Sierpiński triangle
Properties

For integer number of dimensions d {\displaystyle d}, when doubling a side of an object, 2 d {\displaystyle 2^{d}} copies of it are created, i.e. 2 copies for 1-dimensional object, 4 copies for 2-dimensional object and 8 copies for 3-dimensional object. For the Sierpiński triangle, doubling its side creates 3 copies of itself. Thus the Sierpiński triangle has Hausdorff dimension log 3 log 2 ≈ 1.585 {\displaystyle {\tfrac {\log 3}{\log 2}}\approx 1.585}, which follows from solving 2 d = 3 {\displaystyle 2^{d}=3} for d {\displaystyle d}.

The area of a Sierpiński triangle is zero (in Lebesgue measure). The area remaining after each iteration is 3 4 {\displaystyle {\tfrac {3}{4}}} of the area from the previous iteration, and an infinite number of iterations results in an area approaching zero.

The points of a Sierpinski triangle have a simple characterization in barycentric coordinates. If a point has barycentric coordinates (0. u 1 u 2 u 3 …, 0. v 1 v 2 v 3 …, 0. w 1 w 2 w 3 …) {\displaystyle (0.u_{1}u_{2}u_{3}\dots,0.v_{1}v_{2}v_{3}\dots,0.w_{1}w_{2}w_{3}\dots)}, expressed as binary numerals, then the point is in Sierpiński's triangle if and only if u i + v i + w i = 1 {\displaystyle u_{i}+v_{i}+w_{i}=1} for all i {\displaystyle i}.
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