Chapter 17: Partition of Unity

Rotations in 4 and 5 dimensions

Rotation in a single plane

In Chapter 17 of Diaspora, an artifact is discovered near the “rotational pole” of a star in the 5-dimensional macrosphere; this pole is described as the single 2-sphere on the star’s 4-dimensional hypersurface that stays fixed in space as the star rotates. [This has been corrected in the 2013 eBook.]

To see why this is possible (but actually not very likely), first consider the case in 3 dimensions. If a planet in 3 dimensions rotates as a perfectly rigid body, it has a stationary axis running through it which intersects the 2-dimensional surface at two points, the familiar rotational poles. But another way to characterise this motion is by the direction of the plane of the rotation: a slice through the planet along any plane parallel to the equator will reveal the rock to be moving only within that plane. At the centre of each plane the rock won’t be moving at all; taken together, all these fixed centres constitute the axis of rotation.

Rotation in a single plane is also possible in higher dimensions, but in this case there’s room for a larger family of parallel planes, and their centres comprise a fixed region with more dimensions than the familiar 1-dimensional axis. In 4 dimensions, there’s a 2-dimensional family of parallel planes, whose centres make up an entire 2-dimensional plane, perpendicular to the plane of rotation. This fixed plane intersects the 3-sphere of the planet’s surface along a single circle whose radius is as large as the planet’s, and which constitutes a kind of “pole”. This is obviously very different from the case in 3 dimensions, but in 4 dimensions a great circle like this is much less significant than it would be in 3. (You could walk from one point on the surface of the planet to the diametrically opposite point without ever crossing the “pole”, because the surface is 3-dimensional, giving you plenty of room to avoid it.)

In 5 dimensions, there’s a 3-dimensional family of parallel planes, whose centres make up a 3-dimensional fixed volume, again perpendicular to the plane of rotation. This fixed volume intersects the 4-sphere of the planet’s surface to form a 2-sphere for the “pole”. Again, although this 2-sphere has a radius as large as the whole planet’s, there’s more than enough room on the 4-dimensional surface to side-step it when travelling from one side of the planet to the other.

Rotation in two planes

The reason it’s actually unlikely that a star in 5 dimensions would have an entire 2-sphere as its “pole” is that in 4 or 5 dimensions, a rigid body can be rotating in two completely independent planes at once (and if it has suffered random bombardments, this will almost certainly be the case). This is impossible in 3 dimensions; each plane needs two directions to define it, so to keep them completely independent you need a total of at least 4 dimensions.

In 4 dimensions, a rotating star or planet will generally have no pole at all on its surface. Why? Because the fixed plane of the second rotation will be entirely perpendicular to the first; both will pass through the planet’s centre, but they won’t intersect anywhere else. This is similar to the 2-dimensional case, where a disk rotating in a planar universe has only one fixed point, buried in its interior.

In 5 dimensions, there isn’t room for the two 3-dimensional fixed volumes to avoid each other so completely. In general, they’ll intersect along a line, forming a 1-dimensional axis of rotation that intersects the surface at two distinct poles, just like the 3-dimensional case.

In both 4 and 5 dimensions, a rotating star or planet will usually have two distinct rotational periods — a separate “day” for each plane of rotation — and two distinct equatorial circles where the planes of rotation intersect the surface.

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Diaspora / Chapter 17: Partition of Unity / created Saturday, 25 October 1997 / revised Monday, 3 August 1998
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