Closed Curves, an Activity

A closed curve is a shape you can draw, without lifting your pencil, that begins and ends at the same place. An example of a closed curve is the boundary of the shape shown above. In this post Occupy Math is going to use a type of shape he discovered by accident that makes a large number of interesting closed curves. This is one of Occupy Math’s activity posts. For the accompanying booklet with questions for students and fifty shapes click here. Here are the shapes as clip art PNG files in a zip archive. This activity is intended for K-4 students. The rest of the post is about the math used  to build the activity.

Closed curves are, as noted above, shapes beginning and ending in the same place that you can draw without lifting the thing you are drawing with. Mathematicians think of these curves as embeddings of a circle into the plane. The idea of embedding is this: take each point in a circle and have a rule for where to take that point into the plane. The position in the plane must move continuously as we go around the circle. If we think of a circle as going around 360 degrees with an angle named θ, then the closed curve is all the points (X(θ),Y(θ)) as the angle goes around the circle.

In case you are not used to it, the symbol θ is the Greek letter that math people used as a placeholder (or variable) for the number from 0 to 360 degrees that says which way you are facing. The symbols X(θ) and Y(θ) stand for the formulas that compute the x and y position of a moving point from the value of θ as we go around the circle: imagine that you are standing at the center of the circle and turning to look at all of it. This can get complicated as the colored-in example shape above shows.

Occupy Math’s editor rephrased the math above as follows. Imagine you have a magic paint brush that gets longer and shorter depending on which way you are facing and paints a squiggle as it goes. If you know the formula for how long the paint brush should be in each direction, you can use that formula to define a squiggle. I would add that you may have to go around the circle more than once and the rule may be different on different times around the circle. Occupy Math is sad to report that he does not have a supply of magic paint brushes.

While Occupy Math was working on figures for his calculus book, he was trying to create a basket-weave-like figure. A coding bug in the program that was supposed to do this made a figure that looked a bit like a manta ray, shown above, instead. It turns out that the manta ray was part of a whole family of curves with the following formulas:

The numbers N and M are whole numbers and c and d are numbers between zero and 2π. Do not worry if these formulas are above your level — Occupy Math is including them for readers that can use them to make their own shapes.

Facts about these shapes.

Occupy Math found these shapes by accident, but he has figured out some rules about how the four numerical parameters affect the shapes.

• If N and M are both odd, then the squiggle that is drawn is bilaterally symmetric. These shapes have a left and right side that are mirror images of one another. The manta ray shape is an example of this.
• If N and M are both even, then the squiggle that is drawn is rotationally symmetric. For these shapes, rotating them through half a turn leaves them the same shape. The blue shape at the top of this section is an example.
• Mixing even and odd N and M generates shapes with little or no symmetry.
• The number of times the squiggle intersects itself increases as the ratio c/d or d/c gets larger.
• Small values of N and M make simpler shapes: the blue-and-orange picture above uses N=M=2, for example.

The directions from the activity booklet.

The images here are intended to be used to help younger students understand the complexity and pattern of shapes that can be made by sketching a closed shape without lifting their pen or pencil. These images are © Daniel Ashlock on behalf of Occupy Math. These images are licensed by their creator for use in classroom exercises, home instruction, or for use as individual clip art by anyone.

Questions

1. The shapes on the pages of the activity booklet have varying levels of complexity. All of them were drawn with a single curved line that starts and ends at the same place. Given they are all the same in one important fashion, what makes them more or less complicated?
2. There are at least two different types of symmetry in these shapes. Can you figure out what the big categories of symmetry are? Hint: the images are not divided evenly between the two types of symmetry.
3. Think of the lines as fences that make enclosures. What makes these shapes have different numbers of enclosures?
4. Some of the shapes are just abstract, others resemble natural shapes. What is it that makes some of the shapes look like natural objects?

Look at the shapes and think about things you like and do not like about them. Now make your own shape that has the qualities you like.

Have fun with this!

As we saw with the earlier activity post on coloring book math, self intersecting shapes give us a fertile ground for creative expression. Occupy Math’s sister gaming blog even did a post that used these shapes to make potion bottles for use in fantasy role playing games. Some of those are shown in the picture above.

Kardions

 

Here are a couple of examples of different ways you could go with coloring Occupy Math’s new shapes. It is possible that colored pencils, paint, or crayons would be a good way to go as well. The mathematical point here is both to connect art and math — not for the first time in this blog — and also to show how the simple idea of a closed curve can manifest in many different ways.

If you would like a post on how Occupy Math created the marble-like textures used in coloring some of the examples, please say so in the comments or send an e-mail. These textures tile correctly for use as backgrounds. Are there other topics you would like to see? Activities you would like Occupy Math to do a blog on? Please comment or send a note to dashlock@uoguelph.ca!

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics