Origami is a Japanese art of folding paper. Mathematical origami is a special type of origami in which a sheet of paper (a plane model) is folded and then the resulting mathematical attributes of the objects in the plane (straight lines, points) are studied. It was named Origamics by Kazuo Haga and has been proved to be a stronger mathematical tool than an unmarked ruler and a pair of compasses. For example, both the duplication of the cube problem and the angle trisection were solved by folding paper since cubic equations can also be solved in the same manner. The aim of the research project is to set up and prove as many mathematical hypotheses (challenges) originating from the folding of equilateral triangle as possible. More than 30 challenges have been proved in the research.