Geodesic
Geodesic Folding of Regular Tetrahedron
Journal for geometry and graphics, Tome 26 (2022) no. 1, pp. 81-1 Cet article a éte moissonné depuis la source Heldermann Verlag

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We show geometric properties of a family of polyhedra obtained by folding a regular tetrahedron along triangular grids. Each polyhedron is identified by a pair of nonnegative integers. The polyhedron can be cut along a geodesic strip of triangles to be decomposed and unfolded into one or multiple bands. We show that the number of bands is the greatest common divisor of the two integers. By a proper choice of pairs of numbers, a common triangular band that folds into different multiple polyhedra can be created. We construct the configuration of the polyhedron algebraically and numerically through angular and truss models respectively. We discuss the volumes of the obtained folded states and provide relevant open problems regarding the existence of popped-up state. We also show some geometric connections to other art forms.
Classification : 51M20
Mots-clés : Rigid origami, tetrahedron, geodesic bands, basketry 
@article{JGG_2022_26_1_JGG_2022_26_1_a10,
     author = {S. Nishimoto and T. Horiyama and T. Tachi },
     title = {Geodesic {Folding} of {Regular} {Tetrahedron}},
     journal = {Journal for geometry and graphics},
     pages = {81--1},
     year = {2022},
     volume = {26},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JGG_2022_26_1_JGG_2022_26_1_a10/}
}
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S. Nishimoto; T. Horiyama; T. Tachi . Geodesic Folding of Regular Tetrahedron. Journal for geometry and graphics, Tome 26 (2022) no. 1, pp. 81-1. http://geodesic.mathdoc.fr/item/JGG_2022_26_1_JGG_2022_26_1_a10/