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

这种情况下，的超光速飞船绕着这个弯曲的超球体旅行一圈，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， 3d 空中看到的单位球体代表了投射到身上的四元数超球面中唯一未改变的部分。

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.

所说的超球面是所有可能的 4 坐标列表的集合， 其中它的平方和等于 1；这些是 4d 空中距离原点 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.

从几何角度来看，四维生物能够看到刚才描述的这两个垂直旋转，并理解它将你锁定超球面的一个且唯一一个刚性运动中。