Exploring The 4th Dimension Using Animations

The following paragraphs and animations will explore certain aspects of the 4th Dimension using animation techniques. The open source software packages, OpenGL and the OpenGL Utility Toolkit (GLUT), were extensively used to create the animations. The OpenGL and GLUT software are configured to display a Perspective World View (PWV) on the monitor screen. The PWV shows closer objects as larger than more distant objects which makes it easier to visualize what is going on in the animations. First an array of points representing the objects that are being studied is generated and displayed in the PWV. This array of points is then processed using standard Matrix Rotation and Translation methods for an incremental angle and an incremental distance to calculate new locations for all the points and these are displayed in the PWV. The processing and displaying is repeated for a series of incremental angles and incremental distances to animate the objects being studied.

 

 

Introduction To Point World

The Animations take place in a simulated world called Point World. Although the objects in Point World are continuous only discrete Points of objects will be depicted. Objects such as Lines, Circles, Spheres, and Hyper Spheres will be constructed from individual Points. In this way, for example, we don't have to keep track of continuous surfaces but rather just some selected Points on the surface. We want to see how each Point in a more complex object behaves during a 4D Rotation, Translation, or other manipulation.

Point World is drawn in the PWV and consists of a 100 by 100 Unit grid in the xz-Plane with a 100 Unit y-Axis perpendicular to and passing through the center point of the xz-Plane. The y-Axis extends 50 Units up (+direction) and 50 Units down (-direction) from the the intersection with the xz-Plane. The y-Axis can be hidden for 2D animations. Also from this y-Axis intersection, the x-Axis extends 50 Units to the right (+direction) and 50 Units to the left (-direction), and the z-Axis extends 50 Units to the front (+direction) and 50 Units to the back (-direction). Animations for 2D will take place on the horizontal xz-Plane.

In addition to the Points that make up a particular object there is a special Point in Point World called Point Man. This is a single Point that is used as the Observer in Point World. Point Man is a Point observer that can scan across a field of view to look at objects in Point World. Point Man can only see one Point at a time but can remember what has been seen to construct an image for himself. In the animations Point Man scans the scene from his perspective and then sets the Points that he sees to the color Red. Point Man also knows which Points are closer and sets those closer Points to Red first progressively setting more distant Points to Red until the field of view is complete.

This is also a good time to talk about the Eye Location concept. The Eye Location is effectively your location when viewing the PWV. The Eye Location can be moved to any location in the PWV. Moving the Eye Location can help visualize what is happening in an Animation. For most Animations the Eye Location will be above and to the right of the z-Axis.

In the following Animation the Eye Location will Zoom in to Point World and circle around the Point World grid. The Eye Location will then stop while Point Man moves around and Looks at the central Sphere from several different locations. The Eye Location will then Zoom back out. As with all Point World Animations this is a continuous loop Animation so it will start again after Zooming back out.

(Animation1) Introduction To Point World

 

 

Special Considerations For A 2D Point World

Animation1 shows how a Sphere Object might look in Point World. It was constructed using 126 individual Points which are each drawn as a small Ball. In Point World the small Balls represent actual Points. Each Ball is constructed using 72 flat polygons. The shading effect works on the individual polygons to create the overall shading of the Ball. Balls are just markers for where a Point is located. Balls can be drawn packed in more tightly or more loosely. Balls can be packed so tight that they overlap but they still represent Points. The actual diameter of a Ball is not important just the central Point location that it represents is important.

A 2D Point World is considered to be perfectly flat, and therefore it will have identically zero thickness. Not some small thickness, or even differential thickness, but perfectly zero thickness. A 2D Point World exists only in the xz-Plane which means everything is on the Grid, and the y-Axis does not exist. Objects in a 2D Point World have Width and Depth but identically zero Height. Points in a 2D Point World would ideally be drawn as Disks from the Point of View of the 2D Point Man. The 2D Point Man is himself a Disk. But since we are 3D Observers the Points in the 2D Point World will still be drawn as Balls. Balls will represent Points in 2D and the 3D look of the Ball does not mean there is any implied extension into 3D space. Points are depicted as 3D objects because we are stuck in a 3D world and that's what we are familiar with.

Now we need to talk about the common technique of viewing 2D worlds from a 3D perspective. As 3D Observers we have no choice except to imagine that the 2D world we are looking at is embedded in some surrounding 3D space. But this is cheating because a 2D world does not have any real existence in 3D space. We have to imagine some unexplained force holding all the 2D zero Height objects within the plane of the 2D world. There is probably too much false intuition about what existence in a 2D world would be like. We can only presume that the full Width and Depth of things can be perceived through some 2D conscious experience of Vision, but that conscious experience of Vision is probably not the same as our Conscious Light experience. See The Inter Mind for some thoughts on the Conscious Light experience.

In the following Animation Point Man travels around a Circle and stops to look from various locations. The Points and Point Man are shown as Disks and then as Balls. The Eye location moves down and the Grid is hidden in order to view the 2D Point World on edge. Notice how the 2D Point World actually disappears from the on edge perspective when Disks are used. Using Balls enables visibility of the Points even when the 2D Point World is viewed on edge. A 2D Point Man sees his 2D world by scanning left and right so that his visual window into the 2D world is 1-dimensional. The Width and the Depth information are both superimposed onto the 1-dimensional image that he sees. This is similar to how our 3D Depth is superimposed onto our 2D visual image.

(Animation2) Edge View Of A 2D Point World

 

 

Using Axis Sharing To Represent Hyper Points

A Hyper Point is a Point that has a component on a Hyper Space axis. There is a dilemma for a 2D Point Man or a 3D Point Man when they try to physically find the next higher dimension. They are each trapped in their own dimension and are physically unable to look in the direction of the next higher dimension even if it existed. They cannot see the Hyper Points that might be located in the next higher dimension. The Following Animation shows the dilemma for a 2D Point Man.

(Animation3) 2D Point Man Looking At A Sphere

 

If the 2D Point Man could move up into 3D Hyper Space he would be able to look at higher and higher parts of the Sphee until he saw the topmost Point of the Sphere. Similarly if he could move down into negative 3D Hyper Space he would see lower and lower parts of the Sphere until he saw the lowest Point. The following Animation shows the Sphere from Animation3 with Point Man moving up into positive 3D Hyper Space and then down into negative 3D Hyper Space while stopping to look from various levels.

(Animation4) 2D Point Man Moving In 3D Looking At A Sphere

 

But a 2D Point Man is not actually able to go into 3D Hyper Space so he needs a way to view 3D Hyper Points within his 2D world. Point World uses Axis Sharing to let Point Man see Hyper Points from Hyper Space. Since the y-Axis is the Hyper Space axis for 2D we can share the y-Axis with the x-Axis or the z-Axis. Results using the x-Axis as the Shared Axis are similar to the results using the z-Axis as the Shared Axis, and Point World Animations will predominantly use the x-Axis as the Shared Axis. To separate the y-Axis component of a Hyper Point from the x-Axis component the y-Axis component will be multiplied by a Hyper Factor which is usually a value between 2 and 10 depending on the purpose of the Animation.

The following Animation shows the Sphere from Animation3 using Axis Sharing. A 2D Point Man runs an algorithm that draws the various y-Axis components of the Sphere onto the x-Axis to implement Axis Sharing. Point Man can now see the 3D Hyper Points since they are drawn onto the xz-Plane. Point man realizes that when he moves left and right he is effectively moving into negative and positive 3D Hyper Space. Even though he still does not know where 3D Hyper Space is, Point Man can start to understand some things about a Sphere, such as that as he goes further and further into 3D Hyper Space he sees smaller and smaller Circles until there is only a single Point. So he realizes that a Sphere is made out of smaller and smaller Circless locateded further and further into 3D Hyper Space. He also realizes that he would be able to move all around any particular Circle and look at all of it's Points. Note that the 2D Point Man can not see inside a Circle like we can from our 3D perspective. A 2D Point Man sees everything on a 1D edge using his 1-dimensional vision capabilities.

(Animation5) 2D Point Man Looking At Sphere Using Axis Sharing

 

The following Animation shows a Hyper Sphere using Axis Sharing. A 3D Point Man runs an algorithm that draws the various w-Axis components of the Hyper Sphere onto the x-Axis to implement Axis Sharing. The 3D Point Man sees the World like we do and we can see the 4D Hyper Points because they are drawn into xyz-space. We realize that when Point Man moves left and right he is effectively moving into negative and positive 4D Hyper Space. Even though Point Man (and we) still do not know where Hyper Space is, we can start to understand some things about a Hyper Sphere, such as that as Point Man goes further and further into 4D Hyper Space he sees smaller and smaller Spheres until there is only a single Point. So we realize that a Hyper Sphere is made out of Spheres locateded further and further into 4D Hyper Space. We can also see that we can go all around any particular Sphere and look at all of it's Pointss. Note that we can not see inside a Sphere like a 4D Point Man could. We see everything on a 2D edge using our 2-dimensional vision capabilities.

(Animation6) 3D Point Man Looking At Hyper Sphere Using Axis Sharing

 

 

Hyper Factoring Or Zero Factoring An Axis

The Axis Sharing process superimposes a Hyper Axis onto a real Point World Axis. In the case of a 2D Point World the y-Axis is superimposed onto the x-Axis. To help distinguish y-Axis Points from actual x-Axis Points the y-Axis values are multiplied by the Hyper Factor. We say the y-Axis has been Hyper Factored. The Hyper Factor value must be large enough so that the different y-Axis Points do not overlap each other and do not overlap any actual x-Axis Points. The exact value of the Hyper Factor is not important, but a value is needed that is large enough so that the components of the object from Hyper Space do not overlap and also that the components fit on the screen. The following Animation shows a 2D Point World displaying a Sphere using Axis Sharing and varying Hyper Factors.

(Animation7) 2D Point World With A Sphere And Varying Hyper Factors

 

In the case of a 3D Point World the w-Axis is superimposed onto the x-Axis. To help distinguish w-Axis Points from actual x-Axis Points the w-Axis values are multiplied by the Hyper Factor. This Animation shows a 3D Point World displaying a Hyper Sphere using Axis Sharing and varying Hyper Factors.

(Animation8) 3D Point World With A Hyper Sphere And Varying Hyper Factors

 

Any other Point World Axis can be Hyper Factored in order to help visualize particular aspects of the Hyper Space under study. For 2D Animations Axis Sharing for the y-Axis is usually done on the x-Axis and the z-Axis can be Hyper Factored. The Hyper Factor value for the z-Axis is the same as the Hyper Factor value used for the y-Axis using Axis Sharing. This will make it easier to see the relationship of the Points in the yz-Plane. When the z-Axis is Hyper Factored a more circular looking arrangement of the Points results and this is more indicative of the actual relationship in Hyper Space. The circular arrangement is in the yz-Plane where it is understood that the y values are superimposed onto the x-Axis. The 2D Observer probably already suspects that the arrangement of Points for a Sphere in 3D Hyper Space is more complicated than a Circle but is still somehow circular in nature. The following Animation shows how Hyper Factoring the z-Axis will let the 2D Observer begin to get a feel for the shape of a Sphere.

(Animation9) 2D Point World With Hyper Factored Axis

 

For 3D Animations Axis Sharing for the w-Axis is usually done on the x-Axis and the y-Axis or the z-Axis can be Hyper Factored. The Hyper Factor value for the y-Axis or the z-Axis is the same as the Hyper Factor value used for the w-Axis using Axis Sharing. This will make it easier to see the relationship of the Points in the yw-Plane or zw-Plane. The 3D Observer is not physically able to see a Hyper Sphere as it actually is so he must find other compromise methods of visualization. When the y-Axis is Hyper Factored a more circular looking arrangement of the Points results. The circular arrangement is in the yw-Plane where it is understood that the w values are superimposed onto the x-Axis. The 3D Observer already suspects that the arrangement of Points for a Hyper Sphere in 4D Hyper Space is more complicated than a Sphere is still somehow circular in nature. The following Animation shows how Hyper Factoring the y-Axis will let the 3D Observer begin to get a feel for the shape of a Hyper Sphere. Note that the z-Axis could also be Hyper Factored which will give the arrangement a Sphere like look but this generally complicates things and does not add any more understanding to the situation.

(Animation10) 3D Point World With Hyper Factored Axis

 

For a 2D Point World the y-Axis is shared with the x-Axis so some ambiguity can exist when trying to determine if the location of a Point on the x-Axis is due to an x component or a Hyper Factored y component. In order to get a better feel for what is going on with Axis Sharing the x-Axis can be Zero Factored. This just means that the x value of all Points is set to zero. The result of Zero Factoring the x-Axis is that Point arrangements that have x-Axis components collapse to the Hyper Factored y-Axis values. The following Animation shows that when a Sphere is shown using Axis Sharing and the x-Axis is Zero Factored that the resulting arrangement collapses to lines of Points parallel to the z-Axis. This shows that each Circle contains Points that all have the same y-Axis component. Points that originally had a zero y component would collapse to the origin leaving only Hyper Factored y components on the x-Axis.

(Animation11) 2D Point World With Zero Factored Axes

 

For a 3D Point World the w-Axis is shared with the x-Axis so some ambiguity can exist when trying to determine if the location of a Point on the x-Axis is due to an x component or a Hyper Factored w component. The result of Zero Factoring the x-Axis is that Point arrangements that have x-Axis components collapse to Hyper Factored w-Axis values. The following Animation shows that when a Hyper Sphere is shown using Axis Sharing and the x-Axis is Zero Factored that the resulting arrangement collapses to disks of Points parallel to the yz-Plane. This shows that each Sphere contains Points that all have the same w-Axis component. Points that originally had a zero w component would collapse to the origin leaving only Hyper Factored w components on the x-Axis.

(Animation12) 3D Point World With Zero Factored Axes

 

 

Constructing Fundamental Objects In Point World

The Fundamental Objects in Point World are the Line, the Circle, and the Sphere. Each of these Objects can be Empty or Full.

Note that the equations in this section are for people who need to see the details. It is not necessary to understand the equations to understand the different configurations that the Animations portray. The important thing to take away from this section is that Spheres and Circles are sometimes constructed in an odd looking way that makes the Animations for rotations easier to understand.

The size of the Balls that represent Points is variable, and the size of the Balls determine the number of Points that are needed to represent an Object. For example the following equations can be used to construct a Full Line:

NumberOfBalls = (LineLength + BallSize) / BallSize
if(NumberOfBalls is Even) NumberOfBalls = NumberOfBalls - 1
BallOffset = LineLength / (NumberOfBalls - 1)
BallLocation = n * BallOffset
0 <= n <= NumberOfBalls

So if we are constructing a 4 Unit long Full Line and the Ball size is 0.5 Unit then there will be a Ball located at each endpoint and 7 Balls offset from each other by 0.5 Units for a total of 9 Balls ( 9 Points) that represent the Full Line. The number of MPs will be decremented to an odd number if the Ball Size leads to an even number of Points. In this way there is always a Point at the center of a Full Line. The location of this central Point is considered to be the location of a Full Line or an Empty Line even though an Empty Line does not have an actual Point at the center. In Point World a Line can have a Radius that is defined to be the distance from the center to one of the endpoints, which is LineRadius = LineLength / 2.

In Point World Empty Circles can be constructed from Empty Lines, and Full Circles can be constructed from Full Lines. Lines can be used to construct a Circle with Equal Offsets or with Equal Angles as shown in the following Animations for Empty Circle construction and for Full Circle Construction:

(Animation13) Constructing Empty Circles Using Empty Lines

(Animation14) Constructing Full Circles Using Full Lines

 

Note that these construction techniques produce Circles with a rotational symetry around the z-Axis. The construction can also be implemented for x-Axis or y-Axis symetry. This orientation characteristic will prove to be helpful when viewing the Circle rotation Animations later.

Equations For Equal Offset (Full or Empty) Circle Construction:
NumberOfLines = (2 * CircleRadius + BallSize) / BallSize
if(NumberOfLines is Even) NumberOfLines = NumberOfLines - 1
Span = NumberOfLines / 2
DeltaOffset = CircleRadius / (NumberOfLines - 1)
LineOffset = n * DeltaOffset
Angle = asin(LineOffset / CircleRadius)
LineRadius = CircleRadius * cos(Angle)
-Span <= n <= Span

Equations For Equal Angle (Full or Empty) Circle Construction:
BallAngleSize = 2.0 * asin((BallSize / 2.0) / CircleRadius)
NumberOfLines = (Pi + BallAngleSize) / BallAngleSize
if(NumberOfLines is Even) NumberOfLines = NumberOfLines - 1
Span = NumberOfLines / 2
DeltaAngle = Pi / (NumberOfLines - 1)
Angle = n * DeltaAngle
LineOffset = CircleRadius * sin(Angle)
LineRadius = CircleRadius * cos(Angle)
-Span <= n <= Span

These equations will produce Points that have CircleRadius as follows:
CircleRadius = sqrt(LineOffset^2 + LineRadius^2)

Full Circles can also be constructed from Concentric Empty Circles with Equal Offsets or with Equal Angles as shown in the following Animation:

(Animation15) Constructing Full Circles Using Concentric Circles

 

In a 2D Point World Circles can be constructed directly in 3D Hyper Space as shown in the following Animation. The 3D Hyper Space constructions place the Circle component Lines at Offsets along the y-Axis. The component Lines will be perpendicular to the y-Axis. The component Lines can be oriented with the x-Axis or the z-Axis so there are two possible orientations. For the farther construction, where the component Lines are all oriented with the z-Axis, it is intuitely easy to see how the Lines could be brought closer together to form the Circle. The closer construction seems intuitively impossible for forming a Circle, because bringing the Lines closer together overlaps them. This is all an illusion of the Axis Sharing process. If the y-Axis was shared with the z-Axis instead of the x-Axis then the construction in the front would line up on the z-Axis and seem easier to form a Circle and the construction in the back would now not be as intuitive. So we should always be careful how we interpret what we see with Axis Sharing. But the 2D Observer can still realize some interesting things about a Circle in 3D with these constructions. For example the construction in the back depicts the Circle as somehow being Flat for the 2D Observer. Also, the fact that there can be different orientations where the Circle actually looks different seems puzzling, because in a 2D Point World a Circle can only rotate with an xz-Rotation. A Hyper Space rotation is needed to change it's orientation. These things are puzzling to the 2D Observer. A 3D Observer is not puzzled by these things and can easily see the Flatness of a Circle and how it can look different with different orientations. But a 3D Observer is puzzled by the Flatness of a Sphere in 4D Hyper Space as we shall see.

(Animation16) Constructing Full Circles In 3D Hyper Space Using Lines

Equations For Full Circle Concentric Construction:
NumberOfConcentricCircles = (CircleRadius + BallSize) / BallSize;
if ((NumberOfConcentricCircles % 2) == 0) NumberOfConcentricCircles = NumberOfConcentricCircles - 1
DeltaRadius = CircleRadius / (NumberOfConcentricCircles - 1)
ConcentricCircleRadius = n * DeltaRadius
0 <= n <= NumberOfConcentricCircles

In Point World Empty Spheres can be constructed from Empty Circles, and Full Spheres can be constructed from Full Circles. Full Spheres are constructed from Full Circles that are constructed from Concentric Circles. Circles can be used to construct a Sphere with Equal Offsets or with Equal Angles as shown in the following Animations for Empty Sphere construction and for Full Sphere construction:

(Animation17) Constructing Empty Spheres Using Empty Circles

(Animation18) Constructing Full Spheres Using Full Circles

 

Note that these construction techniques produce Spheres that have a rotational symetry around the y-Axis. The construction can also be implemented for x-Axis or y-Axis symetry. This rotational symetry characteristic will prove to be helpful when viewing the Sphere rotation Animations later.

 

Full Spheres can also be constructed from Concentric Full Circles with Equal Offsets or with Equal Angles as shown in the following Animation.

(Animation19) Constructing Full Spheres Using Concentric Circles

 

In a 3D Point World Spheres can be constructed directly in 4D Hyper Space as shown in the following Animation. For 4D constructions the Sphere component Circles are Offset along the w-Axis. The component Circles will be perpendicular to the w-Axis. The component Circles can be oriented with the xy-Plane, the xz-Plane, or the yz-Plane so there are three possible orientations. A component Circle can have two different orientations in each of the Planes because the Lines that make up the Circle can be drawn parallel to either of the two Axes that make up the Plane. Even though there are a total of six possible orientations only one orientation per Plane will be shown to reduce clutter. In the center construction, where the component Circles are all in the yz-Plane, it is intuitely easy to see how they could be brought closer together to form the Sphere. The other two constructions seem intuitively impossible for forming a Sphere, because bringing them closer together overlaps them. This is all an illusion of the Axis Sharing process. If the w-Axis was shared with the y-Axis instead of the x-Axis then the construction in the back would stack up on the y-Axis and seem easier to form a Sphere and the center construction would now not be as intuitive. So we should always be careful how we interpret what we see with Axis Sharing. But we can still realize some interesting things about a Sphere in 4D with these constructions. The front and back constructions depict the Sphere as somehow being Flat. In 4D Space a Sphere is Flat just like a Circle is Flat in 3D Space. So a Sphere can be seen to be Flat in some orientations and Spherical in other orientations. These things puzzle a 3D Observer these things will be explored more in later Animations.

(Animation20) Constructing Spheres In 4D Hyper Space Using Circles

 

Equations For Equal Offset (Full, Empty, Concentric, or Hyper) Sphere Construction:
NumberOfCircles = (2 * SphereRadius + BallSize) / BallSize
if(NumberOfCircles is Even) NumberOfCircles = NumberOfCircles - 1
Span = NumberOfCircles / 2
DeltaOffset = SphereRadius / (NumberOfCircles - 1)
CircleOffset = n * DeltaOffset
Angle = asin(CircleOffset / SphereRadius)
CircleRadius = SphereRadius * cos(Angle)
-Span <= n <= Span

Equations For Equal Angle (Full, or Empty, Concentric, or Hyper) Sphere Construction:
BallAngleSize = 2.0 * asin((BallSize / 2.0) / SphereRadius)
NumberOfCircles = (Pi + BallAngleSize) / BallAngleSize
if(NumberOfCircles is Even) NumberOfCircles = NumberOfCircles - 1
Span = NumberOfCircles / 2
DeltaAngle = Pi / (NumberOfCircles - 1)
Angle = n * DeltaAngle
CircleOffset = SphereRadius * sin(Angle)
CircleRadius = SphereRadius * cos(Angle)
-Span <= n <= Span

These equations will produce Points that have SphereRadius as follows:
SphereRadius = sqrt(CircleOffset^2 + CircleRadius^2)

 

In Point World Empty Hyper Spheres can be constructed from Empty Spheres, and Full Hyper Spheres can be constructed from Full Spheres. Spheres can be used to construct a Hyper Sphere with Equal Offsets or with Equal Angles. A Hyper Sphere cannot be depicted directly. Hyper Spheres can only be indirectly depicted using Axis Sharing. The following Animation shows how Hyper Spheres might look using Axis Sharing:

(Animation21) Constructing Hyper Spheres Using Spheres

 

Equations For Equal Offset (Full or Empty) Hyper Sphere Construction:
NumberOfSpheres = (2 * HyperSphereRadius + BallSize) / BallSize
if(NumberOfSpheres is Even) NumberOfSpheres = NumberOfSpheres - 1
Span = NumberOfSpheres / 2
DeltaOffset = HyperSphereRadius / (NumberOfSpheres - 1)
SphereOffset = n * DeltaOffset
Angle = asin(SphereOffset / HyperSphereRadius)
SphereRadius = HyperSphereRadius * cos(Angle)
-Span <= n <= Span

Equations For Equal Angle (Full or Empty) HyperSphere Construction:
BallAngleSize = 2.0 * asin((BallSize / 2.0) / HyperSphereRadius)
NumberOfSpheres = (Pi + BallAngleSize) / BallAngleSize
if(NumberOfSpheres is Even) NumberOfSpheres = NumberOfSpheres - 1
Span = NumberOfSpheres / 2
DeltaAngle = Pi / (NumberOfSpheres - 1)
Angle = n * DeltaAngle
SphereOffset = HyperSphereRadius * sin(Angle)
SphereRadius = HyperSphereRadius * cos(Angle)
-Span <= n <= Span

These equations will produce Points that have HyperSphereRadius as follows:
HyperSphereRadius = sqrt(SphereOffset^2 + SphereRadius^2)

This can be seen by recognizing that the SphereOffset is on the w-Axis and the SphereRadius is perpendicular to the w-Axis regardless of the particular Point. It is easy in 3D space to see how all the Points of a Circle are at a Radius that can be perpendicular to the y-Axis. In 4D space it is more difficult to see how all the Points on Sphere can be perpendicular to the w-Axis. It must be realized that relative to 4D space our 3D space is flat and the Sphere is also flat. The whole 3D space is perpendicular to the 4D w-Axis so the Points of the Sphere in 4D are all on the same 3D Hyper Plane, which we will discuss later. Therefore the SphereRadius lines are all perpendicular to the w-Axis. The SphereRadius line and the SphereOffset line define a right triangle in 4D space. So it is easy to see that the line from the Point to the origin completes the triangle and is at a distance of HyperSphereRadius.

It is 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 Observer 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 Observer 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 next.

 

 

Basic Hyper Space Rotations In Point World

For a 2D Point World Normal Space is 2D and Hyper Space is 3D, and for a 3D Point World Normal Space is 3D and Hyper Space is 4D. The following Animation shows a 2D Point World with Rotating Circles that have been constructed using Equal Angles, Equal Offsets, and Concentric Circles. This Animation shows that Concentric Circle construction is best for showing Rotation around a central Point (xz-Rotation), but that Equal Offset construction behaves best for showing Rotation around a line (yz-Rotation). The xz-Rotation is the only Rotation that is possible in the 2D Point World. The yz-Rotation is a Hyper Space Rotation for the 2D Point World and Axis Sharing must be used for a 2D Observer. The Concentric constructed Circle spreads out into a visually confusing scatter of Points. The Equal Angle constructed Circle spreads out with easily discernable Lines but the Lines spread out in a nonlinear way that jams them together. The Equal Offset constructed Circle spreads out with easily discernable evenly spaced Lines and, at maximum spread, the centers of the Lines are on the equal spaced red calibration grid lines.

(Animation22) Behavior Of Different Circle Constructions With Rotation

 

The first Animation in each of the following pairs of Animations will show a 2D Point World with an Object (Line, Circle, Sphere, or Hyper Sphere) rotating in 3D Hyper Space using a zy-Rotation. Axis Sharing (y-Axis with x-Axis) will be used to show what the Object is doing because it is physically impossible for the 2D Observer to see the movement of Objects in 3D Hyper Space. The second Animation will show a 3D Point World with the same Object rotating in 4D Hyper Space using a z-Axis to w-Axis rotation. Axis Sharing (w-Axis with x-Axis) will be used to show what the Object is doing because it is physically impossible for the 3D Observer to see the movement of Objects in 4D Hyper Space. The z-Axis will be Hyper Factored in each Animation in order to emphasize the circular nature of the Hyper Space rotation. Each Animation will show a large yellow circle located at the origin in the xz-Plane which will define the Hyper Factor Boundary. This Boundary is an imaginary construct used to emphasize the circular nature of the rotation of Points through Hyper Space. Each Animation also shows a small coordinate system located to the left in the negative x direction that represents a 3D Reference depiction for 3D Observers.

The Software will use an algorithm which makes the Points jump into and jump out of Normal Space which gives them a sticky appearance as they transistion through Normal Space. The Points are actually moving smoothly in the calculation and the stickyness is added to visually emphasize when the Points are going through Normal Space.

The first Animation below shows a Line in 2D Space with a zy-Rotation in 3D Hyper Space, and the second Animation shows the same Line in 3D Space with a zw-Rotation in 4D Hyper Space. The Line is made out of seven Equal Offset Points. The individual Points are color coded for easy observation. Since the Line is Hyper Factored on the z-Axis the points appear streatched out along the z-Axis with the end Points of the Line lying directly on the Hyper Factor Boundary. When the rotation begins the Points (except the center Point) will, with some stickyness, move out of Normal Space and into Hyper Space. The end Points of the Line move around in a cirlce on the Hyper Factor Boundary, and other Points move around in smaller circles. The center Point does not move from the location at the origin and has a rotation that the Observers can not understand or see. The Points have no x-Axis component so the Axis Sharing depiction of apparent x-Axis components are relly all due to y-Axis shared components. With Axis Sharing the 2D Observer is able to see the z-Axis to y-Axis rotation as a circular movement through 3D Hyper Space, and similarly the 3D Observer is able to see the z-Axis to w-Axis rotation as a circular movement through 4D Hyper Space. Note that in both Animations the Line, except for the center Point, would seem to disappear for the Observer in his own Space. The Observers do not know where the Points go when rotating into Hyper Space but is able to see them using Axis Sharing. The 3D Observer however knows where the Points go in the 2D Point World by looking at the 3D Reference. The 3D Observer also understands the rotation of the central Point of the Line in the 2D Point World. The 3D Observer is not so lucky in the 3D Point World with the 4D Hyper Space rotation because the 3D Reference also shows that the Points disappear for that 4D rotation. A 4D Observer would have no problem understaning this kind of rotation.

(Animation23) Line In 2D Space Rotate ZY Share YX Hyper Z

(Animation24) Line In 3D Space Rotate ZW Share WX Hyper Z

 

The first Animation below shows a Circle in 2D Space rotating (z-Axis to y-Axis) in 3D Hyper Space, and the second Animation shows the same Circle in 3D Space rotating (z-Axis to w-Axis) in 4D Hyper Space. The Circle is made out of seven Equal Offset Lines. The individual Lines are color coded for easy observation. There is a direct analogy between the Lines that make up the Circles and the Points that make up the Lines from the previous pair of Animations. However, the Points that make up the Cirlce now also have x-Axis components. The 2D Observer can see that the component Lines that make up the Cirlce rotate around the inside the Hyper Factor Boundary similar to how the Points moved in the previous Animations. The 2D Observer sees that the component Lines maintain a horizontal orientation parallel to the x-Axis throughout the rotation. This is puzzling for the 2D Observer because in 2D Space any rotation would make a Line lose it's orientation with the x-Axis. The 2D Observer also sees that there is a Line that all the other components rotate around so that in some way the Circle is rotatinfg on a Line. The 2D Observer is puzzled again about this because in 2d Space there is only rotation on a Point. This is a new and strange kind of rotation for the 2D Observer. But the 3D Observer can easily see why the Line orientations stay parallel to the x-Axis by looking at the 3D Reference in the 2D Point World. The 3D Observer also completely understands how the Circle can rotate on a Line.

(Animation25) Circle In 2D Space Rotate ZY Share YX Hyper Z

(Animation26) Circle In 3D Space Rotate ZW Share WX Hyper Z

 

A Sphere cannot exist in 2D Point World because it contains y-Axis components. So a Sphere in a 2D Point World must always be in 3D Hyper Space and be shown using Axis Sharing. The first Animation below shows a Sphere in 3D Hyper Space rotating (z-Axis to y-Axis), and the second Animation shows the same Sphere in 4D Hyper Space rotating (z-Axis to w-Axis). Even though a Sphere can be depicted in 3D Space it is more convenient to draw it in 4D Hyper Space and use Axis Sharing to see it. Since the Sphere has to be in 3D Hyper Space in the 2D Point World it makes sense to draw the Sphere in 4D Hyper Space for the 3D Point World just for the symetry of the situation. The Sphere is made out of seven Equal Offset Circles and the Circles are made out of seven Equal Offset Lines. The individual Circles are color coded for easy observation. There is a direct analogy between the Circles that make up the Spheres and the Lines that make up the Circles and the Points that make up the Lines from the previous two pairs of Animations. The Points that make up the Sphere now also have y-Axis components. In the 2D Point World the seven Circle components of the Sphere are spread out from left to right on the x-Axis and are Hyper Factored on the z-Axis to produce an array of Lines. The 2D Observer can see that the component Lines that make up the Sphere rotate around the inside the Hyper Factor Boundary similar to how the Points moved in the previous Animations. The 2D Observer sees that the component Lines maintain a horizontal orientation parallel to the x-Axis throughout the rotation which is again puzzling. The 2D Observer also sees that there is a Line that all the other components rotate around so that in some way the Sphere is rotatinfg on a Line. This is a again a strange kind of rotation for the 2D Observer. But the 3D Observer can easily see why the Line orientations stay parallel to the x-Axis by looking at the 3D Reference in the 2D Point World. The 3D Observer also completely understands how the Circle can rotate on a Line. In the 3D Point World the 3D Observer can see that the Sphere is streatched out and spread accross the x-Axis in the same way as in the 2D Point World but now the component Lines are in 4D Hyper Space instead of 3D Hyper Space. The Lines orbit around the central stationary Line.

(Animation27) Sphere In 3D Hyper Space Rotate ZY Share YX Hyper Z

(Animation28) Sphere In 4D Hyper Space Rotate ZW Share WX Hyper Z

 

The following Animation shows the Sphere drawn in the Normal Space of a 3D Point World. The same z-Axis to w-Axis rotation produces a different effect with different implications. The Sphere is spread out in the z-Axis direction into component Circles. As soon as the rotation begins the Circle components, except for the component at the origin, disappear from Normal Space and enter 4D Hyper Space. The Circles orbit a central Circle that does not move. The 3D Observer sees that the Sphere is rotating on a Circle (more generally a Plane) instead of a Line. This is puzzling to the 3D Observer and is something that will be explored more in later Animations. The Circles are seen to maintain an orientation parallel to the xy-Plane as they orbit around the center. In a 3D Point World the only way a Circle can rotate and stay in the same Plane is if the Circle is rotating around it's center Point. These Circles are not rotating around their centers so this is a puzzling and new kind of rotation for the 3D Observer.

(Animation29) Sphere In 3D Space Rotate ZW Share WX Hyper Z

The following Animation shows a Hyper Sphere in 3D Space rotating (z-Axis to w-Axis) in 4D Hyper Space. There is no usefull 2D equivalent for this Animation. There is a direct analogy between the Spheres that make up the Hyper Sphere and the Circles that make up the Spheres and the Lines that make up the Circles and the Points that make up the Lines from the previous two pairs of Animations. Note that the basic Hyper Sphere cannot exist in 3D Space so the 3D Point World depiction must always show a Hyper Sphere using Axis Sharing.

(Animation30) Hyper Sphere In 3D Space Rotate ZW Share WX Hyper Z

 

 

Mathematical Points And Dimensional Points (Old Version)

When the term Point is used in Point World it is assumed that the reference is to a Mathematical Point. A Point has identically zero diameter and has no dimensional properties. If you let dx be a differential distance between the Points on a line then you can make dx as small as you like and the Points will still never touch. Only when dx is identically zero do the Points touch each other, but they also all collapse onto a single Point. This means you can not arrange Points next to each other in a line configuration where they touch each other. Two Points can only touch each other when they are on top of each other as a single Point.

Another problem with Points is that an n-dimensional Observer will intuitively think of a Point as something more like a tiny Sphere object in that dimension. A 2D Observer thinks the tiny Sphere object is a Circle, and a 3D Observer thinks it's a Sphere, and a 4D Observer thinks it's a Hyper Sphere. We will call this tiny Sphere object a Dimensional Point or nD Point. An nD Point is just the set of all Points in nD Space that are equidistant from the origin using a tiny radius or even a differential Radius (dR), where dR-->0. A 2D Point is an Empty Circle of Points, a 3D Point is an Empty Sphere of Points, and a 4D Point is an Empty Hyper Sphere of Points. The surface of an nD Point is conceptually continuous but in Point World it is constructed from a discrete number of Points. An nD Point can be thought of as the smallest thing that can exist while retaining the n-dimensional characteristic of the space. The definition of an nD Point can also be used to define the 1D Point which is an Empty Line and the 0D Point which is the same as a Point. A Full nD Point can be defined as an nD Point including all internal Points, so a Full 1D Point is a Full Line, a Full 2D Point is a Full Circle, a Full 3D Point is a Full Sphere, and A Full 4D Point is a Full Hyper Sphere.

If a 2D Observer living in the xz-Plane tries to find the y-Axis that goes into 3D Hyper Space he will only be able to travel around the 2D Point at the origin in the xz-Plane. Even if he might be able to conceptually understand 3D Hyper Space he will not be able to physically go there or even look in the 3D Hyper Space direction. The 2D Observer would need to understand the 3D Point before he is able to understand 3D Hyper Space. Similarly, if a 3D Observer (Us) living in xyz-Space tries to find the w-Axis that goes into 4D Hyper Space he will only be able to travel around the 3D Point at the origin in xyz-Space. We might conceptually understand 4D Hyper Space but we can not physically go there or look in the 4D Hyper Space direction even if it did exist. The 3D Observer would need to understand the 4D Point before he is able to understand 4D Hyper Space. The concept to take away from this is that the 2D Point is a different kind of object than the 3D Point and the 3D Point is a different kind of object than the 4D Point and all these objects are different from an actual Point.

The real utility of nD Points is that two or more nD Points can touch. Since the surfaces of nD Points are made out of Points, we can define two nD Points as touching when a Point from one is at the same location in space as a Point from the other one. Multiple nD Points can be arranged like Points to form Lines and Planes, but unlike Points the nD Points can touch. Also, an nD Point has surface structure that can visually be seen to rotate, whereas a Point cannot visually rotate.

From the Object construction techniques in a previous section it is interesting to note that an nD Point can be constructed from smaller and smaller (n-1)D Points. So a 2D Point can be constructed from smaller and smaller 1D Points, a 3D Point can be constructed from smaller and smaller 2D Points, and a 4D Point can be constructed from smaller and smaller 3D Points. Also, since 1D Points are constructed from 0D Points, an nD Point is ultimately constructed from 0D Points wich are just Points.

The construction of an nD Point from smaller (n-1)D Points might seem to violate the definition of an nD Point as being the smallest thing in nD Space. How can there be smaller and larger nD Points? It's because the definition is imposed on the nD Point and not the (n-1)D Points. The (n-1)D Points are just artifacts of the nD Point. Each Point of any (n-1)D Point is still dR from the center of the nD Point regardless of how small the radius of the (n-1)D Point might be.

In the following Animation two 2D Points will move toward each other and touch at the origin. The Balls representing the Points are varied in size to emphasize that the Balls are just markers for Points and that they can be more tightly packed or more loosely packed depending on the purpose of the Animation.

(Animation30) Two Touching 2D Points

 

 

Mathematical Points And Dimensional Points (New Version)

When the term Point is used in Point World it is assumed that the reference is to a Mathematical Point. So a Point has identically zero diameter and has no dimensional properties. Also, if you let dx be the differential distance between the Points on a line then you can make dx as small as you like and the Points will still never touch. Only when dx is identically zero do the Points touch each other, but they also all collapse onto a single Point. This means you can not arrange Points next to each other in a line configuration where they touch each other.

An n-dimensional Observer will naturally think of a Point as something that has n-dimensinal characteristics. A 2D Observer thinks of a Point as a tiny Circle, a 3D Observer thinks it's a tiny Sphere, and a 4D Observer thinks it's a tiny Hyper Sphere. We call these tiny objects Dimensional Points and if the dimension is specified we use the term nD Points. An nD Point is just the set of all Points in nD Space that are equidistant from a central Point with a differential Radius (dR), where dR-->0. A 2D Point is an Empty Circle of Points, a 3D Point is an Empty Sphere of Points, and a 4D Point is an Empty Hyper Sphere of Points. A Full nD Point can be defined as an nD Point including all internal Points, so a Full 2D Point is a Full Circle, a Full 3D Point is a Full Sphere, and A Full 4D Point is a Full Hyper Sphere. An nD Point can be thought of as the smallest thing that can exist while retaining the n-dimensional characteristic of the space. The definition of an nD Point can also be used to define a 1D Point which is an Empty or Full Line and the 0D Point which is the same as a Point.

The real utility of nD Points is that two or more nD Points can touch. Since the surfaces of nD Points are made out of Points, we can define two nD Points as touching when a Point from one is at the same location in space as a Point from the other one. Multiple nD Points can be arranged like Points to form Lines and Planes, but unlike Points the nD Points can touch. Also, an nD Point has surface structure that can visually be seen to rotate, whereas a Point cannot visually rotate.

In the following Animation two 2D Points will move toward each other and touch at the origin. The Balls representing the Points are varied in size to emphasize that the Balls are just markers for Points and that they can be more tightly packed or more loosely packed depending on the purpose of the Animation.

(Animation51) Two Touching 2D Points

 

 

Constructing Dimensional Point Objects

An Object that is constructed from Dimensional Points will be called a Dimensional Point Object or an nD Object. Point World will work with the following Dimensional Point Objects:

Since an nD Point is a representation of the smallest thing in nD Space that retains the dimensionality characteristics of the nD Space it would not be physically compatible, for example, to consider a 2D Point in 3D Space. A 2D Point only has extension and existance in two dimensions and would make no sense in 3D Space. You could not construct 3D Objects in 3D Space with 2D Points. A 3D Observer would naturally think that a Point has some dR radius all the way around in any angular direction, but the 2D Point has dR radius only within the plane of the 2D Point Circle and has identically zero radius in all other directions. A 2D Point is Flat in 3D Space. The 2D Point would need to be replaced with a 3D Point. The dimensionality of the nD Point must be the same as the dimension of the Space. We cannot just take a 2D Point out of 2D Space and see how it behaves in 3D Space. A 3D Point is needed. Similarly it would make no sense to consider a 2D Line in 3D Space. A 2D Line would have only a partial dimensional existance in 3D Space.

 

 

The following Animation first shows a vertical 2D Line constructed from 7 touching 2D Points. Then the 2D Line becomes a 3D Line and then the 3D Line becomes a 4D Line using Axis Sharing.

(Animation31) 2D 3D And 4D Dimensional Lines