Here are some simple animations with parametric equations. What you see below is a function and its inverse. If you click, you will get another semi-random equation.
For week 52 of 52 Weeks of Hands-On Math, I couldn’t resist a play on words: Math rocks! This week I encourage learners to share their math in the neighborhood. Create sidewalk chalk art with Fibonacci hopscotch, paint a math rock garden, make a math obstacle course by your home, or have a piece of outside artwork to share.
Our family created a math rock garden. Rocks are often painted, traded, and shared in our area. We may still do another epic hopscotch course before the summer is over. Math rocks, so let’s share it with our world.
Acrylic non-toxic paint pens were used with these rocks, but we have used watercolors, markers, and ink on rocks in the past.
Like what you see?
Moiré is an interference pattern that can occur in physics, photography, art, math, and more. As a photographer and mathematician, I have always enjoyed seeing these artifacts emerge. Today I played with some Moiré animation for fun in p5.js.
After playing, I of course googled Numberphile and moire and found another wonderful video of theirs:
I love thinking of mirror images when I am block printing. I will never forget the time I printed SPARK backwards on accident for a summer art camp and my kids laughed at the reverse phonics. This week I encourage learners to take a math concept, tessellation, or shape and create a print.
Ways to create plates for printing:
- For younger learners, foam boards easily take impressions.
- For those that are semi-comfortable carving, potatoes, apples, or rubber blocks can provide semi-soft mediums to carve.
- For those that are skilled with sharp objects, wood or lino-blocks may be preferred.
When I teach I say these words at least a few dozen times:
- Do not ever force a blade.
- Do not cut towards yourself or others.
- Keep your tools sharp and cleaned.
- Be in control.
If you are going to carve on a block, foam, or rubber sheet:
- Sketch a design on paper with a pencil. (Keep in mind the size of your carving surface.)
- Transfer the design by rubbing the pencil onto the carving surface.
- Carve your design. (You can either carve in, tracing your lines, or around them.)
- Roll ink on the block with a brayer.
- Place a sheet of paper (or fabric) on the block and burnish (or rub) it with a flat surface to make sure that it makes contact with the block.
- Peel the paper/fabric off. (This can take a few tries to get it right.)
For this post I carved a circle composition with the Fibonacci sequence in mind. I think I run faster with math shirts, so I printed one as well:
One of my children jumped in and we did a Sierpinski potato triangle. To use potatoes: Draw a sketch on the potato, cut out the design, and then treat it like a stamp.
This week I encourage learners to play with their animation skills. Take a math concept, problem, or design and play with ideas to animate it. Start simple to warm up and then build on the ideas. Flip books are fun. I recommend using thinner paper that can be seen through so it is easier to draw on top of the previous frame. Try the paper out before spending the time to draw. Flip books are an analogue GIF.
Here are the basic steps to drawing a flip book:
- Decide on your math topic to animate. (Fractals, projectiles, growth, decay, etc.)
- Draw the starting frame:
- Draw the next frame by tracing and/or using the previous sheet as a reference:
- Repeat the process of making new frames by referencing the previous:
- Once your frames are complete – flip it!
There are other ways to animate as well. There are apps such as iMotion to make stop animation films. I also enjoy using Procreate and Looom to create animated GIFs. Learners may prefer to use technology for their animations. I plan to cover this in some STEAM posts at some point. Please share your animations!
Get up and move! This week learners can dance their favorite equations, math symbols and concepts. Whatever topic is of interest or in the process of being learned is a great one to figure out the dance moves that go with it. I recommend taking 5 to 10 of your favorite moves and making it into a mathematically choreographed dance.
Here is one approach:
- Pile 1: Make flash cards for the vocabulary, plots, or concepts that you want to move to.
- Pile 2: Make flash cards for the body parts that you want to move with (hands, feet, legs, arms, whole body, etc.)
- Draw a card from each pile, turn on the music, and get some moves.
- Note: you can play with sequences, beats, or number of repeats as well. (This works in a virtual environment as well – have a dance break!)
Here is an example generator I made:
If you are a quilter, then you will be a pro with this week’s activity. For the last six months my quilting mother lived with us through chemo and we watched her quilt her heart out. Now that she moved back to her home, I had to laugh because she would have been so much fun making these tiles this week – I should have done this sooner! So many traditional and historical quilts use these squares (a great segue into history – Underground Railroad, folk art, stories, family histories, etc.).
This week I encourage learners to actually sew some tiles, specifically half-square triangles. Another name for these squares is Truchet Tiles. These are tiles that are not rotationally symmetric. The tiles can create a variety of patterns and tessellations. Make some tiles and then play. (The reason I encourage sewing is that the problem-solving, process, and mistakes lead to so much learning.)
If sewing isn’t your thing, then I have a JPG file below you can print and play with. There are also tons of maker ideas for these tiles (Paint squares of cardboard, wood, felt, etc.). A piece of graph paper will also work (shade in half of the squares diagonally).
How to sew:
- Iron the fabric that you wish to use.
- First cut out 25 squares of Color-1 and 50 squares of a Color-2. (You may want to do 2×2″ squares for small tiles or 10×10″ for larger tiles.) You can also do 36 squares of Color-1 and 72 of Color-2. Do you see the ratio? NOTE: Cut these squares carefully (millimeters matter).
- Next cut down the diagonals of 25 of Color-1 and 25 of Color-2 (The same amount of each color if you are doing more than 25 tiles).
- Now, take one triangle of each color and place them face to face and sew them with a quarter inch seam (make sure your seam width is consistent for all of this work).
- Iron the square you just made so that the seam is folded flat to one side and then place it on one of the squares of Color-2. Sew around the border to attach the half-square triangle to the square bottom.
- Now cut the extra fabric, while squaring the tile (make sure you use a grid or corner to make sure it is square as you cut).
- Repeat this for the other 24 tiles (or more if you so chose).
We also made draftboard versions of these tiles to play even more:
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!
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.
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:
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).