Proof that the internet has everything: a site with the best-known packings of identical circles into a square, up to 10,000 circles. Download PDF files with diagrams of those packings or just peruse the table of current best results.
Circle Packings (website)
Here’s some pre-made squares for you to try to fit the requisite number of coins in them. Seven UK two-pence coins should fit into the smaller square, and thirty-one UK two-pence coins into the larger. If you only have access to Freedom Coins, we’ve also made a version for US quarters.
Squares of size 74.25mm and 144.9mm (PDF)
Squares of size 69.5mm and 135.7mm – for US coins (PDF)
Here’s a link to Thomas Hales’ Flyspeck Project, to verify his proof of the Kepler conjecture. And a link to when he announced the completion of the project. Which wasn’t until after I’d finished writing about it in my book. What a jerk.
Flyspeck Project (website)
Announcing Completion of the Flyspeck Project (website)
A printable net of a truncated octahedron, so you can make some for yourself and see how they tile to fill 3D space. You can even pretend you are Lord Kelvin and claim this is the best solution possible. If you want your model to sit flush on a table, you may have to cut some of your shapes in half, but I’m sure you can handle that.
Truncated octahedron net (PDF)
Here are printable nets of the Weaire-Phelan polyhedra, so you can make them for yourself and watch them fill space like champs.
Weaire-Phelan Polyhedra nets (PDF)