The single cut problem It is a geometric mathematical problem that explains how it is possible to obtain some origami plansthat is, complex figures by cutting a sheet along a single line. They are also called floors and their invention dates back to 1721, when the Japanese mathematician Kan Chu sen presented them as stratagems to test mathematical intelligence.
Let’s see how they build and the various forms that can be obtained.
The problem of the single cut and the origami floors
The single cut problem is a mathematical problem who tells us that every figure that is formed only by segments can be obtained bending suitably a sheet several times and then making a single one cut along a single straight line. And the incredible thing is that it is not necessary that the figure is a closed shape, such as the fish you see in the figure below, but it can also be a scheme formed by many parts detached from each other. In short, individuals can be created polygons is concave That convex, polygons disjointed, polygons nestled And even figures that present holes inside them, everything always cut the sheet along a single line!
The scheme seems apparently simple: the aim is to get to coincide with the different segments which make up the perimeter of the or figures that make up our origami, and then cut along their alignment. We can see from the figure above that exist Two different types of lines along which to cut, marked with two types of different strokes: the classic strokes indicate a so -called “upstream fold“, That is, in which the line along which it folds remains high with respect to the sheet that is folded on itself at the bottom, while the point-entry lines indicate the”fold downstream“, in which the fold will be downstream, while the two parts of the sheet will be superimposed upwards.
And here comes the complex part: how to understand what lines must be folded by the sheet? There are several mathematical methods that demonstrate how it is possible and guaranteed to obtain the lines to bend, let’s see the two main ones.
The method of Straight Skeleton and of the disk packing
Before seeing the two methods briefly, let’s start by saying that both are based on very complex and non-intuitive mathematic-geometric explanations to understand. For the most daring, you can find the theorems that make these methods valid at the following link. However, we try to broadly understand what concepts these methods are based.
The Straight Skeleton: fold along the bisectors
The first method, that of the sortTraight Skeleton, generalizes the idea that to match two adjacent segments to match each other, is sufficient Fold the sheet along the bisector of the corner which is formed between the two segments. Clearly this is not enough to obtain our lines, because the corners of the figures are different and the different folds along the bisectors must remain consistent with each other.
For this, lines parallel to the various segments are traced, and the main objective is to always keep them parallel, as if the figure came away to “restricted”. As you can see in the figure, in addition to the bisective lines to the corners, lines appear perpendicular to the various narrowings, which serve to connect the various bisectors together.
The Disk package: Use the circumferences
This method is theoretically a little easier to obtain, because it allows you to perform less lines to obtain our folds, but needs a compass. In a nutshell, to get the folds, we must draw some circumferences with center the leaders of our figure so that the spaces between them have three or four sides. At this point the centers of the adjacent records are combined and the bisectors of the corners that have been formed, as you can see in the figure below.
To be honest, it is not easy to calculate the straight lines that allow us to get the folds, but luckily someone has thought about us! At this link you can find several sheets to print, with which to try to get your origami floors.
