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!