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.
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:
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.
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:
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.
After teaching this a few times this week, I created a video so those that may want to pause and draw at their own pace while playing with isometric paper. Feel free to contact me if you have any questions. I love doodling on Isometric paper – enjoy! Here is a link to the isometric hands-on math activity and here is the paper.
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:
Sometimes the simplest things have wonder hidden within. This week, learners can play with the angles of polygons. How many degrees are in a triangle? In a quadrilateral? In a hexagon? Is there a pattern?
Here is a warm-up activity:
Draw a triangle (any triangle), and cut it out.
Next, rip the corners off:
Now, here is the fun part… put the pointy angles together. What do you get? Try it with lots of triangles and see if you always get a straight line. Rather than lecturing or telling learners that triangles have 180 degrees (or pi radians), let them discover. They can even create art ( I like to make my angles into perspective path doodles.)
Now do the same with a four sided shape. What do you notice? Is it the same for all the ones you can create?
Now do the same with 5, 6 or more sided shapes. There is a rule to be found. Try to discover it if you don’t know. I will put the rule at the bottom of this post.
I did this twice last week with virtual classrooms through the Covid-19 isolation. Students from kindergarten to middle-school ate it up. We used it as a warm up activity (10-15 minutes) prior to doing some loop-doodle math and/or other activities.
The rule for simple polygons is that for n sides there are 180(n-2) degrees. Or you add 180 degrees every time you add a side.
I am trying to tell my 15yr old daughter that an elective high school credit in Graph Theory would be fun next year. Of course I do this as subtly as possible – I start drawing coloring sheets for this post on my iPad and then carefully shade them in. All three of my children slowly sneak up behind me and breath in my ear.
“You know that it will never take more than four colors” I state.
“Really?” I hear my oldest daughter say with a sense of wonder in her voice. “Can I make one?” she asks reaching for my device.
She takes over the iPad. I go for a run. I clean up a bit. She is still designing, thinking, coloring. A wave of gratitude flows over me “Thank God that coloring isn’t just for kindergarten.” We are so blessed to have the abundance and time to be able to color, play, and contemplate.
She finishes her design. “It looks like the beautiful cobbles on our Oregon beaches.” I think, then say.
“That’s what I was going for.” She says. Then gets up and goes back to her school work.
This week I challenge learners to play with coloring sheets. Make your own. Share them. Color them. Contemplate them. Can you restrict the coloring to four colors? It may take some problem solving for more complex sheets.
In graph theory, there is the study of graphs that are made up of nodes (vertices) that are connected with lines (edges). Create a graph for one of your coloring sheets, where the regions are nodes and lines connect the regions that touch.
You could also create a graph with nodes and edges and then the coloring sheet to go with it.
Below are a couple examples (some blank for you, my readers, to use):
Welcome to 52 weeks of math! I will be posting a new activity every Wednesday for 52 weeks of hands-on math. Week 1 is one of my favorites – drawing with thread.
In this activity, learners will play with their rulers (or thread) to create curves with lines. The idea is to have students draw straight lines close together with various slopes to create curves. For younger ages anything goes! For middle school and up, it is a great intro into lines and the Cartesian Plane. Below is an algebra video I made for a class back in 2016. It gives you the basic idea. I also have modifications and additional ideas below.
Possible reflection questions:
Line segments – what are they? How many points do you need to make one?
What is a tangent line? What can have a tangent line?
Do the distances change with each line? Why?
Middle school (use the questions above as well):
What is slope-intercept form?
How can you change the outcome of your art if you change the axes of the graph to have angles other than 90 degrees?
How does the slope change? What observations can you make about the ratios?
How do the slopes change when a quadrant’s set of lines is reflected over an axis?
Algebra (use the questions above as well):
Is there a pattern to observe if the lines are written in standard form or point-slope form?
What type of curve do you think you have approximated?
Can you write a function for the change in slope?
Geometry (use the questions above as well):
When creating reflections over an axis, are there any patterns with sets of parallel or perpendicular lines?
Can you write a function for the change in distance for each line?
Where do you see rotational symmetry, translations or reflections in you art?
Trigonometry (use the questions above as well):
Could you create a similar work of art using polar coordinates?
Can you write a function for the change in angles for your art?
Can you write a trigonometric function for a pattern in your art? Are there any periodic behaviors?
Calculus (Use the questions above as well):
Can you create a function for the slopes? If so, what is this function in relation to the curve you created?
Can you determine the function for the curve you created?