What is the 5th Dimension exactly?

[zeon23445]

Can someone explain to me mathematically, in terms of Axis, and or in word form what the 5Th dimension is? Please don't say its spiritual unless you are somehow speaking on biocentrism >.>

[Daedalus]

Can someone explain to me mathematically, in terms of Axis, and or in word form what the 5Th dimension is? Please don't say its spiritual unless you are somehow speaking on biocentrism >.>

 

Well, it really depends on what we are talking about. The fifth dimension could be just another dimension of space. Such dimensions would be orthogonal (90 degrees) to every other spatial dimension (axes) no different than our 3 spatial dimensions and 1 temporal. However, any set of 5 variables creates a five-dimensional mathematical space. So, it doesn't neccessarily have to be spatial dimensions. If your question is related to physics, then I'll refer to Wikipedia - Five-Dimensional Space:

 

In physics, the fifth dimension is a hypothetical extra dimension beyond the usual three spatial dimensions and one time dimension of Relativity. The Kaluza–Klein theory used the fifth dimension to unify gravity with the electromagnetic force; e.g. Minkowski space and Maxwell's equations in vacuum can be embedded in a five-dimensional Riemann curvature tensor ^{%5B1%5D[unreliable source?]}. Kaluza–Klein theory today is seen as essentially a gauge theory, with gauge group the circle group. M-theory suggests that space–time has 11 dimensions, seven of which are "rolled up" to below the subatomic level. Physicists have speculated that the graviton, a particle thought to carry the force of gravity, may "leak" into the fifth or higher dimensions, which would explain how gravity is significantly weaker than the other three fundamental forces.

Edited September 23, 2013 by Daedalus

[zeon23445]

Can you expand a little bit on the concept of orthogonal space?

[Daedalus]

Can you expand a little bit on the concept of orthogonal space?

 

Sure np. It simply means that all axes as perpendicular or at right angles to each other. If you have every graphed a function in 2 dimensions, [math]x[/math] and [math]y[/math], then you might have noticed that the [math]x[/math] and [math]y[/math] axes are perpendicular / orthogonal / 90 degrees to each other. This is true for higher dimensional spaces too. Just look at 3D space. All axes, [math]x[/math], [math]y[/math], and [math]z[/math], are perpendicular / orthogonal / 90 degrees to each other. The same would hold true no matter how many dimensions of space there are. However, higher dimensions of space beyond three are not easy to visualize.

Edited September 23, 2013 by Daedalus

[HalfWit]

[insert obligatory Up Up and Awaaaaay joke]

 

The first dimension consists of the real numbers, along with the usual addition and multiplication of real numbers. Geometrically, it's often called the real line.

 

The second dimension consists of all the possible pairs (x1, x2) of real numbers, along with the operations of component-wise addition and scalar multiplication. Geometrically, it's known as the plane.

 

The third dimension consists of all possible triples (x1, 2, x3) of real numbers, along with the operations of component-wise addition and scalar multiplication. Geometrically it's generally called 3-space. (Or *real *3-space, in contexts where we are considering n-tuples of complex numbers, etc.)

 

The fourth dimension consists of all possible 4-tuples (x1, x2, x3, x4) of real numbers, along with the operations of component-wise addition and scalar multiplication. Generally it's called 4-space.

 

The fifth dimension consists of all possible 5-tuples (x1, x2, x3, x4, x5) of real numbers, along with the operations of component-wise addition and scalar multiplication. Geometrically it's called 5-space.

 

Dot dot dot. In each case we define the distance of two points as the square root of the sum of the squares of the componentwise differences. We then use the distance function to define a metric; and we use the metric to give us a topology. Then we can have continuous functions, limits, derivatives and all that other good stuff in as many dimensions as we like.

 

Exercise to verify that you've understood this: What is the 475th dimension, mathematically? What's it called geometrically? Write down an explicit expression for the distance between two arbitrary points in real 475-space. Oops I just gave away one of the answers.

 

Moral of the story: Math is not physics. Mathematics is much simpler than physics!

Edited September 24, 2013 by HalfWit

[Amaton]

Can someone explain to me mathematically, in terms of Axis, and or in word form what the 5Th dimension is?

 

What context are we talking here? Because "the 5th dimension" could refer to several things.

 

If you're thinking purely geometric, then you're probably looking for the 5th spatial dimension (as opposed to an extra dimension of time or some application in physics).

 

The general notion with spatial dimensions is surprisingly simple. The first spatial dimension has one axis, typically referred to as the [math]x[/math]-axis. The second dimension has two axes, the [math]x[/math]-axis and the [math]y-axis[/math] forming the [math]xy[/math] plane. Likewise, the third dimension has three axes forming an "xyz space" (very informal wording).

 

When we go beyond the 3rd spatial dimension, things get hard to visualize, but conceptually it all follows the same basic principle. Four spatial dimensions include 4 axes, conventionally wxyz. Five dimensions: five axes; etc. etc. As you can see, for any (simple Euclidean) spatial dimension [math]n[/math], we will have [math]n[/math] axes (given [math]n\ge 0[/math]), each perpendicular to all the others if defined.

 

An interesting bonus fact... The surface of a 5-dimensional hypersphere consists of infinitely many 4-dimensional hyperspheres which themselves each consist of infinitely more spheres. It's somewhat of a word salad, and in a purely mathematical sense, it's really nothing special. Regardless, it's a mindrape in trying to wrap one's head around such an object, as if the 4th spatial dimensional alone wasn't crazy enough.

[studiot]

Multidimensions have application when you come to advanced mechanics.

 

You should look up "generalised co-ordinates"

 

https://www.google.co.uk/#q=generalised+coordinates+in+mechanics

 

I once wrote a paper entitled "The use of the fifth quadrant".