This creative demo illustrates the basic principles of fractals. You will make your own fractal triangle composed of smaller and smaller triangles. Each time the pattern is repeated, the white area decreases because another triangular hole is made. The perimeter however increases because there are more sides that the perimeter measures around.

Fractals appear to be very complicated shapes, but they can be made by repeating simple rules. Complexity can “emerge” from simple rules.

Key Terms

Fractal – a shape that contains a similar or identical pattern that repeats at every scale, large and small

Midpoint – a point in the center of a line; the point that is the same distance from both ends points of a line

Iteration – the repetition of process

Area – the amount of space contained inside of a flat shape.

Perimeter – the lengths of the distance around the outside edge of a flat shape.

Supplies

A Piece of Paper
Ruler
Pencil
Colored Pencils, Markers, and/or crayons

Step 1: Take a Piece of Paper, a Ruler, and a Pencil.

Take a piece of paper, a ruler, and a pencil.

Step 2: Draw an Equilateral Triangle With Each Side Measuring 8 Inches (20 Cm).

Step 3: Find the Midpoint of Each Side and Mark With a Dot. If Using the Recommended Measurements, This Should Be at 4 Inches (10 Cm) on Each Line.

Step 4: Connect the Three Dots to Make Another Triangle in the Middle of the Original. Color the New Triangle If Desired.

Step 5: Continue on This Process of Finding the Midpoint of Each New Triangle to Make More Triangles, Coloring Them in After Each New Size Is Created.

Continue on This Process of Finding the Midpoint of Each New Triangle to Make More Triangles, Coloring Them in After Each New Size Is Created.

Fun Facts:

- There are many types of fractals, this one is named after the Polish mathematician Waclaw Sierpinski. However, similar patterns have been found in medieval roman floors.

- Understanding mathematically how complex fractals form from simple rules can help us to better understand complex forms in nature such as trees, coral, coastlines, and flames.

- The word “fractal” was coined by another Polish mathematician Benoit Mandelbrot.

Further Study and
Exploration:

Fractals Are Typically Not Self-Similar (High School+) [App. 20mins]

https://www.youtube.com/watch?v=gB9n2gHsHN4

What Is A Fractal (and what are they good for?) (Middle to High School) [Approx. 4 mins)

https://www.youtube.com/watch?v=WFtTdf3I6Ug

How fractals can help you understand the universe – BBC Ideas (5th Grade+) [Apprx. 3 minutes]

https://www.youtube.com/watch?v=w_MNQBWQ5DI

https://mathigon.org/world/Fractals

https://cosmosmagazine.com/mathematics/fractals-in-nature

http://theconversation.com/explainer-what-are-fractals-10865

https://mathworld.wolfram.com/Fractal.html