# Cube Folding

## Objective

To experiment with different ways of folding six connected squares to make a cube, as well as to determine which patterns will fold into a cube as well as which patterns (if any) will not.

## Difficulty

Easy (no student should have a problem with the basic concepts).

## Concept

Imagine you are holding a six sided die, or better yet, go grab one from a board game and look at it. The die should most likely be cube shaped. Observe that the die is made up of six squares. Now imagine that you unfolded the die’s sides until all you had left were six connected squares. You may get an arrangement like this:

Or your arrangement may look more like this:

Now, imagine you started from the end rather than the beginning; that is, imagine that you connected six squares together. How many patterns can you make? Will they make a cube? How are they similar to other connected patterns you have already observed? How are they different? Can you create patterns that absolutely cannot be folded into a cube?

## Hypothesis

Based on your answers to the questions in the background and motivation section, do you think that every combination of six connected squares can be made into a cube?

## Materials

You may consider doing this with some construction paper, a ruler, a pencil, and scissors. You could then draw out various patterns of six connected squares and physically fold them into cubes or failed attempts at cubes. Another viable approach is to write a computer program to manipulate the patterns for you. It is recommended that, in either case, you at least demonstrate all the possible arrangements of six connected squares, and then begin discussing which ones work and which ones do not.

## Procedure

First and foremost, you should definitely make a few example patterns and see how they may be folded into a cube or how they fail at making a cube. For instance, observe this example from before and see how it is formed into a cube:

**Important Note**: You should only fold at the edges of the squares, and you should take care to not fold the squares themselves. This image, for example, shows an improper fold:

After you have done this, you should try to form general ideas about how all the patterns will work and what makes various patterns potential cubes versus others that are incapable of being folded into cubes. Once you have done this, you should clearly demonstrate why you think your generalizations hold by looking at various patterns, applying your ideas, and then actually folding them up and seeing if your ideas hold.

## Research

Although this concept is very simple, it would be helpful for you in your analysis if you understood ideas of symmetry in both two and three dimensions. This will allow you to look at fewer examples and still come to a complete conclusion, as information gained from one pattern can be applied to other patterns.

## Analysis

You need to clearly define all possible patterns of the six connected squares, explain why your list of possibilities is complete, and then show which patterns will fold into a cube and which patterns will not. You then need to present your generalizations that you have derived from your observations and explain to which patterns they apply and why the generalizations you have formed allow you to quickly determine whether the pattern is a possible cube or not.

## Conclusion

What do you think will happen? This section is for you to fill in, and will be the final step in your folding experiment.

## Extensions

Instead of trying this with six connected squares, you might try the added difficulty of eight connected triangles and finding which patterns fold together into an octahedron.