Examples

Wulffman is a versatile tool capable of illustrating an infinite variety of crystal shapes. A few examples are given below that demonstrate some of the basic program features listed above.

• Simple Wulff Shape Construction
  
  
  1. The Simple Cube :
     Crystal system: Cubic
     Point Group: Any! Facets: [100]
     Energy: 1.0
     Wulff shape : Cube
     
     
  2. Truncated Octahedron :
     Crystal systems: Cubic
     Point Group: m3m (for example)
     Facets: [100] and [111]
     Energies: 1.0 and 0.85
     Wulff shape : Cuboctohedron
     
     
  3. A hexagonal Pencil :
     Crystal system: Hexagonal
     Point Group: 6/mmm
     Facets: [431], [100], [210]
     Energies: 1.0, 0.86, 0.91
     Wulff shape : Modified Dihexagonal Dipyramid
     
     
  4. A trigonal Lozenge :
     Crystal system: Trigonal
     Point Group: 3_m
     Facets: [14n], n=6-12
     Energies: 0.65, 0.59, 0.55, 0.52, 0.50, 0.485, 0.48
     Wulff shape : Modified Hexagonal Scalenohedron
     
     
  5. Buckyballs! :
     Crystal system: Icosahedral
     Point Group: 235
     Facets: [111], [1 1.62 0]
     Energies: 1.0, 1.02
     Wulff shape : Buckyball
     
     
  6. Custom Flying Saucer :
     Crystal system: Custom
     Point Group: 37-fold roto-inversion around [001]
     Facets: [148], [100], [217], [124]
     Energies: 0.85, 1.63, 0.87, 0.85
     Wulff shape : Flying saucer (?)
     
     
• Dynamic Wulff Shapes : Surface energy anisotropy
  1. Octahedron --> Truncated Octahedron --> Dodecahedron:
     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.
     Graphics: Animated GIF (170k)
     
     
  2. Stars and Icosahedra Forever:
     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.
     Graphics: Animated GIF (270k)
     
     
• Unique Crystal Planes:
  Crystal system: Cubic
  Point group: 432
  Facets:
  Unique ("slice") plane: [112]
  Normal facets: [149], [216], [100]
  Description: Decreasing the energy of the unique plane [112] progressively slices off more and more of the Wulff shape. If the [112] facet had not been unique, 24 equivalent planes would have been generated.
  Graphics: Still GIF (11k) and Animated GIF (308k)
  
  
• Isotropic Surfaces Crystal system: Cubic
  Point group: m3_m
  Facets: [100] (Energy = 0.85)
  Boundary polytope: 500 facets, Skew = 0, Energy = 1.0
  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 end 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.
  Graphics: Still GIF (30k) and animated GIF (335k)
  
  
• Naturally-Occurring Materials
  The following are examples of Wulff shapes found to occur in nature (from Elementary Crystallography , M. Buerger, MIT Press, 1956):
  • Sulfur: An example of development in class mmm
  • Eglestonite : Hg4OCl2, an example of form development in class 43_m.
  • Beryl : Be3Al2Si6O18, Beryllium Aluminum Silicate, a semi-precious mineral, represents form development in class 6mm.
  • Rutile : TiO2, Titanium Dioxide, an example of form development in class 4mm.