What does $S^1\times S^1 \times S^1$ look like?

Question

The primary question is:

What does $S^1\times S^1 \times S^1$ look like? (assuming that it makes sense)

I have feeling that $S^1\times S^1 \times S^1$ sits in 4-D space because one envisions $S^1$ as being a circle on a plane and $S^1\times S^1$ being a torus in 3-D space. Almost an implication to a general rule. Which is probably why I'm having hard time imagining it.

Is there a general rule as implicated above?

It's natural to ask if $S^1\times (S^1 \times S^1)$ is the same as $(S^1\times S^1) \times S^1$?

Thanks in advance.

Answer 1

Another way to think about the $n$-torus is by identifying opposite faces of the $n$-hypercube. E.g.: You get $S^1$ by identifying the ends of the unit interval. You get $S^1\times S^1$ by identifying the opposite edges of the unit square. What you are interested in is what you get by gluing opposite faces of the unit cube.

You can find some nice illustrations here: http://www.geom.uiuc.edu/video/sos/materials/overview/

Answer 2

A good way to visualize the 3-torus is in 3D slices, that rotate and scan the object at various angles.

Using the implicit definition of $T^3$ in Cartesian coordinates, where $R_1 > R_2 > R_3$ :

$$\left(\sqrt{\left(\sqrt{x^2 + y^2} -R_1\right)^2 + z^2} -R_2\right)^2 + w^2 = R_3^2$$

There are 3 distinct planes of rotation that will transform the two torus intercepts: $xz$ , $xw$ , and $zw$

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The equation for rotating on plane $xz$ :

$$\left(\sqrt{\left(\sqrt{\left(x\cdot\cos(t)+z\cdot\sin(t)\right)^2 + y^2} -R_1\right)^2 + \left(x\cdot\sin(t)-z\cdot\cos(t)\right)^2} -R_2\right)^2 + w^2 = R_3^2$$

Rotating at origin by animating $0 < t < 2\pi$:

Holding at these angles while passing through the 3-plane:

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The equation for rotating on plane $xw$ :

$$\left(\sqrt{\left(\sqrt{\left(x\cdot\cos(t)+w\cdot\sin(t)\right)^2 + y^2} -R_1\right)^2 + z^2} -R_2\right)^2 + \left(x\cdot\sin(t)-w\cdot\cos(t)\right)^2 = R_3^2$$

Rotating at origin:

Holding at these angles while passing through the 3-plane:

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The equation for rotating on plane $zw$ :

$$\left(\sqrt{\left(\sqrt{x^2 + y^2} -R_1\right)^2 + \left(z\cdot\cos(t)+w\cdot\sin(t)\right)^2} -R_2\right)^2 + \left(z\cdot\sin(t)-w\cdot\cos(t)\right)^2 = R_3^2$$

Rotating at origin:

Holding at these angles while passing through the 3-plane:

Answer 3

A torus ($S^1\times S^1$) is just a circle rotated around an axis outside the circle. A 'hypertorus' (just my own word for it) is a torus rotated around an external axis, parallel to the $3$D space of the normal torus.