For example, there's an extension of the complex numbers that naturally live up in four dimensions, they're called the quaternions, and they offer a really elegant way to encode and to work with three dimensions.

例如，复数有一个扩展， 自然存在于四维空间中，它们被称为四元数， 并且提供了一非常优雅的方式来编码和处理三维空间。

You want a quaternion type, or you want to add a new posit type, or you want to, so it's hard.

你想添加四元数类型，或者新增一个posit类型，或者你想要… … 这很难。

And Hamilton used a special word for quaternions that had no real part and just ijk components, a word which was previously somewhat foreign in the lingo of math and physics: " vector" .

哈密顿为那些没有实部、仅由ijk分量组成的四元数使用了一个特殊的词，这个词前在数和物理的行话中有些生：“向量”。

That's right, your phone almost certainly has software running inside it that rely on quaternions.

没错，你的手机里几乎肯定有依赖四元数运行的软件。

Like all other non-unit quaternions, It's invisible to us.

和其他所有非单位四元数一样，它对我们来说是不可见的。

What I'll show you here is a way to visualize quaternions in their full four-dimensional glory.

我将向你们展示的是如何以完整的四维视角来可视化四元数。

And indeed, this is how you would tell a computer to perform quaternion multiplication, and the relative compactness of this operation compared to, say, matrix multiplication, is what's made quaternions so useful for graphics programming and many other things.

确实，这就是你告诉计算机执行四元数乘法的方式，而与矩阵乘法相比，这的相对紧凑性使得四元数在图形编程以及其他许多领域中非常有用。

The 20th century also brought quaternions some more love from a completely different direction, quantum mechanics.

20世纪， 四元数也从一个完全不同的方向获得了更多的关注，那就是量子力。

It's something analogous to the equator of a 3d sphere, and it represents all of the unit quaternions whose real part is 0. What Hamilton would have described as unit vectors.

它类似于一个三维球体的赤道，代表所有实部为0的单位四元数。 这相当于哈密顿描述的单位向量。

The thing is, there are other ways to think about and compute rotations, many of which are way simpler to think about than quaternions.

问题是， 还有其他思考和计算旋转的方法，其中许多比四元数简单得多。

So j goes to k, k goes to -j, -j goes to -k, and -k goes to j. This gives us a little table for what the number i does to the other quaternions.

所以 j 变为 k， k 变为 -j， -j 变为 -k，而 -k 变为 j。 这给我们提供了一个小表格， 说明了数 i 对其他四元数的用。

On the other hand, all the unit quaternions with negative real part end up somewhere outside that unit sphere.

另一方面，所有实部为负的单位四元数最终都会落在那个单位球外。

On the one hand, you could just define quaternion multiplication by giving the rules for how i, j, and k multiply together and saying that everything must distribute nicely.

一方面，你可以通过给出i、j和k相乘的规则并说明一切必须分配得当来定义四元数乘法。

But just as a deeper understanding for complex multiplication comes from understanding its geometry, that multiplying by a complex number involves a combination of scaling and rotating, you and I are here for the four-dimensional geometry of quaternion multiplication.

但正如对复数乘法的几何意义有更深的理解，即乘以一个复数涉及缩放和旋转的结合，我们在这里是为了理解四元数乘法的四维几何意义。

So the way that this works is slightly more complicated than a single quaternion product.

所以这方式的工原理比单一的四元数乘法稍微复杂一些。

And multiplying quaternion, q1, by another, q2, has the effect of scaling q2 by the magnitude of q1 followed by a very special type of rotation in four dimensions.

将四元数 q1 与另一个四元数 q2 相乘，相当于先用 q1 的模长来缩放 q2，然后再进行一非常特殊的四维旋转。

For example, i times j is k, which you might think of in terms of i acting on the quaternion j, rotating it up to k. But if you think of j as acting on i, j times i, it rotates i to -k.

例如，i 乘以 j 得到 k， 你可以认为是 i 对四元数 j 进行用，将其旋转到 k。 但如果你认为是 j 对 i 进行用，j 乘以 i， 则会将 i 旋转到 -k。

There's also a rather elegant form of this multiplication rule written in terms of the dot product and the cross product, and in some sense, quaternion multiplication subsumes both of these notions—at least, as they appear in three dimensions.

这乘法规则还有一相当优雅的形式， 以点积和叉积的形式编写，从某意义上说， 四元数乘法包含了这两个概念——至少，因为它们在三维空间中出现。

So after his discovery, he pushed hard for quaternions to be the primary language with which we teach students to describe three-dimensional space, even forming an official quaternion society to proselytize his discovery.

因此， 在他的发现后， 他极力推动将四元数为我们教授生描述三维空间的主要语言，甚至成立了官方的四元数会来传播他的发现。

What you do is not just a single quaternion product, but a quaternion sandwich, where you multiply by q from the left and the inverse of q from the right.

你所做的不仅仅是单个四元数乘积，而是四元数三明治， 即从左边乘以q， 从右边乘以q的逆。