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3x3 Meander Curve

Zuguang Gu ()

2024-09-05

Source: vignettes/meander_curve.Rmd
meander_curve.Rmd

Base patterns

The Meander curve has a different set of base units.

The expansions are very similar as the Hilbert curve, but it also has a flipped version where the units can be flipped horizontally or vertically on various levels.

On each level expansion, there are two traverse paths:

plot_traverse_paths(SFC_RULES_3x3_MEANDER, SFC_RULES_3x3_MEANDER@rules$I[[1]])

Curves

All possible forms of the Meander curve on level 3 taking R(0) as the seed:

draw_multiple_curves(
    sfc_3x3_meander("R", code = "111"),
    sfc_3x3_meander("R", code = "112"),
    sfc_3x3_meander("R", code = "121"),
    sfc_3x3_meander("R", code = "122"),
    sfc_3x3_meander("R", code = "211"),
    sfc_3x3_meander("R", code = "212"),
    sfc_3x3_meander("R", code = "221"),
    sfc_3x3_meander("R", code = "222"),
    nrow = 2, lwd = 2)

The curves on level 3 taking the nine base patterns respectively:

draw_multiple_curves(
    sfc_3x3_meander("I", code = "111"),
    sfc_3x3_meander("R", code = "111"),
    sfc_3x3_meander("L", code = "111"),
    sfc_3x3_meander("U", code = "111"),
    sfc_3x3_meander("B", code = "111"),
    sfc_3x3_meander("D", code = "111"),
    sfc_3x3_meander("P", code = "111"),
    sfc_3x3_meander("Q", code = "111"),
    sfc_3x3_meander("C", code = "111"),
    nrow = 3, lwd = 2)

Unit flipping

Flipping (horizotal or vertical) can be applied by specifying the index of the hierarchical unit on the curve.

p = sfc_3x3_meander("I", 11)
draw_multiple_curves(
    p,
    sfc_flip_unit(p, "1"),  # bottom left
    sfc_flip_unit(p, "2"),  # bottom middle
    sfc_flip_unit(p, c("6", "7", "8", "9")), # top left, four units
    nrow = 2
)

Number of different forms

The total number of different forms of the Meander curve on level \(k\) is composed by two parts: \(9 \cdot 4 \cdot 2^k\) is the number by applying the expanding rules, and \(2^{\frac{9^k - 1}{8}}\) is the number of units on all levels where each unit can be flipped independently.

\[ 9 \cdot 4 \cdot 2^k \cdot 2^{\frac{9^k - 1}{8}} \]

Seed as a sequence 

seed = sfc_sequence("IILILILI", rot = -90, universe = sfc_universe(SFC_RULES_3x3_MEANDER))
plot(seed, grid = TRUE)

sfc_3x3_meander(seed, code = "111") |> plot(lwd = 2)