All Activities U Sierpiński Fold-Out
for parents
for teachers

Sierpiński Fold-Out

Make a beautiful fractal pop-up card by folding and cutting in a repeated pattern.

Ages: 5+ (with help)
Players: 1
Time: 20+ Minutes
Type: arts and crafts
Location: tabletop
Share This Activity
Ages: 5+ (with help)
Players: 1
Time: 20+ Minutes
Type: arts and crafts
Location: tabletop

Instructions

Fold a piece of paper in half to form a rectangle, then cut halfway across the center of the rectangle, starting from the creased edge. It isn't important to be completely accurate in cutting, but you could use a ruler to measure the cuts if that helps. Fold the top half of the paper over to line up with the edge.

blank piece of paper
Piece of paper folded in half
Dotted line showing where to cut
Piece of folded paper with cut down the middle

This next step can be a bit tricky the first time. Open up the paper and push the "folded-over" portion inside. Then re-crease. So far, your paper should look like a fairly boring pop-up card. If you open up the card, a small rectangular part pops out.

Piece of folded paper
Piece of folded paper

Now fold the paper closed again and imagine it as having two smaller rectangles. Repeat the process you just did with each of these. Cut halfway across the center of each rectangle, then fold over the top half of each. Open everything up, push the folded parts inside, and re-crease.

Piece of folded paper
Piece of folded paper
Piece of folded paper
Piece of folded paper

This is a good time to notice that, at first, you made 1 cut halfway through a rectangle. The second time through, you made 2 cuts, because there were two, smaller rectangles. Now repeat the process with 4 smaller rectangles. (Notice a pattern in these numbers?) Cut halfway and fold over the tops. Open everything up, push the folded parts inside, and re-crease.

Piece of folded paper
Piece of folded paper
Piece of folded paper
Piece of folded paper
Piece of folded paper

After this set of cuts and folds, your Sierpiński Fold-Out is looking pretty good! You're welcome to stop here (younger scissor-users will certainly find the next step difficult if you decide to continue).

If you do another round of cuts and folds, this time with 8 rectangles, the results are pretty rewarding. Fold the paper closed again, make 8 small cuts (one for each little rectangle), open everything up, push the new folded parts inside, then re-crease.

Piece of folded paper

Fearless crafters may wish to attempt another set of cuts and folds with 16 rectangles! (Good luck with that!)

Optionally, when you're done cutting, glue your Sierpiński Fold-Out to some paper for backing to form a card.

Piece of folded paper
Piece of folded paper

Don't forget: it's Beast Academy Playground, not Beast Academy Study Hall. Change the rules, be silly, make mistakes, and try again. The Variations and Learning Notes are here for you if you want to dive deeper, but not all of them apply to learners of every age. The most important thing is to have fun.

Variations d

Holiday-Themed Sierpiński Fold-Outs:

Make your fold-out a holiday or birthday card.

Folded paper birthday cake
Birthday
Folded paper heart
Valentine's Day
Folded paper Christmas tree
Christmas

(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.

Piece of folded paper
Piece of folded paper
Piece of folded paper
Piece of folded paper

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.

Piece of folded paper
Piece of folded paper
Piece of folded paper
Piece of folded paper
Piece of folded paper
Piece of folded 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.

Piece of folded paper
Piece of folded paper

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

Piece of folded paper
Piece of folded paper

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.

Piece of folded paper
Piece of folded paper
Piece of folded paper
Piece of folded paper
Piece of folded paper

Classroom Tips d

Make Sierpiński Fold-Outs when studying sequences of numbers, fractions, or lines of symmetry. Make a prototype first to get familiar with the project, and for a model to show the class. After making the fold-outs, make your own Sierpiński Triangle using our printable as a starting point. Assemble students' triangles to make a giant class triangle.

Discussion Questions

  • How many cuts will you have to make on the next step?
  • How many little rectangles will you need to pop out on the next step?
  • What number patterns can you find in your fold-out? (see Learning Notes)
  • Can you make your own "fractal" pattern? (see Learning Notes)

Alignment with Beast Academy Curriculum

  • Level 4, Chapter 3: Exponents
  • Level 5, Chapter 12: Exponents

See Variations and Learning Notes for more ideas on how to adapt this activity and incorporate it into your classroom.

Learning Notes d

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.

Folded piece of paper
The entire shape on the left...
Folded piece of paper
...is duplicated three times on the right.

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.

Triangle drawn on a piece of paper
Triangle drawn on a piece of paper

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

Triangles drawn on a piece of paper

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

Triangles drawn on a piece of paper

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!

Triangles drawn on a piece of paper

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?

Five triangles showing iterations of Sierpinski's Triangle

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

Triangles drawn on a piece of paper

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

Triangles cut out making a Sierpinski 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.

A fractal H and a fractal circle

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.

What do you think of this activity?

We're always looking to improve. Submit your feedback to us below.

Share This Activity
Materials
  • paper
  • scissors
  • glue
Learning Goals
  • shapes
  • patterns
  • counting
  • wonder
Common Core Standards
Image of Ms. Q

Ready to level up?

Keep problem solving with Beast Academy’s full math curriculum for students ages 7–13. Check out our captivating comic book series and immersive online platform.

LEARN MORE

Bring problem-solving to your classroom

Keep your entire class engaged with a full book and online math curriculum, for students ages 7–13. 98% of teachers say they’re satisfied with Beast Academy.

LEARN MORE
Image of a BA book

Recommended Videos

Image of Ms. Q

Ready to level up?

Keep problem solving with Beast Academy’s full math curriculum for students ages 7–13. Check out our captivating comic book series and immersive online platform.

LEARN MORE

Bring problem-solving to your classroom

Keep your entire class engaged with a full book and online math curriculum, for students ages 7–13. 98% of teachers say they’re satisfied with Beast Academy.

LEARN MORE
Image of a BA book