Eclipses in 4D

[PatrickPowers]

This one is easy. Eclipses would never occur in most 4D systems. The 2D ecliptic planes of the sun and moon have to have a line as their intersection, or at least nearly so. This is seldom the case. In those rare systems where eclipses can occur, the frequency of eclipses would be the same as in 3D.

[PatrickPowers]

This one is easy. Eclipses would never occur in most 4D systems. The 2D ecliptic planes of the sun and moon have to have a line as their intersection, or at least nearly so. This is seldom the case. In those rare systems where eclipses can occur, the frequency of eclipses would be the same as in 3D.

Looking at this today I'm wondering whether this is right. This question seems to be analogous as to whether two random great circles on a sphere intersect. Any two such points define a great circle, so have two random pair of points. It seems it should be easy, but right now I can't do it.

[wendy]

In 4d, two random great circled do not necessarily intersect.

The best one can say, is that there is a great circle that is parallel to both of them.

[PatrickPowers]

In 4d, two random great circled do not necessarily intersect.

The best one can say, is that there is a great circle that is parallel to both of them.

Surely this question is answerable. Mainly the question is whether the probability of intersection is zero. Now I'm pretty sure it is. Consider this argument.

I think of this in terms of latitude and longitude. Without loss of generality let's assume that one of the great circles is fixed at latitude zero. There are an infinity of such circles. Chose two points at random for the other circle. The random circle then has a maximum and minimum latitude. Let's further assume that the random great circle crosses the equator, the zone of latitude zero. While doing so it intersects only one of those great circles of latitude zero. The probability of this being our fixed circle is zero. Q.E.D.

With eclipses we are talking about planets instead of points, so the probability is greater than zero. But in the great majority of systems they will never occur. When possible the frequency will still usually be less than here on Earth. A system in which a total eclipse can occur would be seldom seen.