Small crystalline precipitates often form in alloys and have near-polyhedral shapes with round edges. Figure 1 is a transmission electron micrograph showing an example of this where the dark regions, which have shapes between a circle and a square, are Co-Cr alloy particles precipitated in a Cu matrix [1,2]. Why such precipitate shapes form has been explained by the anisotropies of physical properties of metals and alloys originating from the crystal structures [2,3]. Both the Co-Cr alloy particles and Cu matrix have cubic structures. The three-dimensional shapes of the particles shown in Figure 1 are intermediate between a sphere and a cube composed of crystallographic planes {100} as indicated by the Miller indices.

Even if the alloy system such as the Co-Cr alloy particles in the Cu matrix is fixed, the precipitate shapes change as a function of the precipitate size [1,2]. In the case of the Co-Cr alloy precipitates, the spherical to cubical shape transition occurs as the precipitate size increases [2,3]. The size dependence of the precipitate’s equilibrium shape determines the shape transitions [2,3]. When we discuss such physical phenomenon, it is convenient to use simple equations that can approximate the precipitate shapes [2,3,4,5]. In the present study, we discuss a simple equation that gives shapes intermediate between a sphere and various polyhedra.

The solid figure described by

     (1)

expresses a sphere with radius R when p = 2 and a cube with edges 2R as p → ∞ [2,3,4]. It is reported in [6] that the 19th century French mathematician Gabriel Lamé first presented this equation. Intermediate shapes between these two limits can be represented by choosing the appropriate value of p > 2. In [2,3,4], such shapes are called superspheres, and Figure 2 shows the shapes given by (1) for (a) p = 2, (b) p = 4 and (c) p = 20. The parameter R determines the size and p determines the polyhedrality, i.e., the degree to which the supersphere is polyhedron. If |x| > |y| and |x| > |z|, |x/R|^p + |y/R|^p + |z/R|^p= 1 as p → ∞ means |x/R| = 1. This describes the limit for (1) as p → ∞ which gives a cube surrounded by three sets of parallel planes, x = ± R, y = ± R and z = ± R.

Equation (1) can be rewritten as

     (2)

This expression has been extended to describe other convex polyhedra [7]. Although the original superspheres discussed in [2,3,4] are intermediate shapes between a sphere and a cube, now the superspheres can refer to shapes intermediate between various convex polyhedra and a sphere [8].

Superspheres have been used to discuss the shapes of small crystalline particles and precipitates [2,3,5,8,9]. The planes of crystal facets are indicated by their Miller indices. We use this notation in the present study. The cube given by (2) as p → ∞ is the {100} cube composed of six {100} faces. Assuming crystals with cubic structures, the regular octahedron is the {111} octahedron and the rhombic dodecahedron is the {110} dodecahedron [7].

The {111} octahedral superspheres are given by the following equation:

     (3a)

where

     (3b)

The shapes given by (3) are shown in Figure 3.

On the other hand, the {110} dodecahedral superspheres are given by

     (4a)

where

     (4b)

The shapes given by (4) are shown in Figure 4. Equations (2–4) expressed by the spherical coordinate system are shown in [7].

Combined superspheres can be expressed by combining the equations of each supersphere. Combining (2), (3) and (4), we get

     (5)

The parameters a > 0 and b > 0 are those for determining the ratios of the {100}, {110} and {111} surfaces. The shapes of the supersphere given by (5) are shown in Figure 5 when , for two values of p.

The a and b dependences of the shapes given by (5) are understood by examining the polyhedral shapes as p → ∞. Among the three polyhedra given by [h_cube (x,y,z)]^1/p = R, [h_octa (x,y,z)]^1/p = aR and [h_dodeca (x,y,z)]^1/p = bR, the innermost surfaces of the polyhedra are retained to form the combined polyhedron. Figure 6 shows the effect of a and b on the shapes given by (5) as p → ∞.The shape is determined by their location in the quadrilateral surrounded by the points P (a,b) = (3,2), Q (2,2), R (1,1) and S (3/2,1). Various shapes in and around the quadrilateral are shown by the insets in Figure 6 can be summarized as follows:

The boundary between Regions 1 and 2, expressed by the line from P to R, is written as:

     (6)

Figure 6 is essentially the same as Figure 3 in [7,8] where the parameters α = 1/a and β = 1/b are used instead of a and b. In the appendix, the volume and surface area of the polyhedra shown in Figure 6 are written as a function of a and b. The use of the parameters a and b gives a more intuitive diagram (Figure 6), compared with the diagram given by α and β.