Holiday-Themed Sierpiński Fold-Outs:

Make your fold-out a holiday or birthday card.

(For another Valentine's Day idea, see our Möbius Madness activity, under Variations.)

Sierpiński Fold-Out Pyramid:

This one's tough! Be sure you've practiced making a few of the regular Fold-Outs before trying this. Start by making a square of paper and folding it along each center line and along each diagonal. Then fold the square in half to make a rectangle. This rectangle is made up of 2 squares.

Treat each of these squares as the starting rectangle of a Sierpiński Fold-Out. Follow the steps above to make two sets of cuts (or more if you're up for it!).

Important: When you make your cuts, don't cut all the way to the diagonal lines. Leave about 1/8 of an inch. This will also mean your folds don't exactly line up with the edge of the paper.

Now fold your paper together the other way and repeat the process, first with 2 squares, then 4, just as before. Don't let the lines you've already cut distract you, and remember not to cut all the way to the diagonal lines.

When you unfold, you'll see four Sierpiński Fold-Outs pointing toward the center. Wow.

Poke the center up, so the top is like a mountain peak. Then crease the diagonal lines so they also poke upward. Re-crease the folds of each Sierpiński Fold-Out, one for each side of the pyramid.

Number Patterns:

Use the Sierpiński Fold-Out and fractal patterns to explore sequences of numbers. What patterns can you find when making your fold-out? Here are some possibilities:

• Number of rectangles at each step: 1, 2, 4, 8, ...
• Number of layers cut through at each step: 2, 6, 18, 54, ...
• Number of new pieces to "pop out" at each step: 1, 3, 9, 27, ...
• Total number of "popped-out" pieces at each step: 1, 4, 13, 40, ...
• Length of cut at each step (assuming first cut has length 1/2, and assuming your young learner is already familiar with fractions): 1/2, 1/4, 1/8, 1/16, ...

Number sequences can be observed in each of the "Fractal Doodles" below as well.

The Sierpiński Triangle:

The Sierpiński Triangle is an example of a fractal pattern, since smaller versions of the same shape occur within the triangle.

Try drawing your own Sierpiński Triangle. Start with a regular triangle (or use our printable to get started). Draw a dot in the middle of each side.

Connect these dots. They form a smaller upside-down triangle inside the original.

Think of the picture as 3 right-side up triangles with an empty upside-down triangular space in the middle.

Now draw dots in the middle of each of these smaller triangles' sides. Connect these dots (but leave the upside-down triangular space empty). Now you've got 12 even smaller triangles!

Keep going as far as you like, making smaller and smaller triangles! Are you able to see how the Sierpiński Fold-Out resembles this pattern?

When you've had enough, color in all the little right-side up triangles.

If you make three of these of the same size, you can put them together to make a larger Sierpiński Triangle!

Of course, if you make three of these larger Sierpiński Triangles, you could put them together to make a super-large Sierpiński Triangle! Then three super-large Sierpiński Triangles could make a mega-super-large Sierpiński Triangle. You could go on forever!

Fractal Doodles:

Here is a fractal pattern based on the letter "H" and another based on circles.

Can your math beast figure out what the next step of each fractal would look like? Or what the previous step must have looked like? Let your child invent their own fractal patterns, possibly inspired by something personal to them such as the first letter of their name.