Art and Sierpinski’s Triangle



Art and Sierpinski’s Triangle

Sierpinski’s triangle is an example of a fractal figure. A fractal is a geometric shape characterized by self-similarity, the concept of parts of the shape being similar to the shape as a whole or to other portions of the shape. The simplest fractals are formed by starting with an initiator and using a generator to create the figure step-by-step. These steps are called iterations.

To create Sierpinski’s triangle we start with the initiator, a filled-in triangle. The generator is the same triangle with the triangle formed by connecting the midpoints of the sides of the initiator triangle removed.

[pic]

Initiator Generator

At each iteration all filled in triangles are replaced by the generator. Iterations 0, 1, and 2 are shown below.

[pic]

Iteration 0 Iteration 1 Iteration 2

Show iterations 3 and 4 in the space below.

Complete the table below for iterations 0 through 4 in the creation of Sierpinski’s triangle. Let the area of the initiator be 1 square unit. Use the table to find patterns in this construction.

|Iteration |Number of Triangles Left |Area of Smallest Triangle |

|0 |1 |1 |

|1 |3 |1/4 |

|2 | | |

|3 | | |

|4 | | |

List any patterns you see in the table above.

Cut out the triangle on page 4. Use this as a template to draw an equilateral triangle on cover stock paper. The length of a side should be 16 centimeters.

Cut the triangle out of the cover stock paper.

Find and mark the midpoint of each side of the triangle.

Connect the midpoints to create a triangle inside the original triangle.

Use a pencil to pierce the paper inside this triangle. Being careful to leave all of the lines (you do not one the triangle to separate into multiple triangles) cut out this triangle. You should now have iteration 1.

Find and mark the midpoint of each side of the three remaining triangles. Connect the midpoints of each triangle to create 3 new triangles – one inside each of the remaining triangles.

Carefully cut out each of these triangles without cutting through the outside triangle.

You should now have iteration 2 of Sierpinski’s triangle.

Repeat this process with the nine remaining triangles. When done you should have a figure that looks like iteration 3 of Sierpinski’s triangle.

Choose a tissue paper color for each size triangle that you have cut out. Cut out of one tissue paper a triangle that is about one-half inch larger than the large triangle you cut out first. Use the glue stick provided to glue the paper over the triangle. Cut out any part of this triangle that is exposed in another cut out triangle.

Cut out 3 triangles from a different color tissue paper. Make them slightly larger than the next size triangles cut out. Glue them over these triangles. Cut out any parts of the tissue paper that are exposed.

Repeat this process to cover the 9 remaining (smallest) triangles that you cut out.

Tape the four Sierpinski’s triangles made by your team together to form a tetrahedron. Attach string to one corner of the tetrahedron. You now have a piece of mathematical art you can hang in the classroom.

[pic]

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download