Introduction: Origami Cairo Pentagon Tile

A Cairo Pentagonal Tiling of a plane is named after street paving patterns found in Cairo, Egypt.

This Instructable will show you how to fold your own Cairo Pentagon Tile, with angles of 120°, 120°, 90°, 120°, 90°, and side ratios of 1:sqrt(3)-1.

Fold a bunch of these and make your own Cairo Pentagonal Tiling!

Supplies

- Origami paper

I used 6" squares (four different colors) cut into 1/4ths. You can use any size squares and any colors.

- Clear tape (optional)

Step 1: Back Story

On David Mitchell's website: http://www.origamiheaven.com/, you can find instructions for folding an origami Cairo pentagon tile. Those instructions can be found here: http://www.origamiheaven.com/pdfs/cairotile.pdf

His Cairo tile is the top one in the photo.

I set out to create a folding pattern that would yield, what one might consider, a traditional Cairo tile, with angles of 120°, 120°, 90°, 120°, 90°, and side ratios of 1:sqrt(3)-1.

Step 2: Fold #1

Then open the paper.

Step 3: Fold #2

Bring the bottom right corner of the square on to the midline. Make sure that a sharp point is formed in the bottom left corner of the square.

Step 4: Fold #3

Bring the top left corner of the square down and to the right so that the left edge of the square aligns with the crease made in Fold #2.

Step 5: Results of Folds #1 - 3

Open the square. These are the resulting folds.

Step 6: Fold #4 & 5

Repeat Folds #2 & 3, but on the other side of the square.

Step 7: Results of Folds #1 - 5

Open the square. These are the resulting folds.

Step 8: Fold #6

Bring the top right corner of the square onto the crease made in Fold #2. Then unfold.

Step 9: Fold #7

Bring the top left corner of the square onto the crease made in Fold #4. Then unfold.

Step 10: Results of Folds #1 - 7

Open the square. These are the resulting folds.

At this point, notice the intersection points of the folds.

Step 11: Forming the Top Angle of the Pentagon Tile

Fold down the two flaps formed by Folds #6 & 7. This forms a 120° angle at the top of the tile.

Step 12: Fold #8

Bring the bottom left corner of the square up so that the left edge of the square aligns with the folded down flap.

Step 13: Fold #9

Repeat Fold #8 using the bottom right corner and right edge of the square.

At this point, the left and right angles of the pentagon measure 90° each.

Step 14: Fold #10

The next series of folds will help form the desired side ratios of 1:sqrt(3)-1.

Bring the lower left edge of the pentagon up so that it aligns with the upper left edge of the pentagon. Note, these edges are NOT the same length (Yet...).

Step 15: Fold #11

Note that the base of the pentagon hangs over the upper right edge of the pentagon after Fold #10.

Now fold it over the upper right edge of the pentagon.

Step 16: Final Result

Re-open the pentagon.

The first image shows the crease formed by Fold #11.

The second image shows that bottom trapezoid flap folded up. This gives the desired side ratios of 1:sqrt(3)-1. The top right and left edges, and the bottom right and left edges are now all congruent. The base of the pentagon is equal to the those side lengths multiplied by a scale factor of sqrt(3)-1.

If desired, use some clear tape to secure all of the flaps so that the tile will lay flat.

The third image is the Cairo Pentagon Tile flipped over.

These directions started with the colored side of the origami paper facing up, resulting in a white tile. If you wish to make a colored tile, then start with the white side of the paper facing up.

Step 17: Play!

Now make a bunch of these tiles and create the Cairo Pentagonal Tiling.

Are you a math teacher? Share the work by asking each student to make one tile. Then cover a bulletin board or other wall space in your school with this very pleasing tiling.

When making the tiles, ask the students to make some observations about the angles that are formed with each fold.

If you need video instructions, check out the video.

Thanks and Happy Folding!