Youtube educator Vihart et al. published this interesting paper entitled "Computational Balloon Twisting: The Theory of Balloon Polyhedra" which you can access on her website. The purpose of said paper, as explained in the abstract, is to construct "a general mathematical and algorithmic theory for balloon-twisting structures, from balloon animals to balloon polyhedra, by modeling their underlying graphs (edge skeleta)." Furthermore, they provide alogorithms for finding the smallest number of balloons necessary to build the desired graph, or allowing for repeated traversals and shortcuts by using even fewer balloons, ascertaining the total minimum length needed by a certain number of balloons.
As the introduction of the paper asserts, balloon-twisting is a form of sculptural artistry, immediately bringing to mind the inextricable relationship between mathematics and artwork. Indeed, an investigation into said artistry, especially the balloon polyhedra, is the impetus behind the paper which acts as a foundation for a new mathematical school of thought: computational balloon twisting. Being a fun activity, balloon-twisting is a great opportunity to provide a simple yet effective means to clarify, not only important topological concepts, but concepts in chemistry, graph theory, Euler tours and much more to an audience spanning a wide age-range. Furthermore, balloons provide a useful heuristic and motivation for architectural structures known as air beams: "Our approach suggests that one long, low-pressure tube enables the temporary construction of inflatable structures, domes, and many other polyhedral structures, which be later reconfigured into different shapes and re-used at different sites."
"Bloons" are idealized, abstract models of balloons, and one of the central problems of computational balloon twisting is determining which graphs, which are constituted by inflated balloon segments their twisted end vertices, are "twistable" under a host of different conditions. Bloons therefore provide a model for testing which graphs can be twisted and in which ways. Hence, as the paper states: "A twisted bloon is stable if every vertex [ie the endpoint of a bloon] is either tied to another vertex or held at a nonzero bending angle." Additionally, there are two models of balloon twisting: (simple) twisting, in which every subsegment of a bloon between its vertices constitute an edge in the graph, representing a segment of the balloon which is inflated, and pop twisting, in which an intervertical subsegment of the bloon is deflated so that it does not appear in the graph.
There is much more in the paper to mull over and for the sake of brevity as well as the hope that the reader will fully read this outstanding paper the article will stop here. It's incredibly impressive just how useful balloons are in concretizing otherwise abstract tenets and making them highly relevant. The heuristic use of balloons in teaching about math and chemistry is not unique to Vihart et al., but they have probably done the most to systemize their use as well as to formulate a new mathematical branch devoted to their use. If you are a math-lover, then this paper will be highly entertaining and, perhaps, eye-opening.