Constructing Physical Points In Point World
In a 1D Point World the whole world exists on a single line
(the x-Axis) and the surface of the 1D Physical Point is a Mathematical Point Pair at -dR and dR. In a 2D Point World the whole world exists on a single
plane (the xz-Plane) and the surface of a 2D Physical Point is a 2D Circle around the origin with radius dR.
In a 3D Point World the y-Axis is added to the xz-Plane to form a 3D space and a
3D Physical Point is the usual 3D Sphere around the origin with radius dR. In a 4D Point World
a fourth axis (the w-Axis) must be added to the graph, but since the Point World
representation of space is limited to 3 dimensions the w-Axis will have to be handled
using Axis Sharing. A Physical Point in a 4D world is a 4D Hyper Sphere around the origin
with radius dR.
Point World uses a Physical Point construction technique that proves to be useful when analyzing
how Mathematical Points move through Hyper Space. Starting with the trivial 1D case on the x-Axis
it is seen that there is only one way to build the 1D Physical Point which is at x=-dR
and x=dR and which defines the extent of the 1D Physical Point on the x-Axis. Even though there are only two Mathematical Points we can still say that the radius is dR. For the 2D case the z-Axis is added a 2D Physical Point is built by adding 1D Physical Points to the graph at an Offset decreasing by equal -Delta
increments in the -z direction and at an Offset increasing by equal +Delta increments
in the z direction up to z=-dR and z=dR. The radius of the added 1D Physical Point
will become smaller as the Offset increases in a mathematical relationship that
forms the 2D Physical Point. At the largest Offsets of z=-dR and z=dR the 1D Physical Point becomes
a single Mathematical Point that defines the extent of the 2D Physical Point on the z-Axis.
This method of building the 2D Physical Point is called the Equal Offset method. This
method produces a Circle that has Points that are closer together at small Offsets
and further apart at large Offsets. Another method that produces a Circle with Points
evenly spaced around the Circle is called the Equal Angle method. Both these methods produce an arrangement of Points that all have the same Radius to a central Point. Either of these
methods will be used in the Animations depending on the purpose of the Animation.
Also, 3D Spheres can be created from 2D Circles and 4D Hyper Spheres can be created
from 3D Spheres using these methods. The following Animation shows how 2D Circles
can be constructed using Point Pairs and how the Circles look for the two different
methods.
(Animation14)
Constructing 2D Empty Physical PointsUsing 1D Point Pairs
The following Animation shows how 2D Full Circles can be constructed using Point Lines
and how the Circles look for the two different methods.
(Animation15)
Constructing 2D Full Physical PointsUsing 1D Point Pairs
The following Animation shows how 3D Spheres can be constructed using 2D Circles
and how the Spheres look for the two different methods.
(Animation16)
Constructing 3D Physical Points Using 2D Physical Points
The following Animation shows how 4D Hyper Spheres can be constructed using 3D Spheres
and how the Hyper Spheres look for the two different methods. 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.
(Animation17)
Constructing 4D Physical Points Using 3D Physical Points
Equations For Equal Offset Method:
Offset = n * DeltaOffset
Angle = asin(Offset / ObjRadius)
Offset = ObjRadius * sin(Angle)
Radius = ObjRadius * cos(Angle)
Equations For Equal Angle Method:
Angle = n * DeltaAngle
Offset = ObjRadius * sin(Angle)
Radius = ObjRadius * cos(Angle)
Where:
-N <= n <= N.
DeltaOffset is the Equal Offset increment.
DeltaAngle is the Equal Angle increment.
Offset is the Offset of the Pair, Circle, or Sphere component of the object.
Angle is the Angle of the Pair, Circle, or Sphere component of the object.
Radius is the Radius of the Pair, Circle, or Sphere component of the object.
ObjRadius is the Radius of the object (Circle, Sphere, or Hyper Sphere).
Note that these equations will produce Points that have ObjRadius. 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 Point. It is easy in 3D space to see how all the Points of a component 2D Circle are at a Radius that is perpendicular to the y-axis. In 4D space it is more difficult to see how all the Points on a component 3D Sphere can be perpendicular to the w-Axis. It must be realized that relative to 4D space our 3D space is flat and the 3D Sphere is also flat. The whole 3D space is perpendicular to the 4D w-Axis so the Points of the 3D Sphere in 4D are all on the same 3D 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 Point to the origin completes the triangle and is the distance of the Point from the origin which is ObjRadius. So:
ObjRadius = sqrt(Offset^2 + Radius^2)
It is easy to see how 2D Circles are made out of Point Pairs and how 3D Spheres are made out of 2D Circles but it is more difficult to see how Hyper Spheres are made out of 3D Spheres because we can not ever really see a Hyper Sphere. 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 Sphere. 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 Sphere. A higher dimensional space is not just more of the lower dimensional space but rather it is a completely different thing. Still, more can be learned about 4D space especially when it comes to rotations as will be shown later.