Frustration and Curvature: the Orange Peel Carpet

The Orange Peel Carpet

Orange peels are curved: a circular piece of orange has a shorter edge than a pancake of the same size. For an orange, trying to become flat is frustrating: the bigger the piece of orange, the more stretching is needed to flatten it. (This is a practical problem in making flat maps of the curved earth.) Eventually, the orange has to split. The gap can be filled by more orange and the edges sewn together without flaw, but there remains a point of high strain: the edge of the cut is a topological defect. At right is a piece of Pamela Davis Kivelson's photograph at (copyright Pamela Davis Kivelson) (also used in the poster What is Scientific Truth?).

The Frustrated Icosahedron

Packing circles in two dimensions is easy: six circles perfectly fit around a central one to form a hexagon, and this pattern can be continued to cover the whole plane. The triangle formed by three circles fits together into a unstrained hexagon.

Packing spheres in three dimensions is more subtle. Twelve spheres surrounding a center one rattle around a bit. Four spheres form a tetrahedron, but twelve tetrahedra won't quite fit together into an icosahedron. The photo at right is of "Frustration and Curvature", a sculpture built by Pam and a group of physicists: Daniel Rokhsar (Berkeley), me (Cornell), Steven Kivelson (Stanford), and several others (copyright Pamela Davis Kivelson).

Orange peel carpets are a metaphor for many materials in nature. These materials are frustrated too: their local low-energy structures can't be continued to fill space. The blue phases and twist-grain boundary phases of liquid crystals, the exotic Frank-Kasper phases, metallic glasses, spin glasses, superconductors in magnetic fields, and spinning superfluids all are frustrated and all relieve their frustration with regular or irregular arrays of topological defects.

This research was paid for by THE US GOVERNMENT by the NSF.


The Blue Phases.
Pamela Davis Kivelson' web sites, (photograph), and Science and Art #18.
The icosahedron travels as a disassembled set of tetrahedra. It's held together with rubber bands and paper clips. See the animation How do you Build a Frustrated Icosahedron? (thanks to Michael W. Conner and Pamela Davis).

Last modified: March 8, 2002

James P. Sethna,

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).