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
www.neur-on.com (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
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
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
This research was paid for by THE US GOVERNMENT
by the NSF.
The Blue Phases.
- Pamela Davis Kivelson' web sites,
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