Quadrilateral tilings for the construction of renzuru origami

Variations of quadrilateral tilings on a plane can be used to construct conjoined origami cranes known as renzuru. Most variations of renzuru are based on the tiling of squares; however, the squares can be modified into certain other quadrilaterals with inscribed circles. In this paper, I examine three types of tilings that enable the folding of renzuru. The first type consists of periodic tilings with congruent quadrilaterals. The results show that there are ten different tilings of congruent quadrilaterals: eight tilings consisting of vertices of degree four and two tilings consisting of vertices of degree three and six. The second and third types are spiral tilings, the second being formed by congruent quadrilaterals and the third consisting of similar quadrilaterals. The second type is tiled with rhombic quadrilaterals. The third type is constructed with lines which divide the infinite plane both equally and radially and a logarithmic spiral curve. GRAPHICAL ABSTRACT