All points on the surface of a sphere are equally distant from the center.

球心到球面上任何一点距离都一样。

There actually is something called the Riemann sphere that involves dividing by zero by a different method, but that's a story for another day.

事实上，有个叫黎曼球面东西，它涉及到通过不同方法来除以 0，但今天我们且不这个。

That's also of great value scientifically.

这个外球面区域科学研究意义很大。

And then you have another spherical wave coming from the short bank.

然后你会看到另一个来自短波岸球面波。

So, the electric flux is independent of the  radius of the concentric spherical surface?

所以，电通量与同心球面半径无关？

Now, we wanna talk about the product of two spheres.

现在，我们想两个球面乘积。

So the Borsuk-Ulam theorem guarantees that some antipodal pair of points on the sphere land on each other in the plane, which means there must be some necklace division using two cuts that gives a fair division between thieves.

所以博苏克-乌拉姆定理保证了球面上一对对跖点在平面上会重合，这意味着必须存在某种使用两处切割项链分割方式， 能让小偷之间公平分配。

Hubble Space Telescope Program Manager Douglas Broome delivered the troubling news: “The conclusion we’ve come to is that a significant spherical aberration appears to be present in the optics, in the optical telescope system optics, ” he said.

哈勃太空望远镜项目经理道格拉斯·布鲁姆 (Douglas Broome) 发布了这个令人不安消息：“我们得论是，光学望远镜系统光学元件中似乎存在明显球面像差， ” 他说。

The beauty of a spherical reflector is that, by changing the position at which you measure the reflected waves above the dish, you're measuring those waves from different parts of the sky.

球面反射器美妙之处在于，通过改变你在抛物面上测量反射波位置，你就相当于从天空不同部分测量这些波。

The positions of stars and other celestial bodies can be defined by their places on this sphere, if they were imagined to be attached to it.

星星和其他天体方位可通过它们在这个球面上位置来定义，假设它们被想象为附着于其上。

It gives a correspondence between points on the sphere and necklace divisions.

它建立了球面上点与项链分割之间对应关系。

The electric flux which passes through a sphere which is  concentric to a positive point charge equals the charge of the positive point charge  divided by the permittivity of free space.

通过与正点电荷同心球面电通量等于正点电荷电量除以自由空间介电常数。

Say we have a lens that has a spherical face on both sides, known as a convex lens.

假设我们有一个具有球面镜头 两侧，称为凸透镜。

And you can see this when the ball back by the spread-out lines of oil on the ball surface.

当球从球面上油线中滚回时， 你可以看到这一点。

The integral of dA is A; the area of the concentric spherical surface.

dA积分是A；即同心球面面积。

And you could go the other way around; any way to divide up the necklace and divvy up the pieces gives us a unique point on the sphere.

你也可以反过来思考；任何分割项链并分配碎片方法都会给我们球面上一个唯一点。

In other words, going to the antipodal point on the sphere results in reflecting the output through the origin in the output space.

换句话说，到达球面上对跖点会导致输空间中输关于原点反射。

To think about the projection, let's make a little right triangle like this, where what was the height of our spherical rectangle is the hypotenuse, and its projection is one of the legs.

要想考虑投影， 我们可以画一个小直角三角形，其中球面矩形高度是斜边，其投影是其中一个直角边。

The main thing to mention is that there's a more general version of the Borsuk-Ulam theorem, one which applies to higher-dimensional spheres.

需要提到主要一点是， 有一个更一般博苏克-乌拉姆定理版本，适用于更高维度球面。

If the point (x, y, z) on the sphere corresponds to some necklace allocation, what does the point (-x, -y, -z) correspond to?

如果球面上点 (x, y, z) 对应某种项链分配，那么点 (-x, -y, -z) 对应什么？