(International Society of the Arts, Mathematics, and Architecture, 2000) Knoll, Eva
Deltahedra are polyhedra with all equilateral triangular faces of the same size. We consider a class of
we will call ‘regular’ deltahedra which possess the icosahedral rotational symmetry group and have
either six or five triangles meeting at each vertex. Some, but not all of this class can be generated using
operations of subdivision, stellation and truncation on the platonic solids. We develop a method of
generating and classifying all deltahedra in this class using the idea of a generating vector on a
triangular grid that is made into the net of the deltahedron.
We observed and proved a geometric property of the length of these generating vectors and the surface
area of the corresponding deltahedra. A consequence of this is that all deltahedra in our class have an
integer multiple of 20 faces, starting with the icosahedron which has the minimum of 20 faces.