“Balloon twisting is fun” – and may also have industrial and military applications in the construction of so-called Air Beams – but some might say that the mathematics behind balloon twisting has, until recently, been under represented in the academic literature.
To counter this, the authors of the paper ‘Computational Balloon Twisting: The Theory of Balloon Polyhedra’ introduce the idea of ‘Bloons’ – which are mathematical idealisations of real-world balloons (see doggie balloon and its associated bloons above). The research not only provides “…algorithms to find the fewest balloons that can make exactly a desired graph or, using fewer balloons but allowing repeated traversal or shortcuts, the minimum total length needed by a given number of balloons.” But also, in contrast, shows that “…NP-completeness of determining whether such an optimal construction is possible with balloons of equal length.” Although the paper makes great strides towards a full theory of balloon twisting there are still at least two outstanding enigmas.
 It remains open to find interesting polyhedra that fail to have a twisting from a bloon number of equal-length bloons but not by virtue of indivisibility.
 It also remains open whether some symmetric polyhedron can be twisted only from nonidentical units or only from units arranged asymmetrically.
The paper,‘Computational Balloon Twisting:The Theory of Balloon Polyhedra’ was published as part of the Proceedings of the 20th Canadian Conference on Computational Geometry (CCCG 2008), Montréal, Québec, Canada, August 2008.
Also see : Co-author Vi Hart’s extensive webpages on Mathematical Balloon Twisting
BONUS: Co-author Erik Demaine is a member of the Luxuriant Flowing Hair Club for Scientists (LFHCfS).