The art of folding paper into beauty, one crease at a time.

Preface

This project was based around a very classical challenge requiring you to fold a sheet of paper into a sphere (a crumpled ball does not count!); moreover, only using pure origami. This short log documents my approach to this problem and the results I achieved. In total, the CAS experience lasted for about 3 days from 1/12/22 — 3/12/22.

The Rules of Pure Origami

1. Fold the paper without cutting it.
2. Use only one sheet of paper.
3. Do not use glue or any other substances.

The approach

Very quickly, I noticed that folding a smooth sphere using paper would be near impossible so I resorted to creating an approximation. After a little research I settled on this shape:

The Buckminsterfullerene :)

This is the Buckminsterfullerene, also known as the $\text{C}_{60}$ compound.

Why did I choose this shape?

There are many similar approximations, e.g. the icosahedron or dodecahedron but I chose this one specifically due to its more accurate representation of the sphere, and its similarity to the pattern on the soccer ball (known for being a sphere).

Observations

Let us begin with some important observations about the underlying structure of this shape. Note the following:

• 60 vertices
• 90 edges
• 32 faces
  • 12 pentagons
  • 20 hexagons

Note

This can be found using the Euler characteristic

Key Insight

Observe how each vertex is connected to exactly three other vertices, and note that each pentagon and hexagon have the same side-length. This suggests that an equilateral triangle is the most suitable unit for this model.

The Unit

For each vertex, I had something like this in mind:

As we can see they also connect quite nicely:

Notice, however, the connection is structurally extremely weak. To reinforce this, I used another unit to “wrap” over each vertex holding it in place.

This creates “pockets”, which can be used to tuck the extending flaps overall improving the structural integrity of the model. For example, the same connection can be done like so:

Creating the model

From the observations, we know that there are 60 vertices, and therefore I would need 120 equilateral triangles to make this model (two per vertex). Using a sheet of A4 paper, I can make 6 triangles, and thus 3 units. To complete this model I would require 40 sheets of A4 paper.

Finished Product

In the span of two days, (20 units per day) I finished making all the units. I spent the final day assembling the model. This is how it looked.