Week 37: Cantor Set Kirigami

For this week’s activity, learners can play with Cantor Set Kirigami. The Cantor Set is created by drawing a line. Next, remove the middle third of that line (this will create 2 lines). For each of the two lines just created, remove the middle third (this will create 4 lines). Continue with this process until the lines are too thin to work with.

Menger Lux Blox

Some of the fun characteristics to notice is the pattern of the line lengths (1, 1/3, 1/9, 1/27,…), the number of lines generated with each iteration (1, 2, 4, 8, 16, …), the fact that this set is infinite, yet not countable and that it gets smaller and smaller with each iteration.

I created a fun Kirigami Cantor Set and have the template below with a video how-to. Enjoy!

Learners can also draw Cantor Set cities, roads, abstract art, and find many ways to represent this simple fractal. The Menger Sponge is one form of a Cantor set in 3d.

If you need a jpg. of the sheet:

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):

I’m Attracted to Attractors

So many plots and mathematical musings throughout my life have brought on a sense of artistic beauty and awe within my being. In the windowless halls of engineering firms I have smiled at harmonics, or in a homeschooling room squealed in glee when I stumbled upon Pisano periods by trying to play Fibonacci on the piano. Lately I have been playing with attractors. These dynamic systems make me stay up late fiddling with their metamorphic and chaotic beauty.

I came from a Matlab world and have had to teach myself some more cost efficient means of play with javascript and python. The code below is just one of my playgrounds. I don’t know if there is a name for this attractor (please let me know if you know its name). Enjoy:

Attractor1:

With functions you have inputs (x) and outputs (y) that can be plotted on a plane (x,y). With the images below, the x and y values are computed using an initial value of (1,1) and then the next (x,y) is computed using the previous coordinate’s values. The equations are shown below:

x = sin(a1 * oldx) * cos(a1 * oldy) – sin(a2 * oldx);
y = cos(a3 * oldx) – cos(a3 * oldx) * sin(a4 * oldy);

Here is a gallery of some of my outputs:

For the images above, I calculated x and y using a1, a2, a3 and a4 coefficients and the previous x and y values (oldx, oldy). The initial point was (1,1). In the code below, there are only 300,000 points (compared to millions in higher res images). You can play with the values of a1, a2, a3, and a4.

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

I like the p5.js editor. Click here to play. I would say that fiddling with this is a great idea for an “Hour of Code.”

If you like to play with sheets or excel, which is not near as pretty, I made a sheet for you here. This is also handy if you want to see the array of values for (x,y).

Week 35: Yarn-it-up Hyperbolic Space

This week let’s play with yarn! We are going to play with hyperbolic space. You will need some yarn and a crochet hook. You don’t need to know how to crochet, but you will need a little patience and a lot of desire to play. These don’t have to be perfect, and “mistakes” just add to their beauty. There is a great TED Talk on crochet coral that is a great intro into this activity as well (click here), or just watch the videos I put together below. I thought about drawing hyperbolic space as an activity, but decided that having the tactile fluffy math in hands would be much more exciting this week:

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 33: Ultimate Tic-Tac-Toe

Games don’t have to be complicated to require some good thinking skills. We all learn tic-tac-toe when we are younger. We soon learn how to always come to a stalemate with an equal opponent. Once you get the strategy, it can get a little boring… But what if we add a layer (or two) of moves. What does this do to the strategy of the game? Is it so easy to predict your opponents next move?

This week I recommend learners create a tic-tac-toe board that has tic-tac-toe boards in each square. Here is a video of how to play:

My learners contemplated:

  • How many times can you nest the game before it’s too complex?
  • With each layer added, how much longer and more difficult would it be? (how many moves are there?)
  • How is this like a fractal?
  • Could you keep a game going with one move a day for how many days with 2-nested?, 3-nested, 4?
  • What does the game tree look like?
  • How many ways can you play tic-tac-toe vs ultimate tic-tac-toe?(think combinations). What is the combinatorics calculation look like for this?

If you are wondering how I was able to do this in the time of Covid… I use a digital white board and label squares so it is easy to say the next move. You can also use a shared google drawing or a google spreadsheet to play (here is one for you.)

Another blog (Games for Young Minds) that does a great post on this game is here. Math with Bad Drawings also has a great post here. As you can see, this is a fun game with us mathy folks everywhere.

Flipping Origami Class Video

After teaching this a few times this week, I created a video for those that missed it or want to go back. We made two different origami toys that have some flipping fun. Feel free to contact me if you have any questions.

Week 32: Isometric Drawing

Let’s get out our pencils, isometric paper, and thinking caps this week! Isometric drawings are often used in engineering and design as a way to display 3D ideas. They can also be used to create optical illusions and escheresque works of art.

To start, print some isometric paper, or set your digital drawing program to isometric drawing guides. Start by drawing simple objects, like a cube, and play with shading.

Once comfortable with basics, start making skeletons for shapes, linking sides that don’t make physical sense, and thinking about objects that would allow you to go up and down at the same time. Below are some examples and videos to play with:

Opal’s sketch (13yrs old)