as $$\label{eq:graphenergy}
f(x_1, \ldots, x_k) := \underbrace{\frac{\alpha}{2} \sum_{j = 1}^l c_j \sum_{i \in e_j} \|x_i - z_j\|^{\gamma}}_{\text{rubber bands}} + \underbrace{\frac{\beta}{2} \sum_{j = 1}^k \sum_{\substack{i = 1 \\ i \neq j}}^k \frac{w_i \, w_j}{\|x_i - x_j\|^{\delta}}}_{\text{repelling charges}},$$ for $(x_1, \ldots, x_k) \in U$, where $\alpha, \beta, \gamma, \delta > 0$ are constants, and for $1 \leq j \leq l$ the center of hyperedge $e_j$ is defined as $$\smash[t]{z_j := \frac{1}{|e_j|} \sum_{i \in e_j} x_i.}$$

Now, we will generate a visual representation by finding $$\label{eq:mingraphenergy}
{\mathrm{argmin \,}}\{f(x_1, \ldots, x_k) {\ |\ }(x_1, \ldots, x_k) \in U\},$$ where $f$ is the energy function of our hypergraph $G$. Approximate solutions to generated by our algorithm are shown in .

Constants
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We can tinker with $f$ by varying the constants $\alpha$, $\beta$, $\gamma$, and $\delta$. First, we can make a number of observations about $U$, $f$, and their symmetries.

1.  $\{(R(x_1) + x, \ldots, R(x_k) + x) {\ |\ }(x_1, \ldots, x_k) \in U\} = U$ for all $x \in {\mathbf{R}}^d$ and all orthogonal transformations $R \in {\mathrm{O}}({\mathbf{R}}^d)$.

2.  $U$ is an open subset of ${\mathbf{R}}^{d \times k}$ and $U = \theta \, U$ for all $\theta \in {\mathbf{R}}\setminus \{0\}$.

3.  $f(R(x_1) + x, 