Planes/universes/dimensions

[Shrykull]

Ok, so what exactly are the hard definitions of these three terms that are often used very interchangeably? The only definitions I "know" of plane (plane of existence in this case) comes from the AD&D roleplaying game, where they keep changing it, or the geometry definition where a plane is a 2 dimensional surface that theoretically extends in all 4 directions forever, but what is the scientific definition of it? Does a plane contain universes? Or does a universe contain planes? As for dimension, the ones I know are length, width and height, and another the 4th dimension- time, which is not a direction at all, unless you'd consider the past present and future directions

[Darth Wong]

Ok, so what exactly are the hard definitions of these three terms that are often used very interchangeably? The only definitions I "know" of plane (plane of existence in this case) comes from the AD&D roleplaying game, where they keep changing it, or the geometry definition where a plane is a 2 dimensional surface that theoretically extends in all 4 directions forever, but what is the scientific definition of it? Does a plane contain universes? Or does a universe contain planes? As for dimension, the ones I know are length, width and height, and another the 4th dimension- time, which is not a direction at all, unless you'd consider the past present and future directions

A plane is a flat mathematical entity defined by a normal vector. When sci-fi people talk about planes of existence, they are using the AD&D definition, whereby "plane" is just another way of saying "separate universe".

[Shrykull]

A plane is a flat mathematical entity defined by a normal vector. When sci-fi people talk about planes of existence, they are using the AD&D definition whereby "plane" is just another way of saying "separate universe"

hmm, did you ever play that or any other RPG? That was the defintion given in the 3E manual of the planes.

[Nick]

Ok, so what exactly are the hard definitions of these three terms that are often used very interchangeably? The only definitions I "know" of plane (plane of existence in this case) comes from the AD&D roleplaying game, where they keep changing it, or the geometry definition where a plane is a 2 dimensional surface that theoretically extends in all 4 directions forever, but what is the scientific definition of it? Does a plane contain universes? Or does a universe contain planes?

OK, as you say, the standard mathematical definition of a "plane" is a 2D 'slice' through 3D space. Such a plane is generally defined in one of two ways:

Either you give 3 points in 3D space (which cannot be connected by a straight line), and say "the plane which passes through these 3 points". In this manner, a given plane can be defined by any 3 points that it contains (which are not in a line), but any 3 points will define a unique plane. This is the method normally used when first introducing the idea of planes, but for practical mathematics it is too awkward to work with.

The 'normal vector' Mike mentions is a way of defining a plane by a single point on that plane, along with a vector which is perpendicular to that plane. The point defines the location, the vector defines the orientation.

Now, just as you can slice a 2D surface to get a 1D line, and slice a 3D space to get a 2D plane, you can slice any nD space to get a (n-1)D 'hyperplane'.

A (n-1)D hyperplane can be defined by a nD point, along with a nD vector, just as a 2D plane in 3D space is defined by a 3D point plus a 3D normal vector.

However, 'hyperplane' often gets abbreviate to 'plane', with the number of dimensions involved being implied by context.

As for dimension, the ones I know are length, width and height, and another the 4th dimension- time, which is not a direction at all, unless you'd consider the past present and future directions

Generally speaking, 'dimension' is just another term for measurement. However, it does have a specific meaning in terms of coordinate sets - which is what you are referring to above.

Now, in a 1D space (or line), you can uniquely identify a point simply by giving its distance from the origin (you pick a point on the line and declare it to be 'zero').

In a 2D space, you need 2 numbers to uniquely identify a point - and again, you pick an arbitrary point and call it the origin.

In a 3D space, you need 3 numbers.

Each additional number needed to uniquely identify a specific point is a 'dimension'. As you say, the normally terminology used is height, width and length, with the three coordinate axes at right angles to each other.

However, other coordinate systems are possible, even if they aren't used as often (for example, scientists and engineers often use cylindrical or spherical coordinate systems, where some of the dimensions are angles rather than distances). They are still 3D systems, because you still need 3 numbers, but the interpretation of the numbers is different.

Time is considered the 4th dimension, because it is used to identify not only a given point in space, but also the exact time at which we are interested in that point. (E.g. if we considered ground zero of a nuclear blast, it makes a big difference if we are there before, during or after the explosion).

For a long time, it was believed that you couldn't muck around with the time dimension the same way you could with the space dimensions - it was 'different'. General relativity started to poke holes in that assumption - the fact that movement through space could affect the passage of time really messes with our intuitive understanding of the universe.

However, it is clear that time does have some unique characteristics. This is why scientists sometimes say space-time has 3 + 1 dimensions, rather than simply 4. The first 3 dimensions are essentially interchangeable - it doesn't matter if you are moving up & down, forward & back or side-to-side, the equations don't care. Time however, marches forward relentlessly, and doesn't appear to let anything go backwards - it simply doesn't behave quite the same way as the other dimensions do.

The way 'planes of existence' comes into some sci-fi is related to the idea that the universe we experience might be a 4 dimensional hyperplane of a 5 dimensional 'multiverse', where we can't travel along the 5th dimensions at all. (I suspect the actual origin of the term comes from the old cosmological concept of a Flat Earth, with Heaven above it, and Hell below it - in this cosmology, the phrase 'planes of existence' was a literal description of what the diagrams showed).

[Vympel]

A plane is a flat mathematical entity defined by a normal vector. When sci-fi people talk about planes of existence, they are using the AD&D definition, whereby "plane" is just another way of saying "separate universe".

Now see here we have conclusive proof that geekdom and AD&D are distinctly related ...

I've never played pen and paper D&D. I find pen and paper to be quite sad. I am a big fan of Baldur's Gate/Neverwinter Nights/Planescape Torment etc tho