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

所以这种方式的工作原理比单一的元数乘法稍微复杂一些。

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

你想元数类型，或者新增一个posit类型，或者你想要… … 这很难。

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

20世纪， 元数也从一个完全不同的方向获得了更多的关注，那就是量子力学。

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维视角来可视化元数。

Multiplication by any other quaternion is completely similar.

与其他元数相乘完全类似。

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

和其他所有非单位元数一样，它对我们来说是不可见的。

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元数最终都会落在那个单位球外。

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

没错，你的手机里几乎有依赖元数运行的软件。

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义元数乘法。

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.

将元数 q1 与另一个元数 q2 相乘，相当于先用 q1 的模长来缩放 q2，然后再进行一种非常特殊的维旋转。

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，其中许多比元数简单得多。

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分量组成的元数使用了一个特殊的词，这个词前在数学和物理学的行话中有些陌生：“向量”。

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， 自然存在于维空间中，它们被称为元数， 并且提供了一种非常优雅的方式来编码和处理三维空间。

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， 他极力推动将元数作为我们教授学生描述三维空间的主要语言，甚至成立了官方的元数学会来传播他的发现。

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的单位元数。 这相当于哈密顿描述的单位向量。

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 对其他元数的作用。

I'll review the method for thinking about quaternion multiplication described last video, explain why half the angle is used, and why you multiply from the right by the inverse.

我将回顾上个视频中介绍的思考元数乘法的方法，解释为什么使用半角，以及为什么从右边乘以逆元。

So in the same way that complex numbers are represented as a pair of real numbers, each quaternion can be written using four real numbers, and it lives in four-dimensional space.

因此， 就像复数可以用一对实数表示一样，每个元数也可以用个实数来表示，并且它存在于维空间中。

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元数乘积，而是元数三明治， 即从左边乘以q， 从右边乘以q的逆。

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元数乘法的方式，而与矩阵乘法相比，这种操作的相对紧凑性使得元数在图形编程以及其他许多领域中非常有用。