Triangular Faced Origami Polyhedra

Contents
Folding a unit from a rectangle
Folding a unit from a square
Putting together two units
Making a tetrahedron
Making an octahedron
Making an icosahedron
Other examples

Introduction
I designed this unit with Kazuyo Inoue, while at Queen's University, in Spring 1997. It can be used to make various polyhedra with triangular faces, such as the octahedron to the right, or the tetrahedra to the left, and many more, shown below.
There are two versions, one from A4 paper (which will also work from other size rectangles), and the other from square paper. Click on any photo on this page to see a larger version.

Instructions for folding from a rectangle:

1. start with A4 paper, or other rectangular paper.

2. fold in half both ways and open out.

3. fold one short edge to the middle, make a crease on the lower part of the fold, and open out.

4. Make a fold starting at the middle of the top, and made so that the top right corner touches the crease mark shown. To help get the crease starting in the right place, put your finger at the top mid point while bringing over the right corner.

5. Open out so it looks like this:

6. Fold the left half of the top edge to meet the diagonal fold on the right of the paper:

7. open out:

8. Note the shaded hoizontal and diagonal creases:

9. Bring up the diagonal crease indicated in the last picture so that it lies along the horizontal crease also shaded in the last picture:

10. Open out:

11. Repeat on the other side:

12. Open out:

13. Fold over the corners as shown:

14. Fold the short diagonal edges to meet the diagonal creases as shown:

15. Fold in along the diagonal creases as shown:

16. Turn over:

17. Fold down the corners that stick up over the edges:

18. Fold along the creases shown:

19. Turn over. This is the final unit from the front:

20. This is a photo of the final unit. Click on the image for a larger view.

You also need to be able to fold the mirror image version, which is the same to step 12, but then proceeds like this:

Instructions for folding from a square:

1. Start with a square of paper. First make creases that divide the square into quarters.

2. Next fold two opposite sides to the middle line, and make a horizontal crease across the middle:

3. Fold top and bottom edges to the middle line, and open out:

4. Fold the point A to touch the line C, folding from B. Then open out:

5. Do the same for each of the other corners, so you end up with creases as shown:

6. Then open the whole thing out, so you have creases as below:

7. Fold over the top and bottom edges along a line which passes through the points where the creases near the top and bottom of the paper cross:

8. Next fold along the dotted lines shown:

9. Fold over along the line joining the points A and B, and then along the line joining the points C and D. Next make the creases along the lines from R to S, S to T and T to U:

You also need to be able to fold the mirror image version:

Putting two units together

Units fit togther by putting the point of one unit into the pocket of another, as in this example:

Making a tetrahedron

A single unit almost folds into a tetrahedron by itself:

But to keep it in place you need two units, of opposite parity, which fold around each other:

And then you can make lots and lots of them:

Some collections of tetrahedra. (Click for bigger images)

Making an octahedron

1. Take four units, which can be all the same parity, or two of each, as here:

2. for pairs of units of the same parity, join them together to make pyramid shapes with flaps:

3. Put the two pyramid shapes together:

4. Appearance after tucking the red flaps in the green pockets:

5. Turn around:

6. After the green flaps are tucked in the model is complete:

Images of tetrahedra and octahedra:

Making an icosahedron

1. Take 5 units of each parity. Here 5 are shades of green and blue, and the other 5 are shades of red and pink.

2. Take five of the same parity and fit them together like this:

3. Now join the first to the last. Now the model can not lie flat, and it may help to use some paper clips to keep the pieces in place.

4. now the red/pink pieces go in place around the green/blue pieces. Here is the first pink piece in place:

5. To put the red piece in position, first fold it over like this:

6. And slide it into place like this:

7. Then tuck the point into the next blue/green unit:

8. Now take the next pink/red unit:

9. Slide it into place like this:

10. And tuck in the pink point:

11. Next red/pink unit sliding into place:

12. And the point tucked in:

13. Final pink/red unit slides into place:

14. And its point tucks in.

15. Turn over the model:

16. Tuck in a loose pink corner into the next pink/red unit's spare pocket:

17. Then that units point is tucked into the next pocket:

18. And the next point into the next pocket:

19. And the next point into the next pocket:

20. And the final point is tucked into position to complete the model:

And here is the final icosahedra from another view. This icosahedron is made from sheets of A4 paper. There is also a picture of an icosahedron made from the units made from squares in the set of pictures below.

This icosahedra is not very strong when made from paper. But made from sheets of A4 card, with a little glue perhaps, to hold some of the points in place, it is quite a strong model. Or, as an alternative to glue, instead of having the points stick into the upper opening, bend them downwards into the pocket with opening in the same place, but facing the other direction.

Other examples

Some more shapes you can make with this unit (click on image for a bigger picture).

Icosahedral box

Tetrahedra do not tessellate; but if you put 20 together, you get quite close to an icosahedron shape. I've designed an icosahedron shaped box to hold them. When I get time I'll add some instructions for how to fold it. Here are a few images from a few different directions for now.

Icosahedral box (click on image for a bigger picture).