Polyhedra

Mentor: Can you make a fair die that has 3 sides? If you can, how? If you can't, why?

Student: What about this one:

You can construct it from three pieces of paper, gluing or taping the edges together:

Mentor: What other dice can you make this way?

Student: I guess I can make one with 4 sides. And with 5 sides. And... using code; Wait a minute! I can make a die with any number of sides this way! It will look like a blimp!

Mentor: Any number? What about 2?

Student: Yes! It would be like a coin. But I can't do a 1-sided die this way.

Mentor: No, but if you wanted a one-sided die you could make one or find one around you. What can you think of that has only one side?

Student: A Möbius strip!

Mentor: Or a sphere. A marble is a one-sided die.

Student: But you can't make a marble out of paper.

Mentor: What if I told you that we can only use unbending cardboard? Also, each side has to be a figure with straight edges of the same length, and with the same angles (such figures are called regular polygons). In other words, the die should be of the same type as the usual six-sided dice. Can you make a three-sided die this way?

Student: No!

Mentor: Why? What about a die with 4 sides? With 5 sides?

In fact, there are only five types of dice (or polyhedra) that can be built this way.