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。

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，球体三。

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，一个生物能够观察到我刚才描述两个垂直旋转，并理解它们将你锁定在一个且仅有一个刚体运动中，对于超球体来说是这样。

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.

所以我们看到三空间中单位球代了元数超球体中唯一未被改变并投影到我们这里部分。

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-乌拉姆定理也适用于将4空间中超球映射到3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.

我所说超球是所有可能 4 坐标列集合， 其中它们平方和等于 1；这些是 4d 空间中距离原点 1 点。