## 22.5.12

### Build a Fractal tetrahedron (instructions)

I've built several of these fractal tetrahedra before. It's a nice group project and can motivate discussion on geometric ideas. This project takes about 60 minutes with adults, and about 90 minutes with kids. This is appropriate for just about any ages. I get requests for instructions, so I've written them up below, along with some questions to ask during the process to bring out some mathematical ideas. Have fun with this very cool project!

Here's how to do it.

Instructions

Materials:

• 64 x 6 = 384 straws to make one the size shown above. Preferably not the flexible kind, these will work but can cause some problems with bending.
• String
• Tape and/or glue sticks
• Scissors
• 64 sheets of colored paper, printed with triangle template

1. Triangle template. The tetrahedron looks awesome if one face is covered with a piece of colored paper. An easy way to do this is to print out the following triangle pattern on colored paper. You will need to experiment a little to find the right size to fit your straws. I'd suggest making one copy and setting a triangle of straws on top and check to see if it fits well on/in the inner triangle. The idea is that the outside flaps will be folded over the straws and taped or glued to the paper on the inside.
Use your copier's zoom feature to get the right size. It is entirely okay if your straws are a little too long for your paper and one of the corners of your triangle is cut off as shown here:
My straws have always been a little too long to make a template that fits on a standard size sheet of copier paper. Just be sure not to cut off a flap so that it is too short to fold over.

Below is a picture you can use as a template. Clicking on it should make it a big size. I think. You can also just set a straw triangle on paper and draw the template by hand. It doesn't need to be a work of art.
You'll need 64 copies in several different colors. You should make a few more because someone will mess up during construction.

2. Straws and String. Each tetrahedron requires 6 straws and we'll be making 64 tetrahedra all together. You'll need two different lengths of string: 64 long pieces that should be the length of 6 straws, and 64 short pieces that should be 10 cm (4 inches) longer than one straw.

I cut these myself ahead of time by wrapping string around the backs of two chairs that I've spaced so that one wrapping of string is the right length. I simply wrap 64 times and then with one big cut of the scissors all of the strings are the right length. Easy! I wrap around a book to make the short lengths. If you're short on prep time, you can just supply string and have your builders cut their own lengths.

3. Starting the activity. If you're leading the activity it's good to have one tetrahedron built ahead of time to show as a model. Have scissors and glue sticks or tape available and distribute straws, string and paper. I usually have a group of about 30 builders, so everyone gets to make more than one.

4. String 3 straws on the long string and tie together to make an equilateral triangle. The knot must be close to the end of the string so there is still a long tail of string left. If your string is too soft or fuzzy, it may be difficult to get through the straw. If this is the case, the end of the string can be put in the straw and you can suck the other end of the straw and suck the string right through. It works well and is pretty funny.
Questions: What kind of triangles are we making? (equilateral) Why are they called equilateral/what makes these equilateral?

5. Put two more straws on the end of the string and tie the end to one of the corners of the triangle to make a rhombus.

Questions: What shape is this? (rhombus) How do you know it is a rhombus? (all sides the same length) What other properties does this shape have? (it is a parallelogram, it has 60° and 120° angles)

6. Tie the end of the short piece of string to the third corner of the triangle that is not already connected to the other triangle.

7. Slide the sixth straw on this string.

8. Tie the end of this string to the opposite corner of the rhombus. The structure folds to create a tetrahedron!

Questions: Describe this form: How many edges does it have? (6) How many faces? (4) How many vertices? (4) What is is called? (tetrahedron) What does tetra mean? (Greek for 4. There is a connection to the game Tetris: in Tetris all of the shapes are made from 4 squares)

9. Cut out the triangle template. Participants can write their names in the triangle so the name can be seen from the bottom. Set the tetrahedron on top of the paper shape.

10. Fold the flaps over. It may help to pre-crease the flaps. Use tape or glue to attach the flaps to the inside of the triangle.

11. Make groups of 4. Set three tetrahedra together corner to corner as shown and tie the vertices together. There will be loose ends of string that will be useful for this. If not, cut small lengths of string to use.

12. Set the fourth tetraheron on top and tie the corners to the tops of the lower tetrahedra to make a large tetrahedron.

Describe the shape made on the inside of the tetrahedron – how many edges and vertices and faces and what shapes are the faces? (it has 8 triangular faces, 12 edges, 6 vertices. It is called an octahedron, another one of the Platonic solids).

13. Four groups should now come together and connect their four large tetrahedra together in the same way to make an even bigger tetrahedron.

14. There will now be exactly 4 of these even bigger tetrahedron in the room. Connect these 4 together to make the giant full-size tetrahedron.

15. Find a place to hang it!

Questions: How many tetrahedra are there? How many straws? What are different ways to count these? If we made the next larger tetrahedron how many straws would we need? If one small tetrahedron is stage 0 and the next step combines 4 of these to make a stage 1 tetrahedron, and the next composite form is stage 2, can we write a formula to describe the number of small tetrahedron or number of straws in stage n?

16. Extend?

At the end of any math-art activity, we should "LOOK AGAIN." Take time to look at the shape for hidden forms, ideas and connections. What could be changed next time to make the shape more interesting?

• What other materials could be used? With lightweight skewers and tissue paper, maybe this fractal could be a kite and fly!

• What kind of color schemes could be used? A random arrangement of colors always seems to come out nicely, but what kinds of color patterns could be used?

• Research: Sierpinski triangle. Platonic solids.

If you've done this activity and have any good questions to ask or ideas, please post them in the comments! Enjoy!