dimensional, flat-foldable base often serves as an abstract representation of a final three-dimensional shape. Understanding the relationship between crease patterns, two-dimensional bases, and the final three-dimensional form enables origami artists to design intricate shapes. For example, this crease pattern by origami artist Robert J. Lang allocates areas for a creature’s legs, tail, and other appendages. When folded into a flat base, each allocated area becomes a separate flap. By narrowing, bending, and sculpting these flaps, the two-dimensional base transforms into a three-dimensional scorpion.
Now, if we wanted to fold seven of these flowers from the same sheet of paper, we could duplicate the flower’s crease pattern and connect them in a way that satisfies all four rules, creating a tessellation—a repeating pattern of shapes that covers a plane without gaps or overlaps. The ability to fold a large surface into a compact shape has applications ranging from the vastness of space to the microscopic world of our cells.
Using principles of origami, medical engineers have reimagined traditional stent grafts, which are tubes used to open and support damaged blood vessels. Through tessellation, the rigid tubular structure can fold into a compact sheet about half its expanded size. Origami principles have also been applied in airbags, solar arrays, self-folding robots, and even DNA nanostructures—who knows what possibilities will emerge next?
—
This version maintains the original content while removing any unnecessary or potentially sensitive details.
Math – The study of numbers, quantities, shapes, and patterns and the relationships between them. – In math class, we learned how to calculate the area of different shapes.
Origami – The Japanese art of folding paper into decorative shapes and figures. – We used origami techniques to create paper cranes and other intricate designs.
Folds – The bends or creases made by folding something, often used in origami to create shapes. – The folds in the paper allowed us to transform a flat sheet into a three-dimensional figure.
Patterns – Repeated decorative designs or sequences, often used in art and mathematics to create order and predictability. – The quilt was made up of colorful patterns that repeated every few inches.
Shapes – Forms or outlines of objects, which can be two-dimensional or three-dimensional. – In geometry, we studied various shapes like circles, triangles, and squares.
Two-dimensional – Having length and width but no depth, existing on a flat plane. – A square is a two-dimensional shape because it only has length and width.
Three-dimensional – Having length, width, and depth, existing in space. – A cube is a three-dimensional shape because it has length, width, and height.
Tessellation – A pattern made of identical shapes that fit together without any gaps or overlaps. – The artist created a tessellation using hexagons that covered the entire wall.
Angles – The space between two intersecting lines or surfaces, measured in degrees. – We used a protractor to measure the angles in the triangle.
Designs – Plans or drawings produced to show the look and function of an object or work of art before it is made. – The architect’s designs included detailed drawings of the building’s layout.
