Sphere Packing
By William Reyes
Published August 13th, 2018
1 Biography
I am in my final semester at the University of Toronto Scarborough completing a double major in mathematics and statistics. I also have a passion for computer science and wish to pursue a career in web or mobile application development. This abstract is a refined version of a report I wrote for the course Classical Plane Geometries and their Transformations.
2 Sphere Packing
This paper is to provide insight and introduce the concept of Sphere Packing. Sphere Packing is described as the arrangement of non-overlapping identical spheres within a containment space [6]. The main idea is to find the best arrangement to place these spheres to occupy the most space. We will also discuss The Cannonball Problem, associated with Close-Packing of Equal Spheres, The Kepler Conjecture, and Hyperspheres.
2.1 The Cannonball Problem
The Cannonball problem is associated with counting the total number of cannonballs in a square based pyramid. Without counting all the cannonballs individually, Thomas Harriot [1] provided a formula to calculate this number in a square based pyramid with k-layers:
2.2 Close-Packing of Equal Spheres
There are two types of Sphere Packing arrangements to provide maximum density namely; Hexagonal Close Packing [3] and Cubic Close Packing [2]. Hexagonal Close Packing consists of two layers, layer A, includes one sphere surrounded by six others forming a hexagon. The second layer B, includes three spheres forming a triangle and is placed on top of A (Figure 1). In Cubic Close Packing the arrangement consists of three layers. The lower layer A being a hexagon, middle layer B and top layer C are in the form of triangles where B placed on top of A and C is stacked in the depressions of B (Figure 2).
2.3 Kepler Conjecture
It was hypothesized by 17th-century mathematician Johannes Kepler [5] that the two Close-Packing arrange- ments give maximum density for any containment space. This is famously known as the Kepler Conjecture. It was not until 1997 where Thomas Hales proved the Kepler Conjecture using computer calculations, extensive theories of global optimization, linear programming and interval arithmatic. We will not prove the Kepler
Conjecture but instead calculate the packing density of these arrangements which is
(Details of proof are provided in the video).
2.4 Hyperspheres
In higher dimensions an n-sphere or n-hypersphere is defined as the set of n-tuples with
where R is the radius of the hypersphere [4]. Like in 3-spheres, the n-sphere has a volume but instead is referred to as its content. The content of an n-sphere with radius R is given by the following formula:
where Sn is the hyper-surface area of an n-sphere of radius R. In addition, the hypersphere must satisfy
The gamma function is defined by
So,
Thus
The surface area Sn obtains a maximum and then proceeds to decrease as n increases. The exact value of n where Sn obtains maximum surface area cannot be solved but is approximated to be n = 7.25695.
2.5 Hypersphere Packing and Future Developments
Hyperspheres are nearly impossible to visualize and it is unknown what is the best way to pack them. However, in recent studies it has been proven by reseacher Maryna Viazovska [7], the best way to pack spheres in 8 and 24 dimensions is E^8 lattice and the Leech Lattice. The intuition, comes from building the standard way of packing spheres in 3-dimensions into all dimensions. Mathematicians have noticed as the dimension increases the spheres move further and further away from each other. In 8 and 24 dimensions, the spheres move so far away that there is exactly enough space between the spheres to fit in new ones.
This proof enables new symmetries to be discovered bringing a new approach to finding other potential sphere packing arrangements. The next question mathematicians have thought is whether these methods can be adapted to show E^8 and the Leech lattice have universal optimality. In other words, the lattices not only provide best sphere packings but also lowest-energy, in terms of physics. It is known that E^8 and the Leech Lattice are connected to many areas in mathematics and physics, so we can look forward to many more discoveries.
3 References
Darling, D. (2018). Cannonball Problem. [online] Daviddarling.info. Available at: http://www.daviddarling.info/encyclopedia/C/Cannonball Problem.html [Accessed 26 Mar. 2018].
Mathworld.wolfram.com. (2018). Cubic Close Packing – from Wolfram MathWorld. [online] Available at: http://mathworld.wolfram.com/CubicClosePacking.html [Accessed 26 Mar. 2018].
Mathworld.wolfram.com. (2018). Hexagonal Close Packing – from Wolfram MathWorld. [online] Available at: http://mathworld.wolfram.com/HexagonalClosePacking.html [Accessed 26 Mar. 2018].
Mathworld.wolfram.com. (2018). Hypersphere – from Wolfram MathWorld. [online] Available at: http://mathworld.wolfram.com/Hypersphere.html [Accessed 26 Mar. 2018].
Mathworld.wolfram.com. (2018). Kepler Conjecture – from Wolfram MathWorld. [online] Available at: http://mathworld.wolfram.com/KeplerConjecture.html [Accessed 26 Mar. 2018].
Mathworld.wolfram.com. (2018). Sphere Packing – from Wolfram MathWorld. [online] Available at: http://mathworld.wolfram.com/SpherePacking.html [Accessed 26 Mar. 2018].
Viazovska, M. (2017). The sphere packing problem in dimension 8. Annals of Mathematics, 185(3), pp.991-1015.
Chemistry LibreTexts. (2018). Closest Packed Structures. [online] Available at: https://chem.libretexts.org /Core/Physical and Theoretical Chemistry/Physical Properties of Matter/States of Matter/Properties of
Solids/Crystal Lattice/Closest Pack Structures [Accessed 26 Mar. 2018].