Mirrors, Paper Cutting, and Symmetry
Introduction: Mirrors, Paper Cutting, and Symmetry
Experimenting with hinged mirrors is a way to explore captivating kaleidoscopic patterns and their underlying mathematical symmetry. Combining hinged mirrors with paper cutting can capture these patterns, and link to art forms such as quilting, <PERSON>, and similar art forms from many cultures.
Mirrors, Paper Cutting, and Symmetry was developed as part of the Light and Shadow Workshop, using basic tools -- mirrors, lenses, and flashlights -- as the basis for explorations that can help build an intuitive understanding of how light works. Designed for out-of-school-time educators, the workshop is easily adaptable to other educational settings. The activities also work well for remote learning, since most materials are readily available or can be supplied in a teaching kit.
Supplies
Two Mirrors (Approximately 5x7” works well. Plastic mirrors sold for classroom use are safest, but any reasonably stiff mirrored material will work. Be careful of sharp edges.)
Tape
Scissors
Paper
Pen or Pencil
Optional (and easily substituted): Yarn, Embroidery Thread, String, Scraps of Colored Paper Leaves, Assorted Small Items as Available (for example coins, pasta, etc.)
Step 1: Making the Hinged Mirror
Take your two mirrors and lay them on a table, mirror side down. If they are rectangular, put the two shorter sides adjacent. Leave a very small gap (1mm or so) between the mirrors.
Make a tape hinge by taping the backs of the mirrors together. The small gap will allow you to close and open the mirrors like a book after they are taped -- if they don’t close all the way you may need to re-tape them with a larger gap. (If your mirrors have an adhesive back, you may need to cut off a strip of the backer to get your tape to stick.)
If your plastic mirrors have a protective sheet of plastic on the front, remove this now. Note that the surface of many plastic mirrors can be scratched very easily -- better to live with some fingerprints than scratch your mirror by trying to clean it too vigorously.
Step 2: Experimenting With Hinged Mirrors
Put the mirrors, partially open, on a table.
Try putting some small objects between the mirrors and moving the mirrors back and forth.
What do you see? What is surprising?
Spend some time playing with the mirrors, then try making designs with some of the items below. 6
Take photos. Share your results with the group.
Try using:
* Colored paper.
* Yarn, string, or embroidery floss.
* Small household objects, for example, coins, rocks, pasta, you name it.
* Leaves.
What were your favorite designs? Did you try to build any three-dimensional patterns?
What did you notice about using the mirrors?
Step 3: Some Challenges and Puzzles
Here are a few challenges and puzzles to try with your hinged mirrors:
* Try to cut out a piece of paper that will make a square when you place it next to the mirror.
* How about a circle, a star? Choose another shape and try to make it. Try this along the edge of one mirror, and try it touching the corner where the mirrors meet.
* What letters can you draw with half the letter being a reflection in the mirror?
* Put one small object on a piece of paper. Put the mirrors in a straight line in back of the object -- you now see two objects. Can you position the mirrors so you see three? Can you get any number? (The answer depends on whether you count “split” reflections on the centerline.)
* What is the maximum number of reflections you can count?
Step 4: Hinged Mirrors and Papercuts
If you have ever made paper snowflakes, paper dolls, or Paper Picato, you know that papercutting can produce beautiful symmetrical designs. There are many traditions of paper cutting from different cultures to explore.
Sometimes it is fun to make papercuts and be surprised by the results when you unfold the paper. But it can also be fun to be able to plan out a pattern. Hinged mirrors can help you predict the results of a paper cut, and explore its symmetry.
The directions below describe making shapes with one or two axes of symmetry. Of course, you can experiment with more as well!
Step 5: One Axis of Symmetry
Start or by squaring a sheet of paper. Fold the top left corner to the bottom edge and crease. Then cut off the rectangular leftover on the right. 636