# Equilateral triangle packing in the $l^1$ circle v.s. the $l^2$ circle

As established in a previous post Equilateral Triangles In The Taxicab Space there are exactly $$8$$ equilateral triangles (of edge length $$1$$) that can be packed in the $$l^1(\mathbb{R}^2)$$ unit circle, while there are only $$6$$ equilateral triangles (of edge length $$1$$) that can be packed in the $$l^2(\mathbb{R}^2)$$ unit circle.

My suspicion is that if one were to consider an $$l^{1 + \epsilon}(\mathbb{R}^2)$$ unit circle, then for sufficiently small epsilon, the number of equilateral triangles which can be packed in the unit circle is still $$8$$. More so, I suspect that for a specific value of $$\epsilon >0$$ the $$l^{1+ \epsilon}(\mathbb{R}^2)$$ unit circle can only contain $$7$$ equilateral triangles.

Question 1: Is my suspicion correct?

Question 2: Is there a value $$\epsilon >0$$ such that the $$l^{1+ \epsilon}(\mathbb{R}^2)$$ unit circle contains exactly $$7$$ equilateral triangles?

• Perhaps an easier question to consider might be worth approaching. Notice that in the Taxicab space, $l^1(\mathbb{R}^2)$, one can pack exactly 4 circles of radius $1/2$ inside the circle of radius $1$. For the usual Euclidean space, one can only pack $2$ circles of radius $1/2$ into the the circle of radius $1$. Does there exists an $\epsilon > 0$ so that the $l^{1+\epsilon}(\mathbb{R}^2)$ unit circle contains only $3$ circles or radius $1/2$ ? – Drewrl3v Feb 4 '19 at 18:45