NCTM Meeting in San Diego

Sierpinski Pyramid

Paul Kelley
Anoka High School
Anoka, MN 55303
pkelley@informns.k12.mn.us

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Back to NCTM San Diego: Presentation Summary
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Sierpinski Pyramid

To build a Sierpinski Pyramid, follow these steps:
  1. Have enough copies made of the pyramid template to construct the desired size of your pyramid. See the chart below for the number of templates needed for each stage. It is recommended that a thicker, card-stock type paper be used. A sample of the template is enclosed.
    Stage              Number of Templates     Approximate Height
       0                              1             3.5 in.
       1                              4               7 in.
       2                             16              14 in.
       3                             64              28 in.
       4                            256              56 in.
       5                           1024             112 in.
    

  2. Cut each template.

  3. Fold each template to form a small pyramid. Tape the edges together. For lack of a better term, we will call this a Stage 0 pyramid.

  4. Put four of these together to form a pyramid with the "middle" missing. We will call this a Stage 1 pyramid. Make a "brace" from the extra material left over after cutting out the templates, and glue it on.

  5. Put four Stage 1 pyramids together to form another pyramid. Call it a Stage 2 pyramid.

  6. At this point, you will have to move your materials to the place where your completed Sierpinski Pyramid will eventually reside, as going through one more stage will make a pyramid too big to fit through most doorways.

  7. Continue the process described above until you reach the desired size for your Sierpinski Pyramid. A stage 5 pyramid will form an equilateral triangular base that is approximately 140 inches per side.

Materials needed:

Helpful hints or ideas:

  1. Before starting the project, go through in detail with the students exactly what it is you are doing, and the step-by-step procedure. This will help them see the "big picture."

  2. It might be best to have more than one class working on the project. Experience shows that a stage 5 pyramid takes approximately 12 class periods to complete (47-minute periods). If, for example, three classes work on the project, one class can pick up right where the previous class left off, and this will cut the number of days needed to complete the project.

  3. Here is a cooperative learning idea, using four classes and four groups per class (this can be amended to suit personal preference and/or number of classes). Have each of the four groups in your four classes make a stage 3 pyramid. Then, put each class' efforts together, which will form a stage 4 pyramid. Then, put the total effort together, which will form a stage 5 pyramid.

  4. When you are finished, remind the students that the actual Sierpinski Pyramid is formed by removing pieces of an original pyramid, rather than by piecing together individual, smaller pyramids, as they just did. However, since the "real" process is a bit impractical for a classroom construction, it is felt that the method just completed is most convenient.

  5. Be sure to print more templates than you need, as some will get cut improperly, others will get crushed, etc.

  6. To save time, divide the number of templates by the number of students, and have them cut out the templates the night before you start.


Paul Kelley
Anoka High School
Anoka, MN 55303
pkelley@informns.k12.mn.us

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The Math Forum * ** 22 April 1996