Paper bag problem

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Paper bag problem

In geometry, the paper bag problem or teabag problem involves calculating the maximum possible inflated volume of an initially flat sealed rectangular bag which has the same shape as a cushion or pillow, made out of two pieces of material which can bend but not stretch.

The problem is made even more difficult by assuming that the bag is made out of a material like paper or PET film which can neither stretch nor shear.

According to Anthony C. Robin, an approximate formula for the capacity of a sealed expanded bag is:

:$V=w^3 left \left(h/ left \left(pi w ight \right) -0.142 left \left(1-10^ left \left(-h/w ight \right) ight \right) ight \right),$

where "w" is the width of the bag (the shorter dimension), "h" is the height (the longer dimension), and "V" is the maximum volume.

A very rough approximation to the capacity of a bag that is open at one edge is: :$V=w^3 left \left(h/ left \left(pi w ight \right) -0.071 left \left(1-10^ left \left(-2h/w ight \right) ight \right) ight \right)$

(This latter formula assumes that the corners at the bottom of the bag are linked by a single edge, and that the base of the bag is not a more complex shape such as a lens).

The square teabag

In the special case where the bag is sealed on all edges and is square with unit sides, "h" = "w" = 1, and so the first formula estimates a volume for this of roughly:

:$V=frac 1 \left\{pi\right\} - 0.142 cdot 0.9$

or roughly 0.19. According to Andrew Kepert at the University of Newcastle, Australia, the upper bound for this version of the teabag problem is 0.217+, and he has made a construction that appears to give a volume of 0.2055+.

In the article referred to above A C Robin also found a more complicated formula for the general paper bag. Whilst this is beyond the scope of a general work, it is of interest to note that for the tea bag case this formula gives 0.2017, unfortunately not within the bounds given by Kepert, but significantly nearer.

References

*
* cite journal
author=Baginski, F.; Chen, Q.; and Waldman, I.
year=2001
title=Modeling the Design Shape of a Large Scientific Balloon
journal=Applied Mathematical Modelling
volume=25
pages=953–956
doi=10.1016/S0307-904X(01)00024-5

* cite journal
year=2001
title=On the Geometry of the Mylar Balloon
journal=C. R. Acad. Bulg. Sci.
volume=54
pages=39–44

* cite journal
author=Paulsen, W. H.
year=1994
title=What Is the Shape of a Mylar Balloon?
journal=American Mathematical Monthly
volume=101
pages=953–958
doi=10.2307/2975161

* cite journal
author=Anthony C Robin
year=2004
title=Paper Bag Problem
journal=Mathematics today, Institute of Mathematics and its Applications
id=ISSN|1361-2042
volume=June
pages=104–107

* [http://www.ics.uci.edu/~eppstein/junkyard/teabag.html The original statement of the teabag problem]
* [http://maths.newcastle.edu.au/~andrew/teabag/ Andrew Kepert's work on the teabag problem]
* [http://frey.newcastle.edu.au/~andrew/teabag/folding/curvedFold.html Curved folds for the teabag problem]
* [http://www.dse.nl/~andreas/teabag.html A numerical approach to the teabag problem by Andreas Gammel]
* [http://mathworld.wolfram.com/PaperBag.html MathWorld article]

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