How to Make a Modular Origami Stellated Icosahedron

An icosahedron is a polyhedron that has twenty triangular faces. A stellated icosahedron has each of those faces raised to a triangular pyramid. With thirty pieces of square paper, you too can make a sturdy version of this geometric marvel, using no glue at all. If you have three different colors of paper, you can make a version of the model where no two units of the same color touch one another (except at a point).

Steps
Making the units
Start with a single sheet of square paper, with each side measuring approximately 3 inches (about 7.5 cm).

Fold it in half, and make a crease along the fold.

Unfold the previous fold.

Fold in the right and left sides of the square to meet the crease you just made, to make a rectangle. This is often called a cupboard fold, or book fold.

Flip your rectangle over.

Make a diagonal fold where the top right corner meets the left side of your rectangle. Make sure to fold over both ‘doors’ of the cupboard fold.

Turn your paper upside down.

Make another diagonal fold where the top right corner meets the side ofthis shape. Remember to fold over both ‘doors’ of the cupboard fold. This makes a parallelogram.

Make a diagonal fold where the top corner meets the right corner of the parallelogram.

Make another diagonal fold where the bottom corner meets the left corner of the parallelogram. You will end up with a small square.

Flip this square over.

Fold the square in half, making a crease that goes perpendicular to the cupboard fold visible on the square.

Congratulations! You’ve made your first unit!

Putting the units together
Make thirty units. If you have three different colors of paper, make ten of each color.

Now, we can start putting the units together. The surface of the stellated icosahedron is made up of a number of pyramids (in fact, if you take the regular icosahedron – a solid with twenty triangular faces – and make each face the base of a triangular pyramid, you get a stellated icosahedron). So we start by using the units to make a series of connected pyramids.

Start with two units of different colors. The triangular ends of each unit are called ‘tabs’, and the square in the center of the unit contains ‘pockets’ made up by the cupboard fold that goes along the diagonal. Begin by putting the tab of one unit (pictured in orange below) into the pocket of another (pictured in yellow).

Pick a unit of a different color (pictured red) next and put its tab in the ‘orange’ pocket, while also inserting the ‘yellow’ tab in the ‘red’ pocket. Congratulations – you made your first pyramid!

Notice that each unit has two pockets. Continue by taking a new unit (orange in this case), and put its tab in the yellow unit’s second pocket.

Take a new (red) unit as before, and put its tab in the second orangeunit’s pocket, also putting the yellow tab in the red unit’s pocket. Tada! You now have two adjacent pyramids.

Continue adding pyramids in this manner until you have five pyramids that all meet at a point. If you are using three unit colors, make sure to never put the tab of a unit into a pocket of another unit of the same color. This ensures a regular, colorful pattern on your stellated icosahedron.

Keep adding pyramids to your stellated icosahedron, making sure that there are never more than five pyramids meeting at a point. You may need to feel your way around a bit to make sure that you never end up having to put a tab into a pocket of the same color. Don’t worry if you do – you can carefully prise your units apart and rearrange them until you find a proper arrangement.

Proceed, making sure that no more than five pyramids meet at a point, and your model will take shape. The last unit is tricky – you’ll have to make sure that both its tabs go into pockets, and both its pockets are filled by the two remaining free tabs. Proceed with care.

Hooray! You now have a fully formed, colorful, stellated icosahedron.

Tips
When making the units, do your best to make clean and precise folds. Neat, uniform units are much easier to put together.

This model is pretty sturdy, despite the lack of glue. It can also be taken apart relatively easily and put back together again. As a result, the stellated icosahedron makes a good box for a small, light present (such as jewelry, or a note). If you want to make it a gift box, simply insert the item to be gifted into the model before attaching the last few pieces.

Adapt the colors used to make festive displays for Christmas, Easter, orany other holiday or festival.

Link:
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