Within the context of mathematical biology, a sphere packing problem occurred to me. I must note that unlike the typical sphere packing problems, the variant I consider involves minimising the average distance between spheres rather than maximising the density of spheres. Furthermore, the spheres I consider are compressible rather than hard.

I have attempted to solve this problem numerically for a finite number of spheres but I'd also like to learn more about exact solutions in the finite regime as well as the asymptotic regime. The mathematical problem is introduced in a self-contained manner but for those who are curious I decided to share some motivation below.

## Motivation:

A number of excellent scientists including Alan Turing[2] and D'Arcy Wentworth Thompson[3] have turned to mathematical modelling in order to understand morphogenesis. Though I may be slightly biased due to my training in mathematics, it recently occurred to me that early organogenesis may be modelled as a sphere packing process. I think this might partially explain the highly symmetric forms of amphibian eggs.

Consider a mass of spherical cells contained in a small region of space undergoing processes of division and specialisation in order to develop a specific geometrical structure. Now, let's suppose that vascularisation hasn't started so inter-cellular signalling via chemical signals is diffusion-constrained.

Furthermore, let's make the reasonable assumption that the strength of a chemical signal diminishes as $\sim \frac{1}{r^2}$ where $r$ is the euclidean distance from the source cell. We may then conjecture that in order for growth to be coordinated the average pair-wise distance between cells must be minimised at this stage of development.

Allow me to clarify the details of this mathematical problem.

## Finite sphere-packing with soft constraints:

Let us suppose that there are $N$ compressible balls whose centres,$\{r_i\}_{i=1}^N$, are initialised by sampling uniformly from $[-l,l]^3$. Our challenge is to find $\{r^*_i\}_{i=1}^N \in \mathbb{R}^3$ that minimises:

\begin{equation} U_{\text{net}} = \sum_{i=1}^N \sum_{j \neq i} U(r_i,r_j) \tag{1} \end{equation}

where

\begin{equation} U(r_i,r_j)=(\lVert r_i-r_j \rVert^2 - 2l^2)^2 \tag{2} \end{equation}

is the form of each local potential function. Furthermore, my intuition for introducing the constant $2l^2$ is that this would allow cells to synchronise their chemical signalling activity.

As pointed out below, I managed to solve this problem approximately but I am also interested in exact solutions that may be found analytically. Might the general solution for all $N \in \mathbb{N}^*$ be known to pure mathematicians?

I am particularly curious about the geometry and topology of the optimal shape as $N$ increases. Does this geometry become unique as $N\to \infty$?

**Note:** I must add that while the constants $l$ and $2l^2$ may be important in a numerical setting, I don't see how they affect the topology or the number of symmetries of the optimal shape in an analytical setting. To a pure mathematician these constants are probably irrelevant.

## Approximate solutions:

I managed to find approximate solutions in the case where $l=5$ and $4 \leq N \leq 20$ by solving this optimisation problem using gradient descent with the hyperopt library: https://github.com/hyperopt/hyperopt. For a sanity check I found that when $N=4$ I obtain a tetrahedron as expected.

The python code I used is also available on Github: https://gist.github.com/AidanRocke/f48cea76eeaae5c25e11e9e47c9315c7

## References:

- Weisstein, Eric W. "Sphere Packing." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SpherePacking.html
- Turing, A. M. (1952). "The Chemical Basis of Morphogenesis" (PDF). Philosophical Transactions of the Royal Society of London B.
- Thompson, D. W., 1992. On Growth and Form. Dover reprint of 1942 2nd ed. (1st ed., 1917)
- Bergstra, J., Yamins, D., Cox, D. D. (2013) Making a Science of Model Search: Hyperparameter Optimization in Hundreds of Dimensions for Vision Architectures. To appear in Proc. of the 30th International Conference on Machine Learning (ICML 2013).