Weighted Sierpinski triangle surface helpers — sierpinski_triangle_surface

Convenience helpers that take the canonical recursive embedding of the Sierpinski triangle graph and lift it into \(\mathbb{R}^3\). These helpers are intended for benchmark families where the topology is the usual triangular gasket but the intended metric comes from a curved or folded geometry rather than the flat equilateral reference drawing.

Usage

sierpinski.triangle.surface.graph(
  level = 2,
  surface = c("flat", "saddle", "paraboloid", "ripple", "folded"),
  amplitude = 0.75,
  freq_u = 1,
  freq_v = 1,
  x_scale = 1,
  y_scale = 1,
  normalize = c("median", "mean", "none")
)

Arguments

level: Recursion depth. May be 0 or larger.
surface: Triangle surface family. One of "flat", "saddle", "paraboloid", "ripple", or "folded".
amplitude: Finite deformation amplitude.
freq_u: Positive ripple frequency in the first canonical triangle coordinate. Used only when surface = "ripple".
freq_v: Positive ripple frequency in the second canonical triangle coordinate. Used only when surface = "ripple".
x_scale: Positive horizontal scaling applied to the canonical triangle coordinates.
y_scale: Positive vertical scaling applied to the canonical triangle coordinates.
normalize: Normalization applied to the induced edge lengths. One of "median", "mean", or "none".

Value

The coords_surface component is an n x 3 numeric matrix with columns x, y, and z.

sierpinski.triangle.surface.graph() returns a list with components:

edges: the Sierpinski triangle edges,
n: number of vertices,
edge_weights: induced positive edge lengths,
coords_surface: the 3D surface embedding,
coords_param: the canonical 2D triangle coordinates,
weight_scale: the normalization constant applied to the raw edge lengths,
family: always "sierpinski.triangle",
surface: the chosen surface name,
level: the recursion depth,
label: a human-readable family label.

Examples

triangle <- sierpinski.triangle.surface.graph(level = 1)