Icosahedron Sphere (from A4 Papers)

This icosahedron sphere is a fun combination of craft and math (no calculations needed). It's made by connecting many small triangles, with no glue, to create a very stable sphere.

The building blocks are non-curved triangles, and by connecting them into a combination of hexagons and pentagons you create a curved shape. Because of the flexibility, or "hinge", between each 2 connected triangles the result is a more spherical shape.

All you need for this project is:

A4 Paper - 100gr sheets give good stability and are easy to fold.

Scissors or a blade.

I used two colors of paper, because its prettier and easier to understand in the images. In this case you need -

1 1/2 A4 sheets of color A (pink)

1 3/4 A4 sheets of color B (blue)

The paper has to be cut into equal thickness long strips. I chose the pink paper for the triangles and the blue for the connectors.

Pink paper: This is a bit more tedious with a lot more folding and cutting but the end piece size is exactly 1 triangle.

Blue paper: This is less cutting and folding, but you have to be accurate to get the strips equal width.

In order to construct the shapes you need the folds to make exact squares. This is a simple method for getting those fold points accurate (without having to pull out a ruler).

(Anybody who has owned an old printer with continuous feed printer paper has probably done this many times)

Because of the way the pink paper was cut, each piece can now be folded into a triangle. These are equilateral triangles in which each side is the same length. The strips are longer than 3 squares, and this overlap creates the stability of the triangle.

The blue strips need to be cut into connectors- each connector is 4 squares long (cut at the fold marks).

Every 2 triangles are connected by 1 connector piece. The connector goes through both triangles and wraps over the shared sides, and then is tucked in between the 2 triangles.

That way each triangle side is completely wrapped in blue and there is a "hinge" connecting the two triangles.

It is extremely important that when you start connect multiple triangles the "hinge" is always on the same side.

There are 2 basic shapes -

Pentagons - created from 5 connected triangles. These units are curved and their shape is stiff (non bendable)

Hexagons - created from 6 triangles. When no pressure is put on them these units are flat. But these units can be bent along any diagonal.

Triangles are now added one by one to create the shape. Now you need to choose a rule to follow. This seems complicated but it really isn't too bad, as long as you consistently follow the rule you chose you will end up with a sphere.

The rule options:

1. Every 3 pentagons are connected by one triangle. 12 pentagons and 19 triangles.
2. Every large triangle (made out of 4 triangles) has a pentagon at each corner.
3. There is a hexagon attached to the end of each pentagon.

You might notice that these rules are basically the same...because they are!

The only difference is which 'block' your focusing on when you start connecting your pieces (pentagons, large triangles or hexagon-pentagon connection).

I personally follow option 2

Make 1 pentagon and put it to the side.

Start constructing going from one hexagon and adding triangles. As you move up in your chosen rule you will create a kind of semi-sphere.

If you think you got lost, look for that rule that you chose in the connection between pieces.

Keep building until you reach a sphere that is missing a five point star.

At this point you should have left 5 single triangles and the pentagon you put to the side.

When you reach this stage it becomes harder the thread the connectors through a triangle and then pull them back up again.

That is why the last 5 triangles are added with the blue connector already attached.

Each triangle is connected to the 'point' of the missing star. This means 2 sides connect to 2 triangles already in the sphere.

The remaining unconnected side will already have the blue connector in.

Your sphere now has a missing pentagon - but the blue connecting pieces are already there.

Now you have to simultaneously put each connector through one side of the pentagon and fold it over.

2 tips:

1. make sure that the blue connector is already folded over the triangle on the sphere side. That way it won't pull out when you add the hexagon.
2. slightly bend the open connector lengthwise. This creates a smaller width and adds stiffness for threading the connector through the hexagon.

(if you find that you still can't manage with only fingers try a pair of tweezers).

Fold the blue connectors between the triangles... DONE!

Adding hexagons will create bigger (but a little softer) spheres - In this big version I marked one of the curved pentagon and flat hexagon shapes.

Fun fact - if you look at a soccer ball it has the same hexagon-pentagon combination as the sphere you just made.

Science fact - The protein capsules of many viruses share the same 5 and 6 fold morphological units. (for a nice review check out https://doi.org/10.1073/pnas.0405844101)

Still wondering what to do with your sphere:

1. Give it to a mathematician or a cryoEM expert and immobilize them for hours...
2. Combine a 300+ triangle sphere with 1 hell-hound to create a 30 second chew toy... fun!!