A fractal is a geometric object that has an complicated boundary and is self-similar at all scales. One property of fractals is that, as the complexity of the shape increases, the area or volume approaches a finite value and the length of the boundary approaches infinity.
Manipulative 1 is a representation of a Sierpinski triangle. A Sierpinski triangle starts as a equilateral triangle. At each iteration, an area in the shape of an upside-down equilateral triangle 1/4 the size of the triangle is removed from each triangle. At each iteration, the area decreases by 1/4 and the length of the boundary increases by 1/3.
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