In that case, our warp-ship would eventually travel all the way around this curved hypersphere and get back to where it started.

在这种情况，光速飞船最终会绕着这个弯曲球体旅行一圈，最后回到起点。

If that curvature is positive, then it may be that the universe is really the surface of a hypersphere, the 3D surface of a 4D sphere.

如果曲率是正，那么宇宙可能是一个球体表面，即四维球体三维表面。

Geometrically, a four-dimensional creature would be able to look at those two perpendicular rotations that I just described and understand that they lock you into one and only one rigid motion for the hypersphere.

从几何上讲，一个四维生物能够观察到刚才描述两个垂直旋转，并理解它将你锁定在一个且仅有一个刚体运动中，球体来说是这样。

So what we see as a unit sphere in our 3d space represents the only unaltered part of the hypersphere of quaternions getting projected down on to us.

所以看到三维空间中单位球代表了四元数球体中唯一未被改变并投影到这里部分。

As an example, Borsuk-Ulam also applies to mapping hyperspheres in 4d space into 3d space.

作为一个例子，博苏克-乌拉姆定理也适用将4维空间中球面映射到3维空间中。

What I mean by hypersphere is the set of all possible lists of 4-coordinates where the sum of their squares equals 1; those are points in 4d space a distance 1 from the origin.

所说球面是所有可能 4 坐标列表集合， 其中它平方和等 1；这些是 4d 空间中距离原点 1 点。