[fafakxpb] A4 city blocks

We consider a city street layout of concentric circles and radial spokes.  Far from the center, city blocks are roughly shaped like rectangles.  When the aspect ratio of a block exceeds the square root of 2, add a new radial street bisecting the block (and all blocks further out).  Blocks will be roughly square-shaped.

This is similar to the principle of A series paper sizes, e.g., A4, in which cutting a sheet in half results in a similar shape.

First, in the center, is a circle of unit diameter.  It's a traffic rotary.  We choose starting with this circle, and not a circle of unit radius, because it looks prettier.  Second is annulus of thickness 1 divided into 4 because sqrt 0.5 < pi/4 < sqrt 2.  Third is annulus divided into 8 because sqrt 0.5 < 3*pi/8 < sqrt 2.  Fourth and fifth are divided into 16: sqrt 0.5 < 5*pi/16 < 7*pi/16 < sqrt 2.

Xfig source

Counting the center circle as 1, the circles at which new radial rays start are circles 1 2 3 5 8 15 30 59 116 231 462 923...

nextodd(x)=if(x%2,x,x+1)
nn(x)=(nextodd(ceil(sqrt(2)/Pi*2^x))+1)/2
for(i=1,12,print(nn(i)))

Maybe useful as a template for a 2D barcode, though better, as already done in QR, is a square grid of squares.

Maybe for an artistic pixelation.  Or go 囲碁.

It's of course possible to 4-color it, and pretty easily: alternating regions of checkerboards.  Can it be 3-colored?  It cannot be 2-colored.

In 3D, start by projecting a cube onto a sphere.  There is and will always be distortion near the vertices.  Repeatedly subdivide squares into quarters as necessary.  Slabs of aspect ratio 1.4x1.4x1 become divided into 4 columns of aspect ratio 0.7x0.7x1.  Far from the center or vertices, blocks will be roughly cube-shaped.