Week 36: Golden Angle Scavenger Hunt and Drawing Phi-Nominal Phi-lowers

The Golden ratio appears in nature all around us. Flowers and other botanicals often grow at an optimal (Golden) angle of about 137.5 degrees. For the 52-weeks of math activity, I encourage learners to seek out the Golden angle on a scavenger hunt. Take pictures or sketch in a nature journal the pinecones, flowers, and other botanicals that grow in Fibonacci/Golden Ratio spirals. Count the petals, trace the spirals, and collage the scavenger hunt together. Nature is one of the best ways to explore math.

Additionally, I created a Golden Angle grid paper for learners to sketch their own “Phinominal Phi-lowers.” Feel free to print it and play with the spirals and dots. Sometimes seeing flowers, pinecones and succulents can provide inspiration for unique flowers.

For a digital Phi playground and some more background information on Phi (click here).

Golden Angle Grid Paper

Sticky Note Sunflower

What you are seeing is a growth pattern of sticky notes that uses the Golden Angle (137.5 degrees) and then slowly decreases. This angle is commonly found in the plants all around us because it is an optimal angle for growth.

It was a lot of fun playing with the growth angle while creating memorizing spirals in this code, so I created a version for everyone to play with (see below):

https://editor.p5js.org/fractalkitty/present/vThUavVYE (on p5.js – sometimes this network goes down) or:

See the Pen sunflower by Sophia (@fractalkitty) on CodePen.

There is lots of fun to discover here. The fractal starts using the Golden Angle of Phi (137.5) and then decreases. You will find interesting behaviors when the angles approach numbers that divide into multiples of 360 more easily (40, 45, 60, 90, 120, 180, etc…). I wanted to do so much more, but had to stop somewhere.

If you want to know more about Phi, many mathematicians and creative types have presented it better than I (see below):

Week 34: Kirigami

I love paper cutting, so last week I did kirigami with some of my classes. What was so fun about this activity is the amount of play and discovery that happened with two simple supplies (paper and scissors).

Below are the videos I recorded for my classes to be able to go back and work at their own pace. These videos are just a starting place. There are so many methods for folding, cutting, and scoring that can be discovered and explored. My son made dioramas of forests and landscapes that fold with his creations. If you like pop-up books, this is a great place to start.

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:

Week 23: Fibonacci Weaving

If I am not playing with math, then I am tangled in yarn and thinking about math. This week I combined the two and wove A small fibonacci washcloth. Fiber arts are so versatile and an insanely fun way to express mathematics. In future weeks we will look at collage (wool applique), Cantor art, etc.

Fibonacci is a sequence where you start with 0 and 1. You then add those to get 1: {0,1,1}, and then you add 1 and 1 to get 2: {0,1,1,2}, and then you keep adding the last two numbers to get the next number. Here is a comic I created for Marie’s Atlas on Fibonacci:

For weaving, you need a piece of cardboard, small loom or even a notebook card (for thread weaving). The idea here is to create any form of Fibonacci patterns in your work. For mine I did 8 for a lower border, then 1 blue, 1 white, 1 blue, 1 white, 2 blue, 2 white, 3 blue, 3 white, and so on. Below is a sample. There are tons of ways to weave this sequence. You can use different sized looms, patchwork squares with areas that equal the sequence, etc. Have fun!