A reader asked how to find all the hexagon shapes that fit inside a circle. I implemented this diagram with a slider. Red circles are integer distances and black circles are other distances.

Brute force solution:

- You need a distance function. For each hex you need to know the distance from 0,0 to to the furthest corner. I have a javascript version of this in hexagons-in-circle.js (
`distanceToCorner`

). - You need to know all the distances. If you loop over all the hexes, you can calculate
`distanceToCorner`

to each. Then get all the unique distances form this list. Sort them. Little glitch: sometimes two values are very very close together and we probably want to lump them together, so in this code I lumped together values that were the same down to 4 digits. See the calculation of`allDistances`

in hexagons-in-circle.js. Each of the distances in

`allDistances`

gives you a diagram, with hexes with`distanceToCorner(hex) < distance`

. Little glitch: because I'm using 4 digits, sometimes there are rounding errors, and I had to use`distanceToCorner(hex) < distance + 0.00005`

I could not find a closed form formula to find the distances, but I didn't try very hard. I think brute force worked just fine here.