# Superspheres: Intermediate Shapes between Spheres and Polyhedra

## Abstract

**:**

## 1. Introduction

## 2. Cubic Superspheres

^{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.

## 3. {111} Regular-Octahedral and {110} Rhombic-Dodecahedral Superspheres

**Figure 3.**Shapes of the {111} regular-octahedral superspheres given by (3); (

**a**) p = 4 and (

**b**) p = 40.

**Figure 4.**Shapes of the {110} rhombic-dodecahedral superspheres given by (4); (

**a**) p = 6 and (

**b**) p = 40.

## 4. {100}-{111}-{110} Polyhedral Superspheres

**Figure 5.**Shapes of the {100}-{111}-{110} polyhedral superspheres given by (5); (

**a**) p = 20 and (

**b**) p = 100.

_{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:

- Three basic polyhedra
- (a) {100} cube at point P.
- (b) {111} octahedron at point R.
- (c) {110} dodecahedron at point S.

- Combination of two basic polyhedra
- (a) {100}-{111} polyhedra changing from the {100} cube to the {111} octahedron along the line from P to R via Q, by truncating the eight vertices of the cube (The shape at point Q is {100}-{111} cuboctahedron).
- (b) {111}-{110} polyhedra changing from the {111} octahedron to the {110} dodecahedron along the line from R to S, by chamfering the 12 edges of the octahedron.
- (c) {110}-{100} polyhedra changing from the {110} dodecahedron to the {100} cube along the line from S to P, by truncating six of the 14 vertices of the dodecahedron.

- Combinations of all three basic polyhedra
- (a) {100}-{111}-{110} polyhedra with mutually non-connected {110} surfaces in Region 1 (R-1).
- (b) {100}-{111}-{110} polyhedra with mutually connected {110} surfaces in Region 2 (R-2).

**Figure 6.**Diagram showing the variation in the shapes of the {100}-{111}-{110} polyhedral superspheres given by (5) as p → ∞.

## 5. Discussion

#### 5.1. Shape Transitions of Superspheres from a Sphere to Various Polyhedra

^{2/3}, where S is the surface area and V the volume of the supersphere. For a sphere, N = 6

^{2/3}π

^{1/3}≈ 4.84. Figure 7 shows the variations in N as a function of p for the following the superspheres as indicated by the insets:

- (i) the {100} cube type given by (2),
- (ii) the {111} regular-octahedral type given by (3),
- (iii) the {110} rhombic-dodecahedral type given by (4) and
- (iv) the {100}-{111}-{110} polyhedral type given by (5) with and .

**Figure 7.**Dependence of the normalized surface area N =S/V

^{2/3}on p, where S is the surface area and V the volume for various superspheres: (i) the {100} cube type given by (2); (ii) the {111} octahedral type given by (3); (iii) the {110} dodecahedral type given by (4) and (iv) the {100}-{111}-{110} polyhedral type given by (5) with and .

^{2/3}≈ 5.05 has the minimum total surface area for the same V [8,10]. The a and b dependence of N can be calculated easily using the results shown in the appendix.

#### 5.2. Shape of Small Metal Particles

## Acknowledgment

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## Appendix

_{100}, S

_{111}and S

_{110}of the polyhedra shown in Figure 6 are written as a function of a and b. In Region 1, these are given by

^{3}, S

_{100}= 0, S

_{111}= 4√3R

^{2}and S

_{110}= 0 as it should be.

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**MDPI and ACS Style**

Onaka, S.
Superspheres: Intermediate Shapes between Spheres and Polyhedra. *Symmetry* **2012**, *4*, 336-343.
https://doi.org/10.3390/sym4030336

**AMA Style**

Onaka S.
Superspheres: Intermediate Shapes between Spheres and Polyhedra. *Symmetry*. 2012; 4(3):336-343.
https://doi.org/10.3390/sym4030336

**Chicago/Turabian Style**

Onaka, Susumu.
2012. "Superspheres: Intermediate Shapes between Spheres and Polyhedra" *Symmetry* 4, no. 3: 336-343.
https://doi.org/10.3390/sym4030336