Section 1
Understanding Multi-Dimensions
All objects are described as existing in two domains, Time and Space. In array format this would be described as:
Object { Time ( T Dimension ), Space ( S Dimension ) }
The set brackets indicate that the contents of the array may be of different types ( in this case that's true ).
Most people think of time as being another dimension of space, it is not. Given any spatial dimension, time is constant across that space, in other words, one second is one second no matter where you go or what you do in a given spatial dimension.
The reverse is also true.
Given any time dimension, space is constant across that time, in other words, one inch is one inch no matter where you go or what you do in a given time dimension.
These properties are demonstrated by the way the laws of physics are constant. The universe would be near impossible to navigate or understand if the laws of physics constantly changed or varied by where you went or what you did. Predicating an outcome of any action would be a waist of time since the laws covering behavior are constantly changing. When you can't predict an outcome, then the problem is your understanding of the forces affecting the outcome. The laws of physics did not change, you just failed to take into account another force ( variable ) involved.
The basics of spatial dimensions:
Refer to figure - Dimensions.
Dimension 0:
Object types: Point
Number of objects: 1
Dimension 1:
Object types: Point, Line
Number of objects: Infinite, 1
A line is composed of an infinite number of points.
Objects have only a width. In school you where probably taught this dimension as the length of an object.
In order to actually make measurements a discussion about coordinate systems and reference frames ( observer perspective ) must be dealt with.
Coordinate Systems:
Refer to figure - 1-D Space Coordinate Systems.
When communicating in any language and specifically the mathematics of geometry you must establish a standard set of definitions ( a dictionary ) with which to communicate. In geometry the dictionary is referred to as the frame of reference.
In the 1 st dimension a line can exist in any coordinate system.
The 1-D Space Coordinate Systems figure shows a sample of four coordinate systems. A set of principles and conventions defines the 1-D standard coordinate system.
Principles:
Conventions:
Reference Frames:
Refer to figure - 1-D Reference Frames.
It's not enough to have a standard coordinate system, there must also be a standard frame of reference. A frame of reference is a perspective of a world from a single reference point. If two people observe an event, then each observer creates a frame of reference of the observed event from their perspective. That perspective is the reference point that measurements are made.
In view A, no frame of reference is given so there is no way of describing the relation ship between the two systems. When the two observers communicate their measurements, that data is meaningless, because there is no relationship between the measurements. In other words a point on unit 1 of System A has what relationship to any point on System B? System A will report a movement from unit 1 to unit 3. But that is not the same as a movement from unit 1 to unit 3 in System B's coordinates.
To make measurements between the two systems, a standard frame of reference must be established and a translation formula must be used to transfer measurements from another frame of references to the standard frame of reference.
System A and System B coordinates are place above and below the 1-D line-world in 2-D space, to make visual alignment easier ( see view B, I didn't bother to extend them out, I just shifted then so they aligned ). In the real world two frames of reference would not align to each other this neatly. If System A is the standard frame of reference, then a data point in System B would be translated to System A by adding 3. So the translation formula would be:
A = B + 3
Of course to translate from A to B would be just as easy, in this example.
In view C, we use only the standard frame of reference and directly translate System B's data through the formula directly on to System A's coordinate system.
System B reports a movement from point -6 to point -1.
P1 = -6 + 3
P2 = -1 + 3
So the data collected becomes, point moved from -3 ( P1 ) to 2 ( P2 ).
System A can now compare it's observation of the movement. If they are different, then either an error has occurred or their understanding of the event is wrong. This is the importance of having more than one measurement taken from more than one frame of reference. The minimum number of measurements and frames of reference can be determined for any given dimension. The fewer frames to make measurements the more measurements that must be taken and the greater the amount of uncertainty there will be. Measurement from a single frame of reference is only marginally better than a random guess. To fix the largest problem with statistical data, you need more than a single analysis of the data, a clear understanding of the measurement system and an understanding of the data's degree of accuracy.
Dimension 2:
Object types: Infinite ( see figure - 2-D Objects, for a sample )
Number of objects: Infinite
The basic object type is a rectangle. A 2-D object is composed of an infinite number of lines. Objects have a height and a width. In school you where probably taught the height dimension as the width of an object.
Two 1-D reference coordinate systems are placed perpendicular to each other ( at 90 degrees to each other, which is another discussion ). The single point intersection of the two axis is the origin. By convention, the horizontal axis is the X axis ( left to right ) and the vertical axis is the Y axis ( bottom to top ). The width of an object is measured along the X axis and the height of an object is measured along the Y axis.
The space occupied by 2-D space is referred to as a plane. A plane is a square ( all sides are of equal size, and any two intersecting sides form internal 90 degree angles ) encompassing all of the 2-D space. In 2-D space lines are not all straight ( parabola ) or infinite in width ( circle ).
Dimension 3:
Object types: Infinite ( see figure 3-D Objects, for a sample )
Number of objects: Infinite
The basic object type is a box. A 3-D object is composed of an infinite number of planes. Objects have depth, height and width.
Three 1-D reference coordinate systems are placed perpendicular to each other. The single point intersection of the three axis is the origin. By convention, the horizontal axis is the X axis ( left to right ), the vertical axis is the Y axis ( bottom to top ) and the perspective axis is the Z axis ( front to back ). The Z axis is usually tilted at 45 degrees to the X and Y axis so it's easier to visualize. The width of an object is measured along the X axis, the height of an object is measured along the Y axis and the depth of an object is measured along the Z axis. In 3-D space planes are not all flat ( cylinder ) or infinite ( torus ).
Another way to visualize a 3-D object is to create a cube composed of an infinite number of 2-D planes with the 3-D object drawn through as many 2-D planes as needed ( not shown ).
Dimension 4 ( Hyperspace ):
Object types: Infinite
Number of objects: Infinite
The basic object type is a hypercube. A 4-D object is composed of an infinite number of 3-D spaces. Objects have anything, depth, height and width.
There is no name for the 4 th dimension of a 4-D object.
By convention, the horizontal axis is the X axis ( left to right ), the vertical axis is the Y axis ( bottom to top ) and the two perspective axis are the Z and W axis ( front to back ). The Z axis is usually tilted at 22.5 degrees to the X and Y axis so it's easier to visualize. The W axis is usually tilted at 45 degrees to the X and Y axis so it's easier to visualize.
In each dimension the axis are at 90 degrees to each other. But a piece of paper can only show two axis, so all other axis are tilted to the X-Y axis.
One way to visualize a 4-D object is to create a cube composed of an infinite number of 3-D spaces compressed into planes with the 4-D object drawn through as many of these compressed 3-D planes as needed.
The other way is to number crunch the huge number of complex transcendental calculations to translate a hyperspace object into a 3-D object, then translate that into a 2-D object with perspective to give the illusion of a 3-D object in 3-D space.
Perspective?
To understand, how an object may look in a lower dimension is based of the observer's perspective ( frame of reference ).
In figure - Perspective, is illustrated how a 2-D object is seen by an observer in 2-D space. The question is, is that object a square or is it a 2-D slice of a 3-D object that passes through the observer's 2-D space. This can be even more complicated if a circle is seen, since that can be the result of a sphere or cylinder being intersected by the 2-D plane ( not shown ). If a 3-D object was moving through the 2-D space and was not infinite in depth, over a period of time the 2-D object may change or disappear. That would indicate the presence of an intersection.
This problem exist in the 3-D world as well. Unless a 4-D object is passing through the 3-D intersection, there is no way to determine if the object is completely a 3-D object in that 3-D space or an intersection of an object in a higher dimension. If the object is moving and is a hypersphere, then the size of the circle will shrink until it disappears ( it could expand then shrink, depending on the intersection and direction of movement through the 3-D space ). Different frame of references in 3-D space of the 4-D or higher object will provide you with no additional information, since what you are viewing is an infinitely small slice of the 4-D or higher object. As with the square, there is no frame of reference in the 2-D space that provides any additional information about the depth of the square, since it's infinitely small, and therefore not measurable or changeable. However, you can change the 2-D aspect of the 3-D object at that intersection as you can change the 3-D object at it's 4-D intersection. But neither modification will affect the rest of the object in the other planes. An observer in the higher dimension might not even notice a change at the intersection, since it would be an infinitely small modification.
Last updated: Mar 4, 2011