# SIERPINSKI TRIANGLE – Fractals, Colour Schemes, Repetition

Students follow written instructions to draw a Sierpinski triangle and use markers to colour it in a colour scheme of their choice.

80 Minutes

Mathematics
Visual Arts

#### Vocabulary

colour scheme equilateral triangle fractal iteration pattern self-similarity sierpinski triangle

#### Materials

Markers Rulers Pencils Erasers Scissors Glue Sticks Bristol Board - 50.8 cm x 76.2 cm (20" x 30") - 1 per 6 students

## Steps

### Step One

1. Follow the instructions on the How to Make a Sierpinski Triangle worksheet to draw a Sierpinski triangle. (Downloads - Sierpinski.pdf)
2. Keep making smaller and smaller triangles as many times as you like.
3. Be sure to measure carefully.

### Step Two

1. View the colour wheel available on this website to figure out a colour scheme you like.
- Complementary - colours opposite each other on the colour wheel, e.g., red and green; yellow and violet; blue and orange
- Analogous - colours beside each other on the colour wheel, e.g., yellow, yellow-orange, and orange; red, red-violet, and blue; blue, blue-green, and green
- Triad - 3 colours evenly spaced around the colour wheel, e.g., green, orange, and violet; yellow, red, and blue; blue-green, yellow-orange, and red-violet

### Step Three

1. Use markers to colour your triangle.

### Step Four

1. Carefully cut out the triangle.

### Step Five

1. Join your triangle with 5 others.
2. Notice how the triangle pattern can keep growing and growing.
3. A never-ending pattern like this is a fractal.

## Learning Goals

Students will be able to:

• follow written instructions to draw a Sierpinski triangle with at least 3 iterations;
• explain what self-similarity in fractals means;
• explain why they chose their colour scheme;
• demonstrate technical accomplishment and creativity.

## Extensions

Have students:

• use construction paper to create a Sierpinski triangle using an animal or human face instead of a triangle;
• use the cut paper shapes to create an animated film that demonstrates how to make a Sierpinski triangle;
• share the video with others.

## Prepare

1. Download and display the Repetition and Colour Wheel posters available on this website.
2. Download and copy pages 1 and 2 of the How to Make a Sierpinski Triangle worksheet, enough for each student to have one. (Downloads - Sierpinski.pdf)
- For younger students copy only page 3 of the worksheet and have them colour it in a colour scheme of their choice.
3. Teach or review fractals.
- never-ending pattern
- geometric or organic shapes
- one of most important properties of fractals is self-similarity – they look the same no matter how big or small they are
- a small portion of the shape looks the same as the entire shape
- they start with a simple shape that gets more complex and beautiful the more it is repeated
- fractals are found everywhere in nature, e.g., pineapples, ferns, shells, trees, and pinecones
4. Provide time for students to work with rulers drawing and measuring lines and shapes.
5. Teach about the Colour Wheel and colour schemes.
Colour Wheel - a tool for combining colours so they look good together
- first version was designed by Sir Isaac Newton in 1666
- most common version is a wheel of 12 colors based on red, yellow and blue (primary colours)

Analogous - colours beside each other on the colour wheel
- often found in nature and are harmonious and calm

Complementary - colours that are opposite each other on the color wheel
- create a vibrant look and make shapes stand out

Triad - 3 colours that are evenly spaced around the colour wheel
6. Download images of fractals found in nature and art from the Internet, e.g.,
Broccoflower
Leaves
Shell
Animated Construction Sierpinski
Computer Art
Fractal Clock

## Introduction

1. View and discuss the images drawing attention to the self-similarity in the shapes found in nature and the artwork.
2. Explain that mathematics helps us understand the shapes better.
- shapes that appear to be completely random have an underlying pattern that determines how the shapes are formed and what they will look like
- this knowledge can be used in medicine - e.g., studying diabetes; geology - e.g., studying frequency of earthquakes; meteorology - e.g., studying clouds and rain; and all kinds of computer graphics - e.g., when creating textures for computer games
3. Introduce the Sierpinski triangle.
- Waclaw Sierpinski a Polish mathematician described the triangle in 1915
- he did not invent it
- similar patterns are found in 13th-century mosaics and carpets
- is starts with an equilateral triangle
4. View the Animated Construction Sierpinski triangle or demonstrate the process on a large chart paper.
5. Introduce the challenge.

## Activities

### The Challenge

1. Follow written instructions to draw a Sierpinski triangle with at least 3 iterations.
2. Explain what self-similarity in fractals means.
3. Explain why you chose your colour scheme.
4. Demonstrate technical accomplishment and creativity.

### The Process

1. Ensure that everyone understands the challenge.
2. Establish success criteria with your students, for example,
I know I am successful when I have:
measured accurately
- accurately drawn a Sierpinski triangle with at least 3 iterations
- coloured the design using a specific colour scheme
- explained why I chose my colour scheme
- explained what self-similarity in fractals means
- kept the paper in good condition
3. Guide students through the steps outlined in this lesson plan.
4. Observe students as they work.
5. Provide individual assistance and encouragement.

## Sharing

1. Once all the images are complete ask students to share them in partners or small groups.
Look closely at the images and how they are made.
- Share thoughts about the work.
- Talk about how the number of iterations affects the overall design.

- Talk about how the colour scheme used affects the overall design and what makes you think so.
- Tell what was satisfying about making the Sierpinski triangle and explain why.
2. Ask some students to share their ideas with the whole class.
3. Display the individual images in the classroom so students can view them over the next few weeks.
4. Display the giant Sierpinski triangles made by groups of students and invite others to add to it.

## Assessment

1. Observe students as they work – thoughtful focus, discriminating, seeking more information, elaborating, experimenting.
2. Observe students as they discuss their work – active listening, insightful contributions, supporting ideas with evidence found in the artwork and from personal experience.