How it works
Put a light at the centre of the globe and a cube round it: each point casts onto one of six faces.
That is the gnomonic projection, six times over, and it is how
S2 starts. Unfold shows it: the globe swells to the cube, then
the cube opens into a staircase of faces 0 to 5.
Gnomonic cells are much bigger in the middle of a face than at its corners, so S2 stretches each
face with a quadratic before cutting it into a grid. Tick "Without the stretch" to see the
unstretched grid. Each level splits every cell into four.
The cells are numbered along a Hilbert curve that runs through face 0, then 1, and on to 5, so
cells close in number are close on the globe. A cell's id is its face and its path down the
curve; the token is that id in hexadecimal without its trailing zeros.
Below level 10 or so a cell is too small to see on the whole globe. "Close up" restricts the view
to the picked cell and its neighbours on the same face, drawn at their own scale, so Child and
Parent walk all the way down to level 30, a square about 2 cm across. The grid shows the
next two levels inside each cell and the curve runs three levels down.