Fractals from Triangles, Quadrilaterals, Inscribed Circles, and Squares

Fractals are self-similar objects. What does self-similarity mean? If you look carefully at a fern leaf, you will notice that every little leaf - part of the bigger one - has the same shape as the whole fern leaf. You can say that the fern leaf is self-similar. The same is with fractals: you can magnify them many times and after every step you will see the same shape, which is characteristic of that particular fractal. Many scientists have found that fractal geometry is a powerful tool for uncovering secrets from a wide variety of systems and solving important problems in applied science, cartography, astrophysics, and biological sciences.

I created six types of fractals that the children can create from triangles, quadrilaterals, and square arrays.

Triangle with inscribed triangles from midlines:

1. Fill the space with a triangle of any shape (use a ruler).

2. Find the midpoint of each side of the triangle (use a ruler for the first large triangle, then estimate with semi-precision).

3. Connect each of the midpoints with a line called a midline to create the inscribed triangle (notice that this triangle is exactly one fourth of the area of the original triangle).

4. Leave the middle triangle untouched but repeat this process for the three outer triangles until the triangles are too small or you want to move on to the next fractal. 

Notice that the number of triangles of any size will be a power of three: 1, 3, 9…; 3x3x3 = 3^3 =27, 81, 243, 729, 2187…

Triangle with inscribed circles:

1. Fill the space with a triangle of any shape (use a ruler).

2. Inscribe a circle in the triangle (inscribed means that the circle touches each of the sides of the triangle); use a compass if you have one for your first few circles, then free-hand as they get smaller.

3. Continue to inscribe the largest circle possible in the empty space between the first circle and the two sides of the triangle and repeat this until you can see each circle but no empty space.

Apollonian Gasket Fractal

1. Fill the space with a circle (use a compass).

2. Inscribe a three tangent circles in the larger circle (inscribed means that the circle touches each of the sides of the larger circle and two of the smaller circles.

3. Continue to inscribe the largest circle possible in the empty space and repeat this until you can see each circle but no empty space.

Quadrilateral with inscribed parallelograms from midlines:

1. Fill the space with a quadrilateral (a polygon with four sides) of any shape (use a ruler).

2. Find the midpoint of each side of the quadrilateral (use a ruler for the first large quadrilateral, then estimate with semi-precision).

3. Connect each of the midpoints with a segment called a midline to create the inscribed quadrilateral (notice that this quadrilateral always forms a parallelogram [opposite sides are parallel and equal in length]).

4. Leave the outer triangles untouched and repeat this process for the inscribed parallelogram until they are too small. Notice that each of the resulting parallelograms have an area equal to the area of the quadrilateral before it.

Right Triangle with Altitudes forming Right Angles:

1. Fill the space with a right triangle of any shape (use a ruler).

2. Draw an altitude (forms a right angle with the hypotenuse) from the right angle vertex to the hypotenuse (longest side of the right triangle; the side opposite).

3. Repeat this process by drawing an altitude from the right angle vertex until you cannot continue.

4. Each of the resulting triangles are similar (their corresponding angles are equal).

The last two pages of the pdf allow you to create a 4th Iteration Sierpinski's Carpet? I started with a square array of 81 x 81 with a 27 x 27 square in the center surrounded symmetrically by eight 9 x 9 squares, each of which are surrounded by eight 3 x 3 squares, each of which are surrounded by eight 1 x 1 squares.

How many of each sized square can you count? 1, 8, 64, 512. How many would be in the 5th iteration, 6th?