Parallel AND Perpendicular

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

Parallel AND Perpendicular

Postby PatrickPowers » Sat Aug 17, 2019 3:52 am

There is no general definition of "parallel" that is agreed upon. But a good one would be that everywhere the minimum distance between two objects is the same. Then Pringles potato chips in a can would be mutually parallel. Well, maybe not exactly, but close enough.

How about "perpendicular?" A good start would be two objects are perpendicular if at every point of intersection the maximal angle between the two is 90 degrees. If two objects don't intersect, then say if there are natural extensions that are perpendicular then the two objects are perpendicular.

In 3D you can't have two objects that are BOTH parallel and perpendicular. But in 4D and above you can. It's perfectly normal. Consider a sphere. Choose any great circle. Take the great circle that is perpendicular to it. This circle is also parallel to the first.

Now there is a potential problem with these definitions. Consider those Pringles potato chips. Translate them further apart and they aren't parallel any more. But whatever. It's still a useful concept. If this bothers you, a less general definition of parallel can be used. Probably a good idea, but I can't be bothered.
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Re: Parallel AND Perpendicular

Postby anderscolingustafson » Thu Aug 29, 2019 12:25 am

Another definition of perpendicular would be to state that two directions are perpendicular if it is possible for the amount of change in one direction to have no relationship to the amount of change in the other direction.

There are geometries, in which two lines are parallel at particular points, but they are not the same distance at every point. One way to define whether or not two lines are parallel at a particular pair of points would be to say that they are parallel at that particular pair of points if as you approach that pair of points the rate of change in distance between the two lines approaches 0. Now in the geometry of a sphere and the geometry of a hyperbolic plain, while no two lines have the same distance at every point there are equidistant curves, that have the same distance at every point from a particular line even though they are not lines. An example of an equidistant curve would be a circle of latitude, as a circle of latitude is the same distance everywhere from the equator but is not a line, while the equator is a line along the surface of the Earth.
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