This is a very short post on the papasan chair.
http://en.wikipedia.org/wiki/Papasan_chair
It is striking that there are two design elements in a papasan chair (ignoring the base and the cushion): the circular outer frame and the 'spiral' inner part.
The circle is important because it is the curve which, for a given length, maximizes the enclosed area (known as Queen Dido's problem, which was proved by the calculus of variations by Newton):


galileo.phys.virginia.edu/classes/321.jvn.fall02/var_meth.pdf


http://mathematicalgarden.wordpress.com/2008/12/21/the-problem-of-dido/

The spiral is the most compact curve in the sense - within a given area - it covers the greatest area for a given length, and thus provides maximum support.

However, there is some ambiguity because there are many possible spiral curves:

http://en.wikipedia.org/wiki/Spiral

Logarithmic spiral search patterns are known to be most efficient in 2-D although mathematicians are still arguing about them:
 "On the Optimality of Spiral Search" Elmar Langetepe
SODA 2010: Proc. 21st Annu. ACM-SIAM Symp. Disc. Algor., 2010, pp. 1-12

www.i1.informatik.uni-bonn.de/publications/l-oss-09.pdf

Yes, these arguments also neglect the 6 radial spokes, and the fact that the spiral is really a helix because it's
in 3-D.
Agreed: these two arguments are not watertight, and the spiral part may even be wrong, but anyway:
The papasan chair design optimally uses material by its combination of circular and spiral elements.