#### Fractals

Create fractal structures and a large sierpinski pyramid.

*Total: 135 minutes*

# Learning Objectives

- Learn about fractals in nature and what is self-similarity.
- Experience through algorithmic building the powerful idea that by repeating small and simple tasks, greater complexities can be achieved.
- Being able to identify symmetric and asymmetric shapes and observe how stable and aesthetically pleasing they can be.
- Dive into a tangible experience of debugging structures, by scaling and improving weak points.
- Developing communication skills by inventing rules that can be used as building or drawing instructions for yourself and others to use.

*In this lesson, we offer 3 sessions that can be done in sequence or separately.*

# Day 1: Drawing Fractals

*Total: 45 minutes*

# Imagine

*5 minutes*

https://www.flickr.com/photos/evilbu/4576466328

**The IMAGINE part of Drawing Fractals can be assigned to students to take home or experience in the classroom.**- Ask your students:
- What is “similarity”?
- What is self-similarity?
- Can you think of examples of things that are self-similar? Where can you find them?
- What do they think fractals are?

- Show students examples of self-similar structures in nature:
- The leafy structures of ferns
- The river delta system that streams from 1 large river into smaller streams leading into the ocean
- Broccoli
- Blood vessels

- Show students examples of graphical representation of self-similar structures:
- Illustrations of a fern
- Sierpinski triangle
- Koch snowflake
- Mandelbrot and Julia sets

- Example images can be found on the
**Downloads**. - Invite your students to imagine and create their own self-similar shapes or to draw existing ones they really like.

# Create

*10 minutes*

- For this activity, students can draw their own fractals. They can vary in curvy, curly, or straight lines and shapes to create a large pattern.
- There are extensive amounts of tutorials on the internet and you can find a small selection of useful resources that can be found under
**Preparation**below. - If students are struggling to create fractals or confused about what is a fractal, remind them about the Mandelbrot definition: "A fractal is a shape made of parts similar to the whole in some way."
- Challenge yourself to not tell students if their drawing is or isn’t a fractal too early in their process, but provide more images for inspiration to cultivate their growth.

# Play

*10 minutes*

- Allow students to continuously tweak their drawings and allow them to experiment on a different platform like the ones suggested in CREATE.
- Suggest students to change colors, try new shapes, and even translate their drawings, traditional or digital, into other forms of media such as folded and cut paper or constructing with cardboard.

# Share

*10 minutes*

- Have students take a few minutes each to present their collection of fractal drawings and ask them to share how they made each piece if there is more than one and explain parts of their drawing that represents self-similarity.

# Reflect

*5 minutes*

- After sharing drawings, mention how different or similar fractals can be. Ask students what was their inspiration for their fractal.
- Have your students identify if they have seen some of the drawings in nature and in their surroundings and what are the more “abstract” and “impossible” drawings. They may recall more examples seen in passing.

# Day 2: Algorithmic Building

*Total: 45 minutes*

# Imagine

*5 minutes*

**If you choose to start with Algorithmic Building, take some time to do the IMAGINE section of Drawing Fractals first.**- Tell students to think about how they would explain to someone else how to draw exactly the same thing they did. What are the steps that should be taken? Would they prefer to write or to draw?
- Present built examples or pictures of shapes made out of Strawbees and the sequence of steps, called rules or algorithm, that made them possible. Identify with the students which are self-similar shapes and which are not.

# Create

*20 minutes*

- Ask students if they understand the rule by just looking at the images and diagrams. If they understand, ask if they would like to build one or invent their own rules and shapes. If not, invite them to propose a better way to do it! You can find images of Strawbees shapes with a visual representation.
- The teacher can go through some of the provided posters describing the steps on the diagram in a verbal or written way.

- “Zig-Zag”
- STEP 1: Start with any pipe length
- STEP 2: Attach 3-leg Strawbee to one side of pipe
- STEP 3: Attach pipe to leg A
- STEP 4: Attach 3-leg Strawbee to one side of pipe
- STEP 5: Attach pipe to leg B
- STEP 6: Go back to STEP 2

- “Tree”
- STEP 1: Start with a full-length pipe
- STEP 2: Attach 5 Leg Strawbee to one side of pipes
- STEP 3: Next pipes should be shorter
- STEP 4: Attach pipe to all legs B and C
- STEP 5: Go back to STEP 2 (until there is no more material available)

- There are additional posters under
**Downloads**of this lesson. You can provide your students if they are interested. Keep in mind that the posters don’t come with the steps so they might need to figure it out. Below is a rule example for one of those posters.

- “Spiral”
- STEP 1: Start with the smallest pipe length
- STEP 2: Attach 3 Leg Strawbee to one of the extremities of the pipe
- STEP 3: Attach pipe to leg A
- STEP 4: Repeat STEP 2 and 3 twice again
- STEP 5: Next pipe should be longer
- STEP 6: Go back to STEP 2

- Remind them that there is no correct rule and there are many different ways to describe how to build the same thing.

# Play

*5 minutes*

- Allow students to find interesting shapes by breaking the rules they are following to create new versions of it.
- Building fractals is different than drawing because, in the end, you have an object you can throw, wave, spin or wear. Incentivize students to switch between the mindset of following the steps and stepping back and looking at their creations in a playful way.

# Share

*10 minutes*

- Have students partner with another. One student will become a builder and the other the reader. The reader will read their own rules, but do not reveal the originally built model. The builder will try to recreate the model from what they hear.
- Have them share their interpretation before revealing the original. Then switch roles.
- Your students can take this time to edit their rules and try this exercise with another partner.

# Reflect

*5 minutes*

- Ask students:
- If they see a rule that could be made simpler to read or to build without changing the end result.
- If they found something interesting they didn’t know before and if that helped them to build their creations.

- Let your students think about the following:
- Sometimes a slight change can make a big difference in the outcome of the fractal.
- How we write, interpret, and receive feedback.
- Highlight how completely different one build can be from the other by changing, removing or adding one single step.

# Day 3: Sierpinski Pyramid

*Total: 45 minutes*

# Imagine

*5 minutes*

**If you choose to start with the Sierpinski Pyramid, take some time to do the IMAGINE section of Drawing Fractals first.**- Show your students examples of how a fractal looks in three dimensions:
- Computer generated
- Romanesco broccoli
- 3D Sierpinski pyramid
- Menger sponge

- Example images can be found on the
**Downloads**. - When students see the Sierpinski pyramid ask your students the following questions to draw conclusions as to how it was built.
- What is the difference between the Sierpinski triangle and the Sierpinski pyramid?
- How many triangles have in the pyramid? And how did you find this number? Remember that 4 pyramids make another triangle on its own!

# Create

*15 minutes*

Activity

#### Construct a Sierpinski Pyramid

- Your students will collaboratively build a Sierpinski pyramid starting with the 3D shapes, the tetrahedrons from Strawbees and construction pipes.
- You can build tetrahedra ahead of time or right in front of your students as an example, then let them figure out the connectors and pipes needed.

# Play

*15 minutes*

- Once students start to piece the tetrahedrons together, other shape variations can emerge before the desired self-similar pyramid. Allow them to explore the possibilities, unexpected shapes, and properties that can happen.
- During this process, walk inquiring whether students think the structure they are building is strong and stable enough and propose ways to measure its stability.
- As the pyramid grows taller, it might be possible for some students to fit or place objects inside it. Let them decorate the pyramid with lightweight materials paying attention to not exceed the weight capacity.
- Questions to ask your students:
- Can you estimate how big the Sierpinski could grow?
- How many pipes would be needed to add another “floor” to the pyramid?
- What happens if they use pipes with half of the original size?

# Share

*5 minutes*

- As this is a collective build, separate some time for appreciation and stimulate a sense of pride for what they have built together
- If the Pyramid is built on a place that is accessible to other students and teachers, invite them to have a look at what they made and allow students to present their creations.

# Reflect

*5 minutes*

- Mention to your students that fractals are a topic in many knowledge domains:
- Mathematicians describing analytically properties of complex shapes.
- Biologists explaining nature’s behaviors by comparing them with fractals.
- Artists expressing themselves through the intricate patterns that can emerge from repetitive procedures.
- Architects building with modular structures.
- Computer scientists working with recursion.

- Ask students if they can identify something they do in their lives that has to do with fractals or if they know people who they think are doing fractal-related activities.

# Preparation

*30 minutes*

- Ask students to research the following individuals and photo examples of their work:
- Aristid Lindenmayer
- Benoit Mandelbrot
- Wacław Sierpiński
- Helge von Koch

- Have a look at the available tutorials and online resources that could be used in your class. Give preference for media and formats that you have used successfully in the past and make sense in your classroom environment. Some of the resources listed below might require internet access and access to websites that might not be allowed in your school:
- You can also use a website that is made to generate L-Systems and fractals, here is a short list: