What are the possible surfaces that one can construct from a finite set ot triangles?

Question

I am looking for references in discrete differential geometry for a concept I've been interested in.

It is very common to approximate smooth surfaces using discrete triangulations. I am interested in the opposite problem. I start off with edges of prescribed lengths, which I can use to form triangles, and I want to know which discrete surfaces I can from these triangles (I can use a given triangle more than once).

This is similar to a tiling problem. I have a finite collection of possible tiles, but instead of trying to form shapes in the plane, and I want to know which surfaces I can construct from them.

This sounds like something which is probably already studied in discrete differential geometry, but I am not sure what are the relevant terms/names I need to know in order to google some existing works. Has anyone here ever come across this concept? If so, I'd be happy to hear.

Thanks!

Answer 1

I took a course and take part in a research project concerning simplicial surfaces. This structure sounds pretty familiar to your description. However, this research project rather takes an algebraic/combinatorial route when talking about simplicial surfaces rather than a view from differential geometry. For a taste, here is the first introduction of our lecture notes from the course:

Background: In this course we study surfaces composed of triangles. The Platonic solids have been studied since antiquity. Among these, the surfaces of three of these are made up of triangles, namely the tetrahedron, the octahedron, and the icosahedron. These surfaces are our first and most famous examples of surfaces composed of triangles, namely simplicial surfaces. One feature of these three Platonic solids is also that the triangles of these surfaces are congruent to one particular triangle. We call this the control triangle. In this course we will put particular emphasis on simplicial surfaces for which all triangles are congruent to a control triangle. One motivation why we might be interested in these types of surfaces arises from potential applications: If one triangle in a realisation of this surface is damaged and needs replacing, this is easy to do, as one can stockpile spare copies of the control triangle. Surface triangulations are also ubiquitous in many applications, notably in computer graphics. Here it is important that a simplicial surface ap- proximates a given (smooth) surface as good as possible. To achieve such a good approximation, usually the triangulated surfaces consist of a huge number of individual triangles and these triangles are all different. There is a large body of literature devoted to these types of surface triangulations and there are many powerful computational tools, often using sophisticated numerical methods. In this course we will not deal with the question of approximating a given surface by a simplicial surface. We take the opposite approach and study simplicial surfaces, often together with a control triangle, from a purely combinatorial and algebraic point of view. We endeavor to discover the combinatorial principles guiding such surfaces and to understand their structure. We will find algebraic descriptions of such surfaces and aim to classify those with a small number of faces. We ask ourselves how a surface changes, when we change the congruence type of the control triangle. Our explorations are supported by the package SimplicialSurfaces, which is currently under development and is a package for the computational algebra system GAP.

If you are interested in this subject, I could contact someone who can share the current draft of the book they are writing for this subject. For other information, here is the package for GAP, where one can do a lot of stuff with simplicial surfaces.