When we hear the word origami, we probably first imagine a paper model in the shape of an animal or a boat. These simple shapes often serve as the first step into the fascinating world of paper folding. But there are also origami that take several hours or even days of skilful work.
What is fascinating, is that the principle of folding as we know it from origami is also found in the nature around us and in ourselves. A flower’s bud gradually opening up, the gently unfolding wings of insects hidden under elytron (hardened forewings), the complex structure of human DNA, or the walls of the large intestine, all use principles similar to those found in origami. The ability of materials and structures to “fold” and “unfold” as needed is one of the basic building blocks of life.
In recent years, origami has experienced a real boom, not only as a hobby for paper lovers, but also in many artistic and scientific fields. It has influenced architects, furniture designers, artists and scientists. Origami has thus transformed from a mere art of paper folding into a tool that helps shape the future.
Let’s start simply with a strip of paper. Try to create folds on a strip of paper so that it can be folded into a plane and at the same time to get the short ends of the strip of paper together after a few steps (for now it is enough approximately, later we will want the shorter ends to be exactly connected to each other). At the same time, the long edges of the strip must not cross arbitrarily. One long edge of the strip must be visible along its entire length after the shape has been folded. Therefore, the other long edge of the strip of paper will logically be down along its entire length, i.e. adjacent to the flat base on which the resulting shape lies. The following figure shows examples of both incorrect and correct solutions.
Exercise 1. Try folding the correct solution from the previous figure from a strip of paper and create several other variations so that: 1) the strip can be folded flat and the folds are repeated regularly, 2) its shorter edges approximately connect to each other, 3) one of its longer edges is fully visible when viewed from above
Solution. Some possible solutions (including a sample) are shown in the following figure.
If we unfold the paper strips again, we will see the following (see figure).
Exercise 2. For the previous types of solutions, determine the exact pattern of folds of the paper strip so that the two shorter ends meet exactly and can be glued together.
Solution. As usual in origami, we will mark the ridges fully and the valleys in dashed lines. If we draw all the variants in a folded shape the solution is obvious in most cases and follows from the regularity and symmetries that apply in the pattern. The solutions together with the indicated angles are shown in the following figure. The sizes of the remaining angles are obvious from knowledge of supplementary and alternate angles.
However, the situation is more complicated in the third variant (in the previous photos it is the yellow paper version). For the indicated angle \(\beta\) it holds
\[\sin\beta=\frac{v}{c},\]
where \(v\) is the height of the paper strip. If we fix this height, there are still infinitely many solutions depending on the angle \(\beta\), where \(\beta\in(0; 180^{\circ}-45^{\circ})\). Two variants for the angle \(\beta=30^{\circ}\) and \(\beta=15^{\circ}\) are shown in the following figure.
We can also lay the individual strips on top of each other, if we do it in a suitable way, the pattern can be folded flat again. Examples of two variants are in the picture.
All the variants created from by stacking the strips from the previous solution can be found on the worksheet 1 and 2 (if you want to glue the sides, you just need to add overlaps for gluing). The following figure shows the variants from the worksheets folded and glued.
These and other origami “tubes” find application in medicine, design or materials engineering. We will focus on the application of origami in medicine in the next paragraph. Although we will not solve any problems in it, we will introduce another interesting application of origami.
A medical stent is a small, usually mesh-like tube that is inserted into a narrowed or blocked blood vessel or other passageway in the body to help keep it open and allow proper blood or fluid flow. These devices are commonly used when blood flow is obstructed due to narrowing of the arteries. The primary goal of stent placement is to restore normal blood flow.
Stents must be strong enough to withstand the forces in the human body, but also flexible enough to conform to the shape of the vessel.
Recent innovations include the development of stents inspired by the water bomb design, which is known for its excellent expansion properties and flexibility. This design aims to improve the compliance of stents in vessels and represents an innovative approach to minimally invasive cardiovascular interventions. This design allows for a compact stent configuration for deployment and subsequent controlled and uniform expansion after deployment, which increases safety and effectiveness in restoring proper blood flow.
In the third worksheet you will find a pattern for making this model. I recommend this only for true paper folding enthusiasts.