text: In 1878 Josiah Willard Gibbs proposed that a droplet or crystal will arrange itself such that its surface Gibbs free energy is minimized by assuming a shape of low surface energy. He defined the quantity
  Here  represents the surface (Gibbs free) energy per unit area of the th crystal face and  is the area of said face.  represents the difference in energy between a real crystal composed of  molecules with a surface and a similar configuration of  molecules located inside an infinitely large crystal. This quantity is therefore the energy associated with the surface. The equilibrium shape of the crystal will then be that which minimizes the value of .
  In 1901 Russian scientist George Wulff stated (without proof) that the length of a vector drawn normal to a crystal face  will be proportional to its surface energy : . The vector  is the "height" of the th face, drawn from the center of the crystal to the face; for a spherical crystal this is simply the radius. This is known as the Gibbs-Wulff theorem.
  In 1943 Laue gave a simple proof, which was extended in 1953 by Herring with a proof of the theorem and a method for determining the equilibrium shape of a crystal, consisting of two main exercises. To begin, a polar plot of surface energy as a function of orientation is made. This is known as the gamma plot and is usually denoted as , where  denotes the surface normal, e.g., a particular crystal face. The second part is the Wulff construction itself in which the gamma plot is used to determine graphically which crystal faces will be present. It can be determined graphically by drawing lines from the origin to every point on the gamma plot. A plane perpendicular to the normal  is drawn at each point where it intersects the gamma plot. The inner envelope of these planes forms the equilibrium shape of the crystal.
  The Wulff construction is for the equilibrium shape, but there is a corresponding form called the "kinetic Wulff construction" where the surface energy is replaced by a growth velocity. There are also variants that can be used for particles on surfaces and with twin boundaries.