Convex developments of a regular tetrahedron

AbstractThe best-known developments of a regular tetrahedron are an equilateral triangle and a parallelogram. Are there any other convex developments of a regular tetrahedron? In this paper we will show that there are convex developments of a regular tetrahedron having the following shapes: an equilateral triangle, an isosceles triangle, a right-angled triangle, scalene triangles, rectangles, parallelograms, trapezoids, quadrilaterals which are not trapezoids, pentagons and hexagons, and furthermore these cases exhaust all the possibilities of convex developments with sides n⩽6. And we will show that there are no convex n-gons which are developments of a regular tetrahedron when n⩾7. Here, we mean by a development of a polyhedron a connected plane figure, from which one can construct the polyhedron by folding it without getting overlap or gap. In so folding we do not require that the sides of the development should end up as the edges of the polyhedron