Wulffman: Crystal structure visualization


  • Date: 1997
  • Roles played:  User interface designer and developer
  • Development team: Ryan McCormack (Tcl/Tk front-end); Andrew Roosen (C++ back-end); W. Craig Carter (Theory)
  • Development platform: UNIX (IRIX 5.3, SunOS 4.1.3, IBM AIX 3.2)
  • Development language: Interface – Tcl/Tk; Back-end – C++;
  • Download: This software can be download at the NIST Center for Computational Materials Science
  • Samples: See sample crystal shapes and animations below


Wulffman is a program for interactively examining the Wulff shapes of crystals with specified symmetries. The Wulff shape is the shape that possesses the lowest surface energy for a fixed volume, and hence represents the ideal shape that the crystal would take in the absence of other constraints. For a periodic crystal, i.e. one that can be generated by periodic repetition of a simple unit cell, the Wulff shape must be consistent with the crystallographic point group symmetry of the underlying crystal. The point group is simply the set of all point isometries (rotations, roto-inversions, and reflections) that leave the environment around a point unchanged. In the most general case, the Wulff shape will be a convex polyhedron whose faces (facets) correspond to crystal planes that are low in energy.

Wulffman requires the point group symmetry of a crystal of interest, a set of crystal planes (facets), and their respective surface energies. From this information, Wulffman constructs the Wulff shape and sends its output to Geomview, a 3-D visualization tool. Surface energies can be changed at will, and Wulffman reconstructs the lowest energy polyhedron. In this manner, the user can directly and easily visualize the effects of surface energy anisotropy on the equilibrium forms of crystals.

Sample crystal shapes (images)

Crystal system: Icosahedral
Point Group: 235
Facets: [111], [1 1.62 0]
Energies: 1.0, 1.02
Wulff shape: Truncated icosahedron

Description: Carbon-60 molecules with this shape were discovered in the late 80s and named ‘Buckminsterfullerene’ (after Buckminster Fuller). Buckyballs, and the more general class of Fullerenes, have generated a huge amount of interest in the scientific community. The men who first synthesized and studied these molecules were awarded the Nobel prize.

Crystal system: Cubic
Point group: m3_m
Facets: [100] (Energy = 0.85)
Boundary polytope: 500 facets
Wulff shape: Cube + sphere

Description: In the absence of a bounding polytope, [100] facets under cubic symmetry generate a cube Wulff shape. By adding an isotropic boundary polytope with a slightly higher energy, the corners and edges of the cube are cut off and replaced by smooth surfaces. The Wulff shape is effectively the intersection of a sphere and a cube.

Mystery shape!
Symmetry: Icosahedral
Animation: See below
Under the right conditions, a crystal of Sulfur might look something like this shape. It has point group symmetry mmm.
Crystal system: Trigonal
Point Group: 3_m
Facets: [14n], n=6-12
Energies: 0.65-0.48
Wulff shape: Modified Hexagonal Scalenohedron

Animated crystal shapes

Octahedron > Truncated Octahedron > Dodecahedron (GIF, 168K)

Crystal system: Cubic
Point group: m3_m
Transformation: The beginning structure is an octahedron generated by [111] facets. [100] facets are added, and their surface energy is lowered until the truncated octahedron (cuboctahedron) results. [110] facets with low energy are included, and as their energy is decreased relative to [100] and [111], the dodecahedron results.

Stars and Icosahedra Forever (GIF, 264K)

Crystal system: Icosahedral
Point group: 235
Transformation: A general icosahedral form with 60 [132] facets is generated. [111] facets are included and their energy is decreased until the regular icosahedra results.  

Spinning isocube (GIF, 328K)
Animation of the ‘honeycomb’ shape shown above. The animation shows rotation of the Wulff shape.
Wulff pencil (GIF, 144K)
An animated Wulff pencil!
Mystery shape (GIF, 260K)
Animation of the icosahedral mystery shape shown above.