perimään

Perimän is a theoretical measure in geometry designed to quantify boundary complexity of planar shapes under a fixed boundary-refinement rule. It captures how the length of a boundary grows when the boundary is repeatedly subdivided according to a specified procedure.

Definition: For a planar region S with boundary ∂S, select a refinement rule R that replaces each boundary segment by a connected polyline consisting of s pieces, scaled so that the total length changes by a constant growth factor g per iteration. Let P_n be the total length of ∂S after n iterations of applying R. If the limit L = lim_{n→∞} P_n / g^n exists, this limit is called the perimän of S with respect to R and is denoted perimän_R(S).

Computation and interpretation: The perimän depends on R; it is a scale-adjusted measure of boundary complexity. If ∂S is smooth and R is uniform, P_n grows like g^n and perimän_R(S) equals the original perimeter up to a constant factor. If the boundary is fractal and length increases without bound, P_n grows faster than any fixed multiple of g^n and the limit may fail to exist or be infinite; certain fractal boundaries yield a finite nonzero perimän, depending on R.

Examples: A circle under a subdivision that splits each edge into two collinear segments doubles P_n each iteration, yielding perimän_R(circle) equal to its circumference. The Koch snowflake boundary expands by a factor 4/3 per iteration; choosing g=4/3 yields a finite perimän equal to the initial perimeter. A smooth shape refined with a different rule may yield the same perimän as the initial boundary.

Limitations: The value of perimän is not intrinsic to the shape alone; it depends on the chosen refinement rule. Therefore, comparisons require a shared, fixed R and normalization.

See also: Perimeter, Fractal dimension, Minkowski content.