Constructing Physical Points In Point World
We start with the assumption that a 0D PP is a single MP at the origin. There cannot be an actual PP in a 0D Point World because there are no dimensions to extend into. This Point World consists of nothing because a single MP is really nothing and is a degenerate PP . Next we add the x-Axis to make a 1D Point World and put an MP at x=-dR and at x=+dR to make a 1D PP at the origin. We have used 2 0D PPs to make the 1D PP. The 1D PP consists of only two MPs on its surface and has zero surface area which also makes it a degenerate PP.
In a 2D Point World the z-Axis is added perpendicular to the x-Axis of the 1D Point World to form a 2D Point World (xz-Plane) and the surface of a 2D PP is a 2D Circle of MPs located around the origin with radius dR. We cann use two different methods to assign the MPs that are located on the 2D Circle. The first method will be called the Equal Offset method. We decide how many 1D PPs we want to construct on each side of the x-Axis and use the Equations for the Equal Offset Method below. This
method produces a 2D PP defined by a Circle that has MPs that are closer together at small Offsets
and further apart at large Offsets. This method constructs a 2D PP that doesn't look good but behaves very well in the animations. The second method is called the Equal Angle method. Again we first decide how many 1D PPs we want to construct on each side of the x-Axis and use the Equations For Equal Angle Method below. This
method produces a 2D PP defined by a Circle that has MPs that are evenly spaced around the Circle. This method constructs a 2D PP that looks good but does not behave as well as the Equal Offset method in the Animations.
The PPs in n-dimensional Point World will be constructed from the PPs of (n-1)-dimensional Point World. This construction process will produce PPs with well behaved MPs as seen in later Animations.
Note that the equations are included for completeness and for those who need to know every detail. It is not necessary to understand the equations in order to understand the Animations.
So far we have been describing the construction of Empty PPs but there can also be Full PPs. A 1D Full PP is a line of MPs from the MP at -dR to the MP at dR. The following Animation shows the construction of 2D Empty PPs using 1D Empty PPs.
(Animation14)
Constructing 2D Empty Physical Points Using 1D Empty Physical Points
The following Animation shows how 2D Full PPs can be constructed using 1D Full PPs.
(Animation15)
Constructing 2D Full Physical Points Using 1D Full Physical Points
In a 3D Point World the y-Axis is added perpendicular to the xz-Plane of the 2D Point World to form a 3D space (xyz-Space) and the surface of a 3D PP is a 3D Sphere of MPs located around the origin with radius dR. We can use the Equal Offset method or the Equal Angle method to assign the MPs that are located on the 3D Sphere. We first decide how many 2D PPs we want to construct on each side of the xz-Plane and use the Equations For Equal Offset Method or Equal Angle method below. The following Animation shows how 3D Empty PPs can be constructed using 2D Empty PPs. 3D Full PPs are constructed similarly and are not shown in an Animation.
(Animation16)
Constructing 3D Empty Physical Points Using 2D Empty Physical Points
In a 4D Point World
a fourth axis (the w-Axis) must be added perpendicular to xyz-Space, but since the Point World
representation of space is limited to 3 dimensions the w-Axis will have to be handled
using Axis Sharing. The surface of a 4D PP is a 4D Hyper Sphere of MPs located around the origin
with radius dR. We can use the Equal Offset method or the Equal Angle method to assign the MPs that are located on the 4D Hyper Sphere. We first decide how many 3D PPs we want to construct on each side of the xyz-Space and use the Equations For Equal Offset Method or Equal Angle method below. For 4D Hyper Spheres
the Equal Angles cannot properly be shown when using Axis Sharing so both methods
show the Offsets. Note how the Offsets are not equal for the Equal Angle method. The following Animation shows how 4D Empty Hyper Spheres can be constructed using 3D PPs. 4D Full PPs are constructed similarly and are not shown in an Animation.
(Animation17)
Constructing 4D Empty Physical Points Using 3D Empty Physical Points
Equations For Equal Offset Method:
DeltaOffset = dRadius / (2 * N)
Offset = n * DeltaOffset
Angle = asin(Offset / ObjRadius)
Offset = dRadius * sin(Angle)
Radius = dRadius * cos(Angle)
Equations For Equal Angle Method:
DeltaAngle = (Pi/2) / (2 * N)
Angle = n * DeltaAngle
Offset = dRadius * sin(Angle)
Radius = dRadius * cos(Angle)
Where:
N is the number of (n-1)-dimensional PPs on each side of the PP under construction.
-N <= n <= N.
DeltaOffset is the Equal Offset increment.
DeltaAngle is the Equal Angle increment.
Offset is the Offset of the (n-1)-dimensional PP of the n-dimensional PP.
Angle is the Angle of the (n-1)-dimensional PP of the n-dimensional PP.
Radius is the Radius of the (n-1)-dimensional PP of the n-dimensional PP.
dRadius is the differential Radius of the n-dimensional PP.
Note that these equations will produce MPs that have dRadius. This can be seen by recognizing that the Offset is on the Hyper Axis (y or w) and the Radius is perpendicular to the Hyper Axis regardless of the particular MP. It is easy in 3D space to see how all the MPs of a component 2D PP are at a Radius that is perpendicular to the y-axis. In 4D space it is more difficult to see how all the MPs on a component 3D PP can be perpendicular to the w-Axis. It must be realized that relative to 4D space our 3D space is flat and the 3D PP is also flat. The whole 3D space is perpendicular to the 4D w-Axis so the MPs of the 3D PP in 4D are all on the same 3D Hyper Plane and therefore the Radius lines are all perpendicular to the w-Axis. So the Radius line and the Offset line define a right triangle in the n-dimensional space and it is easy to see that the line from the MP to the origin completes the triangle and is the distance of the MP from the origin which is dRadius. So:
dRadius = sqrt(Offset^2 + Radius^2)
It is easy to see how 2D PPs are made out of 1D PPs and how 3D PPs are made out of 2D PPs but it is more difficult to see how 4D PPs are made out of 3D PPs because we can not ever really see a 4D PP. We can only theorize about it and see it using Axis Sharing or some other method. We actually need a different kind of Brain. We need a 4D Brain and Visual System. Similarly a 2D Point Man can not ever really see a 3D PP. He needs a 3D Brain and Visual System. Even as 3D beings we can not show a 2D Point Man how to see a 3D PP. A higher dimensional space is not just more of the lower dimensional space but rather it is a completely different thing. However it's not hopeless, because more can be learned about 4D space especially when it comes to rotations as will be shown next.