Week 28: Apollonian Gaskets

Apollonian Gaskets are a creative way to play with circles, fractals, and mindfulness in math. Students can cut out circles and place them within circles or practice their drafting skills with a compass and ruler. The idea is to draw a large circle and then fit smaller and smaller circles inside as closely as possible (tangent circles). For a great resource on gaskets, click here. For the theorem behind them click here (Descartes’ Theorem). This is a great time to review circle geometry. For middle and high school students, learning Descartes’ Theorem can be a fun. I have found that learners’ desire to be exact in their art has lead them to want to learn the math. However, these sketches need not be perfect; just have fun.

Step 1 – Draw a circle with a compass. Mark the center and sketch a diameter line. It is nice to use pencil and ink for these steps to be able to erase some lines and ink in your circles.

Step 2 – Add marks on the diameter that divide it into fouths. Draw two circles that have a diameter of half of your original circle by placing your compass center at the 1/4th and 3/4ths marks.

Step 3 – Draws circles that fit in the largest two spaces, (1/3 of the radius of the original). Once you set your compass to 1/3 the size of the original circle, zero in on the center of the next circle by using the perpendicular line to the original diameter and moving the compass around until it only touches the outer circle and two inner circles.

Step 4 – If you want to be exact, then you need to use Descartes’ theorem to calculate the size of each circle to proceed. You can also use a circle template or stencil set to eyeball tangent circles. The idea is to continue to fill in the spaces with more and more tangent circles. The sketch below is just approximated for a “quick sketch” and not done with the precision of Descartes’ Theorem.

Before writing this entry, I played with cutting an Apollonian Gasket from acrylic. Here is my design for a math toy. My files are on Etsy.

Friggatriskaidekaphile

frig·ga·tris·kai·dek·a·phile
/fɹɪɡə,triskīˌdekəˈ,fīl/

noun
a person who has an extreme love for the friday the thirteenth.

Week 27: Patterns in the Paper Weaving

I love fiber arts and weaving. So, I have one more weaving post for this series, but this time it’s with paper. This activity is great for all ages and can be done with ribbon, bias tape or strips of paper. I like to use origami paper strips.

The idea here is to play with repeating patterns and find where you can create secondary patterns, tessellations, and other shapes. Learners can experiment with over/under weavings and see what amazing patterns emerge. Make sure to have lots of colors, and encourage experimentation (diagonal, skipping, color patterns in warp and weft, gradients, etc.)

Math is beautiful. Math is playing with patterns and abstract thoughts. This is a wonderful activity to tickle the math parts of our brains.

Weavings above are done by my family and friends. My daughter and son really made a week of weaving papers.

Some questions to ponder:

  • Can you create a matrix or array that can represent your pattern?
  • For precalc and above – what would operations on your matrices result in if you mapped colors to numbers?
  • Can you create curves or other optical illusions with weaving techniques?
  • How can weaving relate to our numbers? (number line, even/odd, etc.)
  • Can you weave a function? What is the input and output?

Week 26: Musical Math

This piece was created using the Pisano Periods created by dividing the Fibonacci sequence by 12 and 24.

You don’t have to be a musician to play with music and math. This week, I encourage learners to experiment with sound and patterns. Below is a list of ideas to experiment with:

  1. Create a rhythm as an individual or a class that follows a sequence and build on it, (drums can be hands on tables or buckets).
    • Drum a Fibonacci set or other mathematical beats, (0, 1, 1, 2, 3, 5, repeat), with various instruments.
    • Drum a decay rhythm of holding notes, (ex: 8 beats, 4 beats, 2 beats, 1 beat, 1/2 beat, 1/4 beat, repeat).
    • Drum in a circle where learners explain the pattern of a selected drummer with math. Take turns creating and guessing patterns.
  2. Use a tuning app to study notes on an instrument in Hz. Plot the notes of an octave – what do you see? (This is better for learners that use the Cartesian coordinates.)
  3. Take a concept that is being studied and represent it with music. (Addition, subtraction, variables, exponents, etc.)
  4. Create a map from a sequence or set to a melody on an instrument. I did this with the Pisano Periods a couple of years ago and had a lot of fun with it.
    • To do this:
      • Determine the set of numbers you would like to use: {3,1,4,1,5,9,2}
      • Map the range of the set to a note: 1 = C, 2 = C#, 3= D, 4 = D#, etc.
      • Play your melody: