This picture by Toby Hudson shows the densest known packing of the regular heptagon. Of all convex shapes, the regular heptagon is believed to have the lowest maximal packing density.

We listed some competitors in the last post:

The disk has a maximum packing density of:

$$ \frac{\pi}{\sqrt{12}} = 0.9068996 \dots $$

The regular octagon has a maximum packing density of:

$$ \frac{4 + 4 \sqrt{2}}{5 + 4 \sqrt{2}} = 0.9061636 \dots $$

If we smooth the corners of the regular octagon in a specific way, we get a maximum packing density of only:

$$ \frac{ 8-4\sqrt{2}-\ln{2} }{2\sqrt{2}-1} = 0.902414 \dots $$

This is conjectured to be the lowest possible for a centrally symmetric convex shape.

But if we drop the constraint of central symmetry, the regular heptagon enters the game, and apparently beats the smoothed octagon! The densest *known* packing of the regular heptagon, shown above, has density

$$ \frac{2}{97}\left(-111 + 492 \cos\left(\frac{\pi}{7}\right) – 356 \cos^2 \left(\frac{\pi}{7}\right)\right) = 0.89269 \dots $$

Greg Kuperberg and his father Włodzimierz showed this is the densest ‘double-lattice packing’ of the regular heptagon. A **lattice packing** of a shape is one in which all copies of that shape are translates of a fixed copy by lattice vectors. A **double-lattice packing** of a shape is the union of two lattice packings such that a 180° rotation about some point interchanges the two packings:

• Greg Kuperberg and Włodzimierz Kuperberg, Double-lattice packings of convex bodies in the plane, *Discrete and Computational Geometry* **5** (1990), 389–397.

However, it has not been proved that this is the densest packing of the regular heptagon!

The image here was created by Toby Hudson and put on Wikicommons with a Creative Commons Attribution-Share Alike 3.0 Unported license.

*Visual Insight* is a place to share striking images that help explain advanced topics in mathematics. I’m always looking for truly beautiful images, so if you know about one, please drop a comment here and let me know!

Could you do a rounded corner heptagon and get an even lower max density? If the octagon is a guide, then maybe.

It’s not obvious that rounding corners of the heptagon will reduce the maximum packing density: note that rounding the heptagon’s corners ever so slightly will allow us to compress the stacks of heptagons in the picture. That doesn’t happen for the octagon, which has an even number of sides.

I hope that whoever conjectured the heptagon was the worst possible convex shape for dense packing would have considered small perturbations like this.

I’m 3 years late to this, but better late than never. I believe you’re talking about me (“whoever conjectured the heptagon was the worst possible convex shape for dense packing”). I did in fact consider small perturbations and I was able to prove that all sufficiently small perturbations allow for denser packing (see https://ykallus.github.io/gt2014.html). I also made a little javascript interactive where you can try to come up with a counterexample: https://ykallus.github.io/demo/shape.html

Hi, Yoav! I guess at the time I didn’t know, or had forgotten, who conjectured that. I’m glad you checked small perturbations. And it’s great to let people try to come up with counterexamples!