# Does folded cube have a name?

Folding a square to produce a torus has an analogy with "folding" a cube to produce what? Is there a name for the resulting 4D structure?

## 2 Answers

Yes. Perhaps unfortunately, such an object is also called a torus. If you want to be precise, you might use the word "hypertorus" or say explicitly "$$n$$-dimensional torus". In your case, you would be interested in the $$3$$-torus embedded in $$4$$D.

In general, if we write $$S^1$$ for the circle, then $$S^1 \times S^1 \times \cdots \times S^1$$ is called a torus (by analogy with the classical case $$S^1 \times S^1$$).

In particular, a square is $$[0,1] \times [0,1]$$. So when we glue together the boundaries, we're glueing the $$[0,1]$$s into circles. So we wind up with $$S^1 \times S^1$$.

What about a cube? It's $$[0,1] \times [0,1] \times [0,1]$$. Again, we glue opposite sides of the cube together, and we wind up with $$S^1 \times S^1 \times S^1$$.

I hope this helps ^_^

Such an object is also called a genus-3 handlebody.