I'm sure you already know this - but, a doughnut is not a sphere . After all, it's got a hole right through the middle (producing a 3D ring shape that mathematicians call a 'torus').
However, this has some interesting mathematical consequences, which are explained in the following video:
Here are some examples of the differences between the two (all of which are illustrated by visuals in the video):
There's actually an entire branch of mathematics, devoted to studying what's possible on different kinds of surfaces. It's known as 'topology', and it's kind of like geometry without the concepts of size and distance: if one shape can be smoothly transformed into the other (and back again), without cutting or gluing the material, then the two shapes are topologically equivalent to one another. For example: a cube, a pyramid, and a tetrahedron are all topologically equivalent to a sphere - because they can all be smoothly deformed into a sphere (and back again). However, a torus is not topologically equivalent to a sphere, because of the hole in the middle: you can't deform a sphere into a torus and back again, without cutting and gluing the material. (In fact, there's an old joke about a topologist who's enjoying a doughnut and a coffee. The poor guy can't tell the difference between the doughnut and the coffee cup, because they're topologically equivalent to one another (the doughnut has a hole in the middle, and the coffee cup has a hole in the handle!)
Anyway, does anybody have any questions, or anything to add?
However, this has some interesting mathematical consequences, which are explained in the following video:
Here are some examples of the differences between the two (all of which are illustrated by visuals in the video):
- If you're on a sphere, and you start walking in a straight line, then you'll always get back to where you started after one complete circuit. However, if you do this on a torus, this doesn't necessarily happen: you might have to do several circuits before you get back to where you started (or perhaps you'll never get back there at all!)
- Imagine a group of people holding hands and making a circle. If these people are standing on a sphere, then they can all walk forwards without breaking their circle, until they get to a single point (or, at least, until they get squished ). However, if these people are standing on a torus, this won't necessarily be the case.
- If there was a torus-shaped planet, then it would be possible for wind to be blowing in the same direction at every point on a planet. However, this is not possible on a sphere-shaped planet: there will always be at least two points with no wind. (This is a consequence of the 'Hairy Ball Theorem': if you attempt to comb a hairy ball, then you'll always end up with at least two 'bald spots'!)
- Imagine you have three homes, and three utilities (for example: water, fire, electricity). You need to connect each home to each utility, in such a way that your connections do not cross one another. This is a well-known problem in graph theory - and, if you're working on a flat plane, then it can't be done. In fact, if you're working on a sphere, it still can't be done: no matter how you connect the houses to the utilities, you'll always have at least one crossing. However, on a torus, you can connect them all without any crossings - and the video shows how it can be done.
There's actually an entire branch of mathematics, devoted to studying what's possible on different kinds of surfaces. It's known as 'topology', and it's kind of like geometry without the concepts of size and distance: if one shape can be smoothly transformed into the other (and back again), without cutting or gluing the material, then the two shapes are topologically equivalent to one another. For example: a cube, a pyramid, and a tetrahedron are all topologically equivalent to a sphere - because they can all be smoothly deformed into a sphere (and back again). However, a torus is not topologically equivalent to a sphere, because of the hole in the middle: you can't deform a sphere into a torus and back again, without cutting and gluing the material. (In fact, there's an old joke about a topologist who's enjoying a doughnut and a coffee. The poor guy can't tell the difference between the doughnut and the coffee cup, because they're topologically equivalent to one another (the doughnut has a hole in the middle, and the coffee cup has a hole in the handle!)
Anyway, does anybody have any questions, or anything to add?
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