Every intersection point corresponds  with some quadruplet of points.

每个交点对应于某四个点的组合。

Each configuration of these sliders is a point in 4D space: A quadruplet of numbers.

每个滑块的配置都是4维空间中的个点：组四个数字。

But if you use different types of assisted reproductive technology that make your ovaries excrete more eggs, you might get pregnant with triplets or quadruplets.

但如果用其他类型的辅助生殖技，巢排出更多的子，就可能怀上三胞或四胞。

Maybe if you're a little bit lucky,   here's one thing you might notice; every intersection point is uniquely associated  with a quadruplet of points on the exterior.

也许如果你有点幸运， 你会注意件事；每个交点都唯地与圆外的四个点相关联。

For a given quadruplet, you look at the  two kind of diagonal cords between them and those will intersect within the circle  and it goes the other way around.

对于给定的四元组，你看看它们之间的两种对角线弦，这些弦会在圆内相交，情况也会反过来。

Borsuk-Ulam says that if you try to map that set, all those special quadruplets, into 3-dimensional space, continuously associating each one some triplet of numbers, there must be some antipodal collision.

博苏克-乌拉姆定理说，如果你试图将那组特殊的四元组连续映射三维空间中，并将每个四元组关联个三元数字组，那么必定会发生些对跖点碰撞。

The function that answers it is called " n choose 4" which by definition  counts the number of quadruplets from a set of size N and the way to calculate it is similar  to what we saw before.

解答这个问题的函数叫做“n选4”，根据定义，它计算的是从大小为N的集合中选取四个元素的方法数，其计算方式与我们之前看的类似。

There is no doubt that the ability to frame analytic facts geometrically is very useful for math But it's all a tease - because when you start asking questions about quadruplets or quintuplets or 100 tuples of numbers It's frustrating.

毫无疑问，能够用几何方式来表达分析事实对于数学非常有用。但这都是个诱惑——因为当你开始询问四元组、五元组或百元组这样的数字组合时，就会感沮丧。