More relaxing and artistic Wednesdays. Today, I wanted to show you a couple of incredible origami pieces and also to tell you a little about the mathematics behind this type of art. The practice and study of origami has very interesting mathematical applications and problems in it. Also, I have to state that all the images are by Robby Kraft. I have chosen the more mathematical pieces for this post, but you can find out more on his Instagram page.
One of the mathematical problems encountered in origami is the problem of flat-foldability (whether a crease pattern can be folded into a 2-dimensional model). Most of the time the construction of origami models is sometimes shown as crease patterns. The major question about such crease patterns is whether a given crease pattern can be folded to a flat model, and if so, how to fold them; this is an NP-complete problem; you know how important and hard this problems are. Related problems when the creases are orthogonal are called map folding problems.
Moreover, origami can be used to construct various geometrical designs not possible with compass and straightedge constructions. For instance paper folding may be used for angle trisection and doubling the cube. These problems are stated to be impossible to solve.
Angle trisection is a classical problem of compass and straightedge constructions of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge, and a compass. The problem as stated is generally impossible to solve, as shown by Pierre Wantzel in 1837. Wantzel’s proof relies on ideas from the field of Galois theory—in particular, trisection of an angle corresponds to the solution of a certain cubic equation, which is not possible using the given tools. Note that the fact that there is no way to trisect an angle in general with just a compass and a straightedge does not mean that there is no trisectible angle: for example, it is relatively straightforward to trisect a right angle (that is, to construct an angle of measure 30 degrees).
Doubling the cube is another ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first, using only the tools of a compass and straightedge. The Egyptians, Indians, and particularly the Greeks were aware of the problem and made many futile attempts at solving what they saw as an obstinate but soluble problem. However, the nonexistence of a solution was finally proven by Pierre Wantzel in 1837, applying the contemporary development of abstract algebra by Galois.
Don’t forget that maths is everywhere! Enjoy! ~LThMath