Introduction
Fractals, introduced in 1975 by Benoit Mandelbrot, are patterns that repeat infinitely. They can be used for computer graphics,
economics, antennas, and other things. Many fractals are created with math, but they can also be found in nature. When displayed as a
picture, a fractal equation becomes a work of art. Throughout this Webquest, you will be provided with activities and information about
fractals, as well as the Golden Ratio and Fibonacci Sequence. Beginning in the next tab, follow the directions to complete the tasks.
Task
To begin with, fill in the first set of numbered questions in the "Process" section, using the link featured in the "Process" section. The answer key for the questions is in the "Evaluation" section.
Next, draw a Koch Curve fractal on a piece of paper using the examples and instructions from the first two links in the "Process" section. Your drawing does not have to be infinite.
Now, fill in the second set of numbered questions in the "Process" section using the next three links, also featured in the "Process" section. The answer key for the questions is in the "Evaluation" section.
Finally, create a necklace using the Fibonacci Sequence. The instructions can be found in the "Process" section.
Process
Use the link below to answer the following questions about fractals and check your answers in the "Evaluation" section.
https://fractalfoundation.org/fractivities/FractalPacks-EducatorsGuide.pdf
1. There are three main types of fractals. Where can each of them be found?
- _______________
- _______________
- _______________
2. What are two main types of fractals in nature?
- _______________
- _______________
3. Fractals in ______________ use equations.
4. Fractals in ______________ use repeating shapes.
5. A simple fractal equation is _______________.
Draw a Koch Curve Fractal using information from the previous link and the following link.
https://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html
Use the links below to answer the following questions about the Fibonacci Sequence and Golden Ratio. Afterwards, check your answers in the "Evaluation" section.
https://fractalfoundation.org/OFC/OFC-11-1.html
https://fractalfoundation.org/OFC/OFC-11-2.html
https://fractalfoundation.org/OFC/OFC-11-3.html
1. The first ten numbers in the Fibonacci Sequence are ___, ___, ___, ___, ___, ___, ___, ___, ___, ___
2. The Greek letter ________, also known as φ or the Golden Ratio, represents a number that, when rounded to the nearest hundredths place, is approximately equal to ______.
3. The Golden Ratio can be used to form shapes such as the Golden ____________, which can, in turn, form the Golden __________.
4. A seashell is an example of a __________ in nature.
5. Using an irrational fraction for the __________ ____ ______________ in a spiral allows for a more complicated pattern.
Use the instructions below to create your Fibonacci Sequence necklace. You will need a marker, string, scissors, and beads. Make your necklace colorful. The following link gives a partial example of a necklace. Fibonacci Sequence Necklace Read the instructions once all the way through for better understanding before making the necklace.
1. Cut a piece of string that is plenty long enough to fit over your head when it is tied into a circle. Be sure to make it extra long so that there is plenty of space for knots.
2. Fold the string in half and mark the half-way point.
3. Tie a knot at the half-way point.
4. Fold the string so that the knot at the half-way point touches a point near one end of the string. Mark that point and the fold (one of the quarter points of the string).
5. Cut a shorter piece of string that will hold one bead with a knot at each end.
6. Tie one end of the new string, slide a bead on, and tie it to the necklace on the marked side of the half-way knot.
7. Repeat steps 5 and 6, tying the second string on after the first one.
8. Repeat steps 5 and 6 with a string that holds two beads. Attach this in sequence after the other strings. Make sure that the beaded strings are evenly spaced along the main string.
9. Repeat steps 5 and 6 with 3 beads, 5 beads, 8 beads, 13 beads, and so on until you reach the quarter mark. Depending on the thickness of the string, additional knots or other techniques may be necessary to insure that the various strings stay in placeon the main string. By now, a quarter of the main string should be filled with the smaller beaded strings.
10. After reaching the quarter point, begin to put less beads on the necklace again. For example, if you previously put on 1, 1, 2, 3, 5, 8, and 13 beads, you should now put on 8, 5, 3, 2, 1, and 1 beads. Remember to space the strings evenly along the main string; youshould arrive back at one bead when you reach the mark that you made near the end of the string.
11. Now you should have half of the string filled with smaller beaded strings in a triangular shape, and the other half of the string should still be blank.
12. Fold the string at the half-way knot and mark the quarter point and the point near the end on the blank side, using the already beaded side as guidelines.
13. Repeat steps 5-10 with the second half of the necklace. By the end of step 9, 3/4 of the necklace should be filled with beads.
14. Now, the whole main string should have smaller strings attached to it except for the two ends. Tie these together, and the necklace is complete. It should have two triangular-shaped Fibonacci sequences: one to hang over your chest and one to hang over your back.
Evaluation
Answer Key #1:
1. nature, geometry, algebra
2. branching, spiral
3. algebra
4. geometry
5. Z = Z² + C
Answer Key #2:
1. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34
2. Phi, 1.62
3. rectangle, spiral
4. spiral
5. angle of rotation
Conclusion
After completing the tasks and studying the provided information, you should now have a better understanding of what fractals are, the
different types of them, and what they look like. Although this Webquest only covers the basics of fractals, there is much more to be
learned about them.
Credits
Additional information about fractals can be found in the following sources:
https://fractalfoundation.org/resources/what-are-fractals/
https://users.math.yale.edu/public_html/People/frame/Fractals/