This problem has been studied by many people in a general setting of convex bodies.

Given a convex body $K$ in $\mathbb R^d$ and an $\alpha > 0$, find the maximal number $H_{\alpha}(K)$ of nonoverlapping translates of $\alpha K$ touching K.

$H_{\alpha}(K)$ is often referred to as the generalized Hadwiger number (or the generalized kissing number).

L. Fejes Tóth studied the asymptotics of $H_{\alpha}(K)$ for polytopes in 1970s. His result was extended to the general case by K. Böröczky Jr., D. Larman, S. Sezgin, and C. Zong ("On Generalized Kissing Numbers and Blocking Numbers", Rend. Circ. Mat. Palermo, 65 (2000), pp. 39-57) who showed that $$H_{\alpha}(K)\sim \frac{C}{\alpha^{d-1}}$$ for any convex body $K\subset\mathbb R^d$. The constant $C$ can be calculated more or less explicitly in terms of $K$.

This problem has been studied by many people in a general setting of convex bodies.

Given a convex body $K$ in $\mathbb R^d$ and an $\alpha > 0$, find the maximal number $H_{\alpha}(K)$ of nonoverlapping translates of $\alpha K$ touching K.

$H_{\alpha}(K)$ is often referred to as the generalized Hadwiger number (or the generalized kissing number).

L. Fejes Tóth studied the asymptotics of $H_{\alpha}(K)$ for polytopes in 1970s. His result was extended to the general case by K. Böröczky Jr., D. Larman, S. Sezgin, and C. Zong ("On Generalized Kissing Numbers and Blocking Numbers", Rend. Circ. Mat. Palermo, 65 (2000), pp. 39-57) who showed that $$H_{\alpha}(K)\sim \frac{C}{\alpha^{d-1}}$$ for any convex body $K\subset\mathbb R^d$. The constant $C$ can be calculated more or less explicitly in terms of $K$.