We have a binary choice of operations.
First look at the examples after 3 steps.
Some parameters of the curve can be adjusted to see different aspects of the result
The metric referes to the way the quarter circle arcs are drawn, using either the Euclidean, supremum, or taxicab metric
increasing the number of iterations of the process allows us to see further steps on the way to fractal space filling curves
Warning: It takes a while to load the program for more operations, but it's not too slow once compiled
Adjusting parameters may give clearer results
MAIN RESULT:
Given the initial formula f₀(ℓ,m) for the edge colouring of edge (ℓ,m) of the initial tiling, then after k+1 iterations of the sequence of operations, there is a formula for the edge colours as follows:
f_k+1(ℓ,m)=f_k⋅T(ℓ,m) if ⌊ℓ/2⌋+⌊(m+1)/2⌋ is even
f_k+1(ℓ,m)=f_k⋅R(ℓ,m) otherwise
where
T(ℓ,m)=(⌊ (ℓ-m+1)/2⌋,⌊ (ℓ+m)/2⌋)
and
R(ℓ,m)=(⌈ (ℓ-m+1)/2⌉,⌈ (ℓ+m)/2)⌉
This is for the 0 operation at iteration step k+1.
R and T are exchanged for operation 1.

Space Filling Smith Truchet Variations