## 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, Papel Picado, 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. 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. (You could also use a piece of origami paper, or just use a rectangular sheet, but starting with square paper makes it easier to make a quilt out of the paper cuts later.)

Fold your hinged mirror open, so it acts like one long mirror.

Place the fold of the paper against the mirror.

Draw a shape on the paper with a single line, starting with your pen touching the mirror and returning to the mirror at the end. Looking in the mirror will show you what the shape will look like when you cut it out and unfold the paper.

Now try cutting out the shape following the line, and unfolding your paper.

The crease from the fold will show the line of symmetry, and you can see by refolding that the two sides are the same. Many plants and animals have this sort of symmetry -- can you think of examples?

Can you use the mirror to make some other interesting shapes?

## Step 6: Two-axis of Symmetry

Now try arranging your mirrors at a right angle to each other.

Fold a square sheet of paper in half, and then in half again.

Set the hinged mirrors at a right angle to each other, and put the corner of the paper where the folds all come together into the corner of your mirror.

Draw a shape with a single line on the paper, starting with the pen touching the mirror on one side, and ending by touching the mirror on the other side. Looking into the mirrors will show you the full pattern.

Cut out the shape you draw along the line, and unfold your paper.

Again, the creases show the axis of symmetry, but this time there are two axes -- you can fold the paper along either fold, and the halves will mirror each other.

Can you think of examples of things that have this sort of symmetry?

Keep experimenting!

See what other designs you can make. Here are some ideas if you need them:

- Try making more folds of the paper. What happens?
- Can you make a leaf or tree design?
- Snowflakes are 6 pointed, can you make a snowflake?
- What happens if you put different corners of the paper into the corner of the mirror?
- What happens if you fold your paper differently?

## Step 7: Quilts and Other Connections

There are many traditions of paper cutting from different cultures. Papel Picado cutouts are often used as decorations for the Day of the Dead celebrations in Mexico. Research other traditions.

Quilting is another widespread art form that often involves levels of symmetrical patterns. In fact, hinged mirrors are sometimes called “quilting mirrors”, because quilters use them to envision how a fabric pattern will look when it is reflected into a symmetrical quilt square (search the web for “Using a Quilting Mirror”).

Different cultures have different quilting traditions; for example, Hawaiian quilting often uses organic leaf and flower shapes. Research other traditions. Quilts may have many axes of symmetry. Look at different quilt patterns; each square may be symmetrical and you may find there are other axes of symmetry in the whole quilt. There are also asymmetrical quilts -- sometimes called “crazy quilts”.

If you make paper cuts with square sheets of paper, you can then assemble them into a quilt.

Have students research, or ask parents, grandparents, or other relatives about any paper cutting, quilting or other traditions from their cultures that involve symmetrical designs.

These mirror activities investigate one type of symmetry sometimes called, not surprisingly, mirror symmetry (or reflectional, or line symmetry). This sort of symmetry is very common in the natural world, as well as in art and architecture. You can search for other types of symmetry on the web -- just be ready for many, sometimes competing, terms. The terms are not critical for this activity, just keep in mind that some patterns used in quilting and papercutting display additional types of symmetry such as Rotational Symmetry or Translational Symmetry.

## Step 8: Resources

Hinged Mirrors and faces:

https://www.exploratorium.edu/snacks/corner-reflec...

More on hinged mirrors, nature, and kaleidoscopes:

https://www.instructables.com/Plant-Prisms-Cardboa...

Paper Snowflakes:

https://www.instructables.com/Paper-Snowflakes-1/...

(Search Instructables for “Snowflake” for many other ideas)

Quilting:

https://www.instructables.com/3D-Quilt-Codes-the-...

**This work is made possible by support from STAR, a Biogen Foundation Initiative. The team at Lesley supporting this initiative includes faculty and staff in the Lesley STEAM Learning Lab, Science in Education, the Center for Mathematics Achievement, and other related Lesley University departments and programs.**

## Comments