Week 40: Tensegrities

In math we find balance and equilibrium. We balance equations. We keep balance by using properties of identity (multiply by 1 or add zero), Properties of Equality (mirroring operations), and by using the many other ways to manipulate and play with structures in math.

This week’s math is about equilibrium through building tensegrities (see photo above). Buckminster Fuller coined the term “Tensegrity” by combining “tensional” and “integrity.” He described the structure as, “Islands of compression in an ocean of tension.”

For this week’s activity, grab building toys, hot glue and sticks, or straws and string to create tensegrities. Learners can work on the simple design in the photo above, or on polyhedrons, bridges, sculptures, and more. Some challenging shapes would be polyhedral structures or towers. The goal is to create a structure that uses the tension of strings and the weight of the objects in positions that reach equilibrium.

For high school or middle school students, force diagrams may be a fun activity as well. Think about the moments and forces that balance in each structure created. There is a great “Beyond the Brick” video here. For strings at angles, there are some great trigonometry applications to play with.

Week 39: Toothpicks

A box of toothpicks can lead to an afternoon of entertainment. This week learners can play with the toothpick sequence. The sequence produces really interesting geometries and lines as it grows. I recommend watching Numberphile’s Youtube video on this sequence here. There is also OEIS’ website that allows for play with variations and many iterations. Grab a box of toothpicks and let’s begin:

Start by placing a single toothpick:

And then place toothpicks centered at each end:

And then place toothpicks centered at each end again:

Repeat this process at the ends that are available:

I also made a GIF in Procreate (stop animation is a wonderful way to play with all sorts of math):

Allow for play with the toothpicks to see what other mathematical patterns and tessellations are created.

Another option is to use graph paper to draw this sequence. Have fun!

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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

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 19: Math Haiku

Poetry forms are like a puzzles. You have to take the words you want to say and rearrange them, find synonyms, and reformulate them until they can fit in a form. This problem solving is so similar in math.

One of the first forms to play with is the Haiku. It is a three line poem with no rhyming scheme that fits a syllable pattern of 5/7/5. Traditionally there is a season mentioned (Kigo) and a cutting word to compare two ideas (Kiru). Learners can try to do a traditional Haiku, or they can just work with the syllable pattern to start. This can be done in any classroom to contemplate the concepts that are being learned in a different way. When we relate these abstract ideas to our inner beings, we remember.

Once poems are complete, maybe a work of art can complement it.

Here are some that I wrote. Please share yours!