4x4 Meander Curve

When the mode of the curve (e.g. 2x2, 3x3 or nxn) increases, the number of different forms of the level-1 units will also increase. Here as an example, we only demonstrate the Meander type of the 4x4 curve.

The level-1 unit of the 4x4 Meander curve can be extended from the corresponding unit of the 3x3 curve, shown as the red segments in the right two plots.

With the same method, actually we can extend the Meander unit to higher modes (\(n \ge 5\)).

Let’s check the expansion rules for these two types of base units on 4x4. Note we only constructe the rules for I/R/L.

draw_rules_4x4_meander(type = 1)

draw_rules_4x4_meander(type = 2)

And the 4x4 Meander curves on level 2:

draw_multiple_curves(
    sfc_4x4_meander("I", "11", type = 1),
    sfc_4x4_meander("I", "12", type = 1),
    sfc_4x4_meander("I", "11", type = 2),
    sfc_4x4_meander("I", "12", type = 2),
    nrow = 2
)

Let’s use a spiral sequence as the seed:

spiral_seed = function(loop) {
    paste(rep(paste0("R", sapply(0:loop, function(i) strrep("I", i))), each = 2), 
        collapse = "")
}
seed = spiral_seed(10)
draw_multiple_curves(
    sfc_sequence(seed),
    sfc_4x4_meander(seed, 1),
    nrow = 1
)

Zuguang Gu (z.gu@dkfz.de)

2024-09-05