Paper folding has gained significant value in science over the last fifty years. With straight and single folds and their intersections, lines and points in the model of the Euclidean plane are defined, but origami is not only associated with conventional Euclidean geometry. It helps us solve various problems that stem also from areas like algebra, number theory, projective geometry, analysis, and many others. In this paper, we will define the set of origami numbers and constructions that can be performed through paper folding. We will examine how origami can help solve problems where Euclidean tools are powerless, focusing particularly on the question of construction of regular heptagon, angle trisection and the construction of the distance $\sqrt[3]{2}$. We will get to know Haga's theorems, fold tangents to conics, solve cubic and quartic equations, and finally, from various perspectives, learn about Alhazen's optical problem and the origami construction of its solution with help of dual conics.
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