How is the Sierpiński arrowhead curve constructed?

Construction rule: A to B-A-B, B to A+B+A, Rewrite the string n times, then draw: A and B are steps forward, + and − are turns of 60°.. n is the order; the base runs from (0, 0) to (1, 0).

What is the arc length of the Sierpiński arrowhead curve?

Arc length: L_n = (3/2)^n. n is the order; the base runs from (0, 0) to (1, 0).

Fractal curves

Sierpiński arrowhead curve

One unbroken path whose limit is the Sierpiński triangle, drawn with nothing but 60° turns.

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Formulas and key values

Construction rule: $A \to B{-}A{-}B,\quad B \to A{+}B{+}A$
Rewrite the string n times, then draw: A and B are steps forward, + and − are turns of 60°.
Arc lengthIn the graph ≈ 11.39: $L_n = \left(\frac32\right)^n$
Fractal dimension: $D = \frac{\log 3}{\log 2} \approx 1.585$

n is the order; the base runs from (0, 0) to (1, 0).

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01 · Definition

What is the Sierpiński arrowhead curve?

The Sierpiński triangle, described by Wacław Sierpiński in 1915, is usually drawn by cutting holes. The arrowhead curve reaches the same shape as a single path: two rewriting rules, A → B−A−B and B → A+B+A, with every + and − a 60° turn, produce an order-n path whose limit is the triangle.

Each order replaces every segment by three half-length segments, so the length grows by 3/2 each time while the endpoints stay fixed. The limit has the triangle’s fractal dimension, log 3 / log 2 ≈ 1.585: more than a line, less than an area.

02 · Properties

Key properties

Order n has 3ⁿ segments of length 2⁻ⁿ (base length 1).
Total length (3/2)ⁿ, growing without bound.
Rules: A → B−A−B, B → A+B+A, with turns of 60°.
The limit is the Sierpiński triangle, of dimension log 3 / log 2 ≈ 1.585.
The path never crosses itself.

03 · Questions