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，超光速飞船最终会绕着这个弯曲超球体旅行一圈，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 空中超球面映射到 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，那么宇宙可能是一个超球体表面，即四维球体三维表面。

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，在 3d 空中看到球体代表了投射到身上四元数超球面中唯一未改变部分。

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.

所说超球面是所有可能 4 坐标列表集合， 其中它平方和等于 1；这些是 4d 空中距离原点 1 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.

从几何角度来看，四维生物能够看到刚才描述这两个垂直旋转，并理解它将你锁定在超球面一个且唯一一个刚性运动中。