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Basic Concepts Sierpinski Triangle
A Sierpinski Triangle can be made by connecting the midpoints of each side
of a triangle to form four separate triangles. Then the colored triangle in the
center is cut out.
Step 1 Step 2 Step 3
The number of remaining triangles in step 1, 2, and 3 is 3, 9, and 27
respectively. Thus, in the Sierpinski triangle, the number of triangles in each
step increases by a factor of three.
Example The colored squares are cut out in each step below. How many
squares will remain in Step 3?
Step 1 Step 2
Class Notes
Since 1 big square is divided into 9 squares, and 1 square in the center is cut out, the
number of remaining squares in Step 1 is .
In Step 2, there are squares in each of the remaining squares from Step 1. Thus,
the number of the remaining squares in Step 2 is , which increases by a factor
of .
Therefore, the number of remaining squares in Step 3 will be , which increases by a
factor of .
95Rules for Shapes
A Sierpinski Triangle can be made by connecting the midpoints of each side
of a triangle to form four separate triangles. Then the colored triangle in the
center is cut out.
Step 1 Step 2 Step 3
The number of remaining triangles in step 1, 2, and 3 is 3, 9, and 27
respectively. Thus, in the Sierpinski triangle, the number of triangles in each
step increases by a factor of three.
Example The colored squares are cut out in each step below. How many
squares will remain in Step 3?
Step 1 Step 2
Class Notes
Since 1 big square is divided into 9 squares, and 1 square in the center is cut out, the
number of remaining squares in Step 1 is .
In Step 2, there are squares in each of the remaining squares from Step 1. Thus,
the number of the remaining squares in Step 2 is , which increases by a factor
of .
Therefore, the number of remaining squares in Step 3 will be , which increases by a
factor of .
95Rules for Shapes
