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

在这种情况下，我们的超船最终会绕着这个弯曲的超球体旅行一圈，最后回到起点。

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

例如，Borsuk-Ulam 还适用于将 4d 空的超球面映射到 3d 空。

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.

如果曲率是正的，那么宇宙可能是一个超球体的表面，即四维球体的三维表面。

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.

因此，我们在 3d 空到的单位球体代表了投射到我们身上的四元数超球面唯一未改变的部分。

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.

我所说的超球面是所有可能的 4 坐标列表的集合， 其它们的平方和等于 1；这些是 4d 空距离原点 1 的点。

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.

从几何角度来，四维生物能够到我刚才描述的这两个垂直旋转，并理解它们将你锁定在超球面的一个且唯一一个刚性运动。