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Journal for Geometry and Graphics 26 (2022), No. 1, 081--100
Copyright Heldermann Verlag 2022

Geodesic Folding of Regular Tetrahedron

Seri Nishimoto
The University of Tokyo, Tokyo, Japan

Takashi Horiyama
Hokkaido University, Hokkaido, Japan

Tomohiro Tachi
The University of Tokyo, Tokyo, Japan

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.

Keywords: Rigid origami, tetrahedron, geodesic bands, basketry.

MSC: 51M20.

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