[00:03] Sure, [music] we usually think about and draw them on two-dimensional surfaces, kind of thing [music] in a zoo, doesn't mean it is a zoo. This is a circle, but this is [music] not. A circle is pure circumference. There's nothing inside of
[00:18] you only need one number to locate any point in it. Now, if you want to talk and its interior, well, now you're talking about a completely different geometric object that mathematicians sometimes call a disk. [music]
[00:31] Unlike a circle, a disk has two dimensions because you need two numbers to locate a point in it. You need to know how far around the point is and how deep inside it is. Similarly, only two numbers are needed to identify any
[00:43] particular point on a sphere like this one, latitude and longitude. So, this is The only thing here that is three-dimensional would be the sphere [music] and the space inside it, but that's not a sphere. That's what
[00:57] mathematicians call a ball. Circles and spheres have a lot in common. For >> when embedded in one dimension more than they are, they're both just the collection of all points equally distant from a center point that's [music]
[01:11] outside of them. So, in other words, circles and spheres are the same geometric concept [music] just extended to different dimensions. And for that reason, mathematicians sometimes just give them the same name.
[01:23] They call this a two-sphere and this a one-sphere. [music] Similarly, spheres and the spaces inside them are called balls. So, a disk would be a two-ball and this would [music] be a three-ball. The skin of a
[01:36] four-dimensional ball would be a three-dimensional sphere. The skin of a five-ball would be a four-sphere and so on. But, can we imagine a zero-sphere?
[01:48] No problem. Of course, we can. Just as we so often do with two-spheres and zero-dimensional sphere in in space that is one dimension So, a one-dimensional Simply choose a center and a radius, and the zero-dimensional sphere will be
[02:02] every point that is that far away from the center. So, these two points. That's it. This is a zero-dimensional sphere. Two separate dots. Zero-spheres are freaky. They contain two different locations, but nowhere to go. And of
[02:17] bound, we get ourselves a one-dimensional ball. This means that every line segment [music] is a one-ball. If you imagine pairing up every mathematical point in your body, and then connecting those points with a
[02:30] line segment, everything will finally make sense. You have been a one-dimensional ball pit your entire life.