Geodesic Folding of Tetrahedron

TL;DR

This paper explores the geometric properties of polyhedra formed by folding a regular tetrahedron along triangular grids, revealing how they can be decomposed into bands and related through common triangular folds.

Contribution

It introduces a novel family of polyhedra generated by folding a tetrahedron and analyzes their decomposition into geodesic bands based on integer parameters.

Findings

Polyhedra can be decomposed into geodesic bands with a number equal to the GCD of two parameters.

Different polyhedra can share a common triangular band despite being formed by different foldings.

The folding process creates a family of polyhedra with specific geometric and combinatorial properties.

Abstract

In this work, we show the geometric properties of a family of polyhedra obtained by folding a regular tetrahedron along regular 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 (homeomorphic to a cylinder). The number of bands is the greatest common divisor of the two numbers. By a proper choice of pairs of numbers, we can create a common triangular band that folds into different multiple polyhedra that belongs to the family.