# 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 3D Perspective View on the monitor screen. The 3D Perspective View shows closer objects as larger than more distant objects which makes it easier to visualize what is going on in the Animations. The Animations are driven by a Translational and Rotational Math Engine (TRME) that is constantly updating positions and orientations of the displayed objects. First an array of Points representing the objects that are being studied is generated and displayed in the 3D Perspective View. 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 3D Perspective View. The processing and display operation is repeated for a series of incremental angles and incremental distances to animate the objects under study. Note that the TRME can calculate full 4D Translations and Rotations which are displayed in the Animations using a method called Axis Sharing as will be fully described.

We will use the classic analogy approach where an Animation for a 2D Observer studying 3D Space will be shown and then an analogous Animation for a 3D Observer studying 4D Space will be shown. The Axis Sharing method combined with the special way objects are constructed effectively produces slices or cross sections into the Hyper Spaces under study. Using analogies and slices is not new but the Animations present these methods in a new and dynamic way. Also, the concept of Dimensional Points allows Observers to graphically see the Hyper Space connections of objects, that must be understood when thinking about higher dimensions. And as a final note, this study is not about the Time dimension from Relativity, where they like to say that Time is the fourth dimension.

The Animations are Animated GIF files. The GIF format is a good cross-platform choice but may load slow on some systems. If you have a good modern high speed Internet connection then the load time should only be a few seconds.

Thank You
4/24/15

Last Update 4/25/15

CONTENTS:

Introduction To Point World

Using Axis Sharing To Represent Points in Hyper Space

Hyper Factoring And Zero Factoring An Axis

Constructing Fundamental Objects In Point World

2D And 3D Hyper Space Rotation Analogies

Sphere In 2D And Hyper Sphere In 3D Rotation Analogies

Mathematical Points And Dimensional Points

Rotations On A Line And On A Plane Of Dimensional Points

Flatness of nD Space relative to (n+1)D Space

## 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 3D Perspective View 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 the Observer. This is a single Point that is used as the Observer in Point World. The Observer is a Point Observer that can scan across a field of view to look at objects in Point World. The Observer can only see one Point at a time but can remember what has been seen to construct an image for himself. In the animations the Observer scans the scene from his perspective and then sets the Points that he sees to the color Red. The Observer 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 3D Perspective View. The Eye Location can be moved to any location in the 3D Perspective View. 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 the Observer 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 shows how a Sphere Object might look in Point World. It was constructed using 203 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 Absolutely 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. Of course a 2D Observer does not know there is no Height. Points in a 2D Point World would ideally be drawn as Disks from the Point of View of the 2D Observer. The 2D Observer 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 Balls 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 the Observer travels around a Circle and stops to look from various locations. The Points and the Observer 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 Observer 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.

Similarly, our 3D Space is absolutely Flat when viewed by a 4D Observer. The 4D Observer would probably opt to use 4D Hyper Balls (Hyper Spheres) to depict Points in Animations instead of Balls. We will devote some time in later sections exploring how our 3D Space could ever be viewed as Flat.

## Using Axis Sharing To Represent Points in Hyper Space

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. Throughout this analysis analogies will be made between how a 2D Observer might be able to understand 3D Space and how a 3D Observer (you and me) might be able to understand 4D Space. For a 2D Observer 3D Space is a puzzling thing and for a 3D Observer 4D Space is likewise a puzzling thing. However, a 3D Observer can easily understand 3D Space since this is his Normal Space. The thinking is that if we can get some insight into why 3D Space is so difficult for the 2D Observer to understand then we might apply this insight to the problem of understanding 4D Space.

Since 2D Space is Flat a 3D Observer can easily see how to construct a 3D Hyper Space for the 2D Observer. If the y-Axis is the 3D Hyper Space axis then we could start the creation of a 3D Space by constructing another 2D Space one unit up in the positive direction on the y-Axis. This new Space must be perpendicular to the y-Axis and parallel to the original 2D Space. Now there are two 2D Spaces making up the whole Space. This second 2D Space is a mystery for a 2D Observer because he physically can not go there or even look in that direction. Now lets fill in the space between the two 2D Spaces by constructing an infinite number of these 2D parallel Spaces. We now have a chunk of 3D Space. The chunk of 3D Space that was created can conceptually be extended in the positive direction out to infinity by constructing more 2D Spaces. Next we can construct the other side of the 3D Space by proceeding on the negative y-Axis. There is now a separate and unique 2D Space located at each Point on the y-Axis. Even though 3D Space is not actually made from layered 2D Spaces, it makes the Animations easier to understand if we look at it this way.

The 2D Observer can conceptualize that other 2D Spaces exist outside of his 2D Space but he can never really understand how to physically move into these Spaces. The 2D Observer is not quite able to understand the construction that we have just done because he is unable to understand the Flatness of his own 2D Space. From the 2D Observer's point of view the universe is an infinite expanse of Width and Depth. Although he can comprehend the Width of a thing as being Flat he is unable to comprehend how the Depth that he experiences could ever be flattened out with the Width of a thing. He is always trying to understand the Flatness of 2D Space by looking from within 2D Space, so he can never succeed. The 2D Observer must leave 2D Space and look back at it, but this is physically impossible.

For this analysis the 2D Observer will conceptually move up or down the y-Axis into other parallel 2D Spaces. The 2D Observer will only be able to exist in and observe things in one 2D Space at a time. He cannot physically look out from one 2D Space into another one. Since the 2D Observer does not know where any of the other 2D Spaces are located he can not move to them in the same way he would move within any 2D Space. He senses that these other Spaces are out there somewhere because of the mathematics, and to move to another 2D Space he conceptually has to disappear from his 2D Space and reappear in another 2D Space. He can move around and observe things in the new 2D Space as easily as in his original 2D Space.

Now we can propose a procedure for constructing a 4D Hyper Space from Flat parallel 3D Spaces. It is completely analogous to the 2D case. If the w-Axis is the 4D Hyper Space axis then a 3D Observer could conceive of 4D Space as being constructed from an infinite number of layered Flat 3D Spaces in the positive and negative directions on the w-Axis. There is a separate and unique 3D Space located at each Point on the w-Axis.

Similar to the 2D Observer the 3D Observer can conceptualize that other 3D Spaces exist outside of 3D Space but he can never really understand how to physically move into these Spaces. He is not quite able to understand the construction for 4D Space because he is unable to understand the Flatness of 3D Space. From his point of view the universe is an infinite expanse of Width, Height, and Depth. Although he can comprehend the Width and Height of a thing as being Flat he is unable to comprehend how the Depth that he experiences could ever be flattened out with the Width and Height of a thing. The 3D Observer is always trying to understand the Flatness of 3D Space by looking from within 3D Space. A 3D Observer must leave 3D Space and look back at it, but this is physically impossible. A way to understand the Flatness and parallelness of 3D Spaces will be explored in later sections.

Similar to the how the 2D Observer conceived of moving into other 2D Spaces, the 3D Observer will conceptually move up or down the w-Axis into other parallel 3D Spaces. The 3D Observer will only be able to exist in and observe things in one 3D Space at a time. He cannot physically look out from one 3D Space into another one. Since the 3D Observer does not know where any of the other 3D Spaces are located he can not move to them in the same way he would move within any 3D Space. He senses that these other Spaces are out there somewhere because of the mathematics, and to move to another 3D Space he conceptually has to disappear from his 3D Space and reappear in another 3D Space. He can move around and observe things in the new 3D Space as easily as in his original 3D Space.

There is a dilemma for a 2D Observer or a 3D Observer 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 Points that might be located in the next higher dimension. The Following Animation shows the dilemma for a 2D Point Man.

If the 2D Observer could move up into 3D Hyper Space he would be able to look at higher and higher parts of the Sphere 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 the 2D Observer moving up into positive different 2D Planes of 3D Hyper Space and then down into negative 2D Planes of 3D Hyper Space while stopping to look from each 2D Plane.

But a 2D Observer is not able to physically go into 3D Hyper Space he can only conceptually go there. He needs a way to view Points that exist in other 2D Planes. Point World uses Axis Sharing to let the 2D Observer see Points from 3D 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 Point in Hyper Space 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. Because of the Hyper Factor the y-Axis is stretched out on the x-Axis. For example, if the Hyper Factor was 5 then you would have to go 5 Units on the x-Axis for every Unit on the y-Axis, and a Point located +1 Unit on the y-Axis would be located at +5 Units on the x-Axis. This Point located at +1 Unit on the y-Axis is also in a 2D Plane located at +1 Unit on the y-Axis. If we place two more Points in that 2D Plane at -1 Unit and at +1 Unit in the x-Axis direction then these Points will be shown at 4 Units and 6 Units respectively on the x-Axis. Now lets place another Point at +2 Units on the y-Axis and then place two more Points at -1 Unit and +1 Unit in the x-Axis direction which would also be on the 2D Plane located at +2 Units on the y-Axis. These Points would show up on the shared x-Axis at 9 Units, 10 Units, and 11 Units. So even though y-Axis distances are stretched out the x-Axis distances are not. The Hyper factor of 5 separates the two groups of three Points on the x-Axis so they don't overlap.

The following Animation shows Points placed as described above plus another set of Points placed at -4 on the y-Axis. Each Point is depicted twice, once on the y-Axis itself, and then on the shared Axis. Each Point is also color coded to make it easy to see how it looks on the actual y-Axis and on the shared Axis. Then each group of Points is Rotated at a different rate and the point group at y = 2 is moved out to y = 3 and then back to y= 2. The behavior the Points on the actual y-Axis should be compared to corresponding Points on the shared Axis. Remember that the 2D Observer can only look at the Axis Shared depiction of the Points and the Points on the y-Axis are there for our reference.

The following Animation shows the same groups of Points as in the previous Animation but does not Rotate them. Instead a 2D Observer moves to each group of Points and scans left and right parallel to the x-Axis. Even though it appears that the 2D Observer is scanning directly on the x-Axis he is only scanning parallel to the x-Axis because he is offset onto the y-Axis. In this way the 2D Observer is able to see the Points from 3D Hyper Space. They are arranged in a sideways kind of way compared to how we know they are arranged on the actual y-Axis. The 2D Observer is not fooled by this sideways arrangement and knows that each group of Points is at a different location on the y-Axis.

The following Animation shows the Sphere from Animation3 using Axis Sharing. A 2D Observer runs an algorithm that draws the various y-Axis components of the Sphere onto the x-Axis to implement Axis Sharing. The 2D Observer can now see the Points from other 2D Planes since they are all drawn in the xz-Plane. The 2D Observer realizes that when he moves left and right he is effectively moving into negative and positive 3D Space. Even though he still does not know where 3D Space is, the 2D Observer can start to understand some things about a Sphere, such as that as he goes further and further into 3D 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 Circles located further and further into 3D Space along the y-Axis. He also realizes that he would be able to move all around any particular Circle and look at all of it's Points from the corresponding 2D Space.

Note that the 2D Observer can not see inside a Circle like we can from our 3D perspective. A 2D Observer sees everything on a 1D Edge using his 1-dimensional vision capabilities. Note that Point World uses a color scheme to indicate if the Observer is in Normal Space or Hyper Space. The Observer is colored Blue when he is located in Normal Space and the Observer is colored Green when he is located in Hyper Space. The Scan Lines are colored Red in Normal Space and Green in Hyper Space. When a Point is scanned it turns Red in Normal Space or Hyper Space.

Next next Animation will depict an analogous situation for a 3D Observer looking at Points from 4D Hyper Space. The Points are placed as in the above Animations with two additional Points placed in each group of Points on the w-Axis. Each Point is depicted only once on the Axis Shared area because there is no way to depict Points in 4D as a reference. Each group of Points is Rotated at a different rate and the point group at w = 2 is moved out to w = 3 and then back to w= 2. Note that groups of Points are rotated in a different angular directions this time to show that in 3D Space there are three possible angular directions.

The following Animation shows the same groups of Points as in the previous Animation but does not Rotate them. Instead a 3D Observer moves to each group of Points and scans left, right, up, and down parallel to the xy-Plane. In this way the 3D Observer is able to see the Points from 4D Hyper Space. They are arranged in a sideways kind of way but we know from the 2D analogous case that they are probably not sideways on the w-Axis. The 3D Observer is not fooled by this sideways arrangement and knows that each group of Points is at a different location in on the w-Axis.

The following Animation shows a Hyper Sphere using Axis Sharing. A 3D Observer runs an algorithm that draws the various w-Axis components of the Hyper Sphere onto the x-Axis to implement Axis Sharing. The 3D Observer sees the World like we do and we can see the Points from other 3D Spaces since they are drawn into xyz-space. The 3D Observer realizes that when he moves left and right he is effectively moving into negative and positive 4D Space. It must be emphasized that the sense of left and right is an artifact of the Axis Sharing Process and it must be remembered that the direction is really positive and negative 4D Space. Even though the 3D Observer still does not know where 4D Space is, he can start to understand some things about a Hyper Sphere, such as that as he goes further and further into 4D Space he sees smaller and smaller Spheres until there is only a single Point. So he realizes that a Hyper Sphere is made out of Spheres located further and further into 4D Space. He also realizes that he would be able to move all around any particular Sphere all 3D angular directions, and look at all of it's Points from the corresponding 3D Space. Note that we can not see inside a Sphere like a 4D Observer could. We see everything on a 2D Plane using our 2-dimensional vision capabilities.

## Hyper Factoring And 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 groups of Points from different 2D Spaces at different locations on the y-Axis do not overlap each other. 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.

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.

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.

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.

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.

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.

## 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.

• Empty Line: Points located at the two ends of the Line.
• Empty Circle: Points located at discrete locations on the circumference of the Circle.
• Empty Sphere: Points located at discrete locations on the surface of the Sphere.
• Full Line: Points located at discrete locations over whole length of the Line.
• Full Circle: Points located at discrete locations over whole area of the Circle.
• Full Sphere: Points located at discrete locations over whole volume of the Sphere.

Lines are probably the most useful Objects in Point World. Circles will be constructed using Lines, Spheres will be constructed using Circles, and Hyper Spheres will be constructed using Spheres. So everything is ultimately constructed using using Lines. Objects will have an Orientation when they are constructed in this way, which makes the component Points, Lines, and Circles move in a visually instructive manner. Objects can be Oriented on any of the four major Axes. Each of the lower level component Objects cannot be Oriented with the same Orientation as any higher level Object. So if we are going to construct a Sphere Oriented on the z-Axis then the component Circles will have to be Oriented on the x-Axis, the y-Axis, or the w-Axis. In other words the component Circles will have to be perpendicular to the Orientation of the Sphere. If we Orient the Circles on the x-Axis then the Lines that make up the Circles will have to be on the y-Axis or the w-Axis. Orientations are specified by a set of Axis designators where there are no repeated Axis designators. There are 24 valid Orientations for a Sphere in 4D Space, examples of which are xyz, xzy or wxy. A Hyper Sphere would need a four coordinate Orientation and a Circle only needs two coordinates. This will become more clear after seeing the Animations that will follow.

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 Points 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:

Note that these construction techniques produce Circles with a rotational symmetry around the z-Axis and in this case it is a zx-Orientation. The construction can also be implemented for x-Axis or y-Axis symmetry. 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
-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
-Span <= n <= Span

These equations will produce Points that have CircleRadius as follows:

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

So far we have constructed Circles in 2D Space, but in a 2D Point World Circles can also be constructed directly into 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 closer construction, where the component Lines are all oriented with the z-Axis, it is intuitively easy to see how the Lines could be brought closer together to form the Circle. The farther 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 back would line up on the z-Axis and seem easier to form a Circle and the construction in the front 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.

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

Note that these construction techniques produce Spheres that have a rotational symmetry around the y-Axis. The construction can also be implemented for x-Axis or y-Axis symmetry. This rotational symmetry 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.

So far we have constructed Spheres in 3D Space, but in a 3D Point World Spheres can also be constructed directly into 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 intuitively 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 and these things will be explored more in later sections.

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
-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
-Span <= n <= Span

These equations will produce Points that have SphereRadius as follows:

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:

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
-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
-Span <= n <= Span

These equations will produce Points that have HyperSphereRadius as follows:

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.

## 2D And 3D Hyper Space Rotation Analogies

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.

The Animation software can display Rotations in Point World in many different ways. All combinations of different Rotations, Object Orientations, Axis Sharing Axes, and Hyper Factored Axes were used. There are 6 rotations, 4 Object Orientations, 4 Axis Sharing Axes and 4 Hyper Factored Axes possible and with some mutual exclusion rules the software generated 936 different Animations. It was found that the visual results were duplicated for each of the different Rotations, so many Animations were redundant or not useful. It was decided after a lot of review that a zy-Rotation for 2D Point Worlds and a zw-Rotation for 3D Point Worlds gave the best results. The zy-Rotation is a Rotation into 3D Hyper Space for 2D Space and the zw-Rotation is a Rotation into 4D Hyper Space for 3D Space. Axis Sharing (y-Axis with x-Axis) for 2D Space and (w-Axis with x-Axis) for 3D Space will be used because it is physically impossible for the 2D Observer or 3D Observer to see the movement of objects in their respective Hyper Spaces. 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 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 following Animations will show basic objects (Line, Circle, Sphere, or Hyper Sphere) under rotation in a 2D Point World or a 3D Point World. The Animations will show that there are analogous visual results for Rotations into 3D Hyper Space and 4D Hyper Space. We will use the standard method of Showing how things look for a 2D Observer in 2D Space and then showing how things look for a 3D Observer in 3D Space. You should pause and single step these Animations to take note of the locations of individual Points, Lines, and Circles, in the 3D Reference area as compared to the location of these Objects in the Axis Shared area. As you become familiar with the Axis Sharing process these motions will become increasingly easier to understand. It is important to be at least a little familiar with these basic Hyper Rotations in order to understand topics in later sections.

Note that 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 transition 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 into 3D Hyper Space, and the second Animation shows the same Line in 3D Space with a zw-Rotation into 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 stretched out along the z-Axis with the end Points of the Line lying directly on the Hyper 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 orbit around in a circle on the Hyper Boundary, and other Points orbit 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 really all due to y-Axis shared components. With Axis Sharing the 2D Observer is able to see the zy-Rotation as a circular movement through 3D Hyper Space, and similarly the 3D Observer is able to see the zw-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 understanding this kind of rotation.

The first Animation below shows a Circle in 2D Space with a zy-Rotation into 3D Hyper Space and the second Animation shows the same Circle in 3D Space with a zw-Rotation into 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 Animations. However, the Points that make up the Circle now also have x-Axis components. With Axis Sharing the 2D Observer can see that the component Lines that make up the Circle orbit around inside the Hyper Boundary similar to how the Points orbited in previous Animations. The 2D Observer sees that the component Lines maintain a horizontal orientation parallel to the x-Axis throughout the orbit. 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 orbit around so that in some way the Circle is rotating 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. But the 3D Observer is still puzzled by zw-Rotations into 4D Hyper Space. He sees that, except for one line, all the Lines disappear from 3D Space as the Rotation progresses. And with Axis Sharing the 3D Observer can see the same visual movement as the 2D Observer, except this time the movement is in 4D Hyper space instead of 3D Hyper Space. The third Animation shows how the Circle behaves with the same Rotation but with a different Orientation. In this case the Lines still orbit around a central stationary Line but the component Lines are oriented with the y-Axis instead of the x-Axis. This seems to show the situation in a more natural way for the zw-Rotation. But it should be remembered that the two different 3D Orientations below are just depicting the same zw-Rotation in different ways.

The Animations below show another way to depict Circles using Axis Sharing. In the first Animation a Circle is constructed directly into 3D Hyper Space for a 2D Point World and in the second Animation a Circle is drawn directly into 4D Hyper Space for a 3D Point World. It is interesting how the Points of the Circle all expand out and are distributed across the Axis Shared view and Rotate as one big Circle. The Circle Rotates on a Point at the origin in both cases. In the 2D case the Axis Sharing area shows that there is a Line in 2D Space on the z-Axis and the rest of the Circle is in 3D Hyper Space. In the 3D case the Axis Sharing area shows that there is a Line in 3D Space on the z-Axis and the rest of the Circle is in 4D Hyper Space. The 3D Reference for the 3D case also just shows a Line because the rest of the Circle is in 4D Hyper Space. Also the Line in the 3D Reference in the 3D case disappears when the Rotation begins with different Points from Hyper Space popping in and out of the 3D Reference from 4D Hyper Space. In these Animations an array of separate Points are seen to orbit one central Point.

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 a 2D Point World constructed into 3D Hyper Space with a zy-Rotation, and the second Animation shows the same Sphere in a 3D Point World constructed into 4D Hyper Space with a zw-Rotation. 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 symmetry 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 previous 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 orbit around inside the Hyper 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 orbit around so that in some way the Sphere is rotating 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 stretched out and spread across 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. The third Animation shows how the Sphere behaves with the same Rotation but with a different Orientation. In this case the Lines still orbit around a central stationary Line but the component Lines are oriented with the y-Axis instead of the x-Axis. This seems to show the situation in a more natural way for the zw-Rotation. As with the Circle it should be remembered that the two different 3D Orientations below are depicting the same zw-Rotation in different ways. The fourth Animation shows another depiction that can be made for a 3D Point World. The Sphere is now drawn in the Normal Space of a 3D Point World. The same zw-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.

The following Animation shows a 3D Point World with a Hyper Sphere constructed into 4D Hyper Space with a zw-Rotation. There is no useful 2D Point World 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 previous 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.

## Sphere In 2D And Hyper Sphere In 3D Rotation Analogies

In previous Animations the Rotation behavior of an object was shown in a 2D Point World and then in a 3D Point World. In all cases the same object was used in the 2D Point World and in the 3D Point World for the comparison. In this section we will explore the Rotation of a Sphere in a 2D Point World versus the Rotation of a Hyper Sphere in a 3D Point World. Also in previous Animations the objects being studied were oriented on the z-Axis. This orientation is optimal when showing analogous behavior for 2D and 3D Rotations using the same object. For 2D Animations the z-Axis is the only choice because the x-Axis is used for Axis Sharing. Because we were interested in analogous behavior the 3D Animations also used z-Axis orientations. But for 3D Animations the y-Axis orientation of objects can be more useful. In the Animations below, for a 2D Point World a Sphere will be directly drawn oriented on the y-Axis and for a 3D Point World a Hyper Sphere will be directly drawn with component Spheres oriented on the y-Axis. For the 3D Point World the y-Axis will be Hyper Factored instead of the z-Axis to properly show the y-Axis orientation of the objects. For the 2D Point World the z-Axis will still be Hyper factored. The perspective from the 3D Point World will be used in both Animations.

In the Animations below

## Mathematical Points And Dimensional Points

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. It is simply a location in Space. 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 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-dimensional 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 will call this tiny object a Dimensional Point and if the dimension is specified we use the term nD Point. 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). 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. 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.

An nD Point can be thought of as the smallest Object that can exist while retaining the n-dimensional characteristics of the Space. For 2D Space a Point is not an existent Object. It is a Mathematical concept. It has no existence or extension into any dimension. The basic Object in 2D Space is the 2D Point because it has extension in 2 dimensions. A 2D Point has a differential Area whereas a Point has identically zero Area. However a 2D Point has no real existence in 3D Space because there is no extension into 3D Space. A 2D Point has identically zero Volume. When we work in 3D Space we will need to use 3D Points, which have extension into all 3 dimensions and have a differential Volume. Similarly, a 3D Point will not have any real existence in 4D Space because there is no extension into 4D Space and the Hyper Volume is identically zero. We will need to use 4D Points when working in 4D Space. Also note that a higher Dimensional Point can not fully exist in a lower dimensional Space. Only a cross section of the higher Dimensional Point can exist in the lower dimensional Space.

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 existence 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.

In the following Animation two Empty nD Points will be constructed on the y-Axis, which is the Hyper Axis for a 2D Point World. A 2D Point is first constructed at the origin and then another one 1 Diameter away in the positive direction on the y-Axis. We use Empty nD Points because we are mostly interested in the surface behavior of these Objects. The 3D Reference shows the actual placement and the Axis Shared area shows what a 2D Observer would see. A 3D Observer sees that the two 2D Points are not touching. In fact they can not touch no matter how close you bring them. They can only touch when they are at the same location because a 2D Point has zero extension in the direction of the y-Axis. A 2D Point is Flat in 3D Space. Circles are added to form two 3D Points. The 3D Points can now touch. The 3D Points touch at a single Point as indicated. The 2D Observer can not directly see the touching 3D Points that make up the Line but must interpret what is depicted in the Axis Sharing area. The 2D Observer sees the component Circles of the 3D Points displayed across the x-Axis. The 2D Observer imagines that each component Circle is in a separate parallel layer of 2D Space somewhere. The Axis Sharing area depicts a kind of sideways rendition of the actual 3D Space because as you go Up or Down in 3D Space you go Right or Left in the Axis Sharing Area.

In the following Animation two Empty nD Points will be constructed on the w-Axis, which is the Hyper Axis for a 3D Point World. A 3D Point is first constructed at the origin and then another one at 1 Diameter away on the w-Axis. We again use Empty nD Points because we are mostly interested in the surface behavior of these Objects. The 3D Reference can only show the 3D Point that is in our 3D Space. The 3D Observer sees that the two 3D Points are not touching. Assuming an analogous geometric situation to the 2D case we can say that they can not touch no matter how close you bring them. They can only touch when they are at the same location because a 3D Point has no extension in the direction of the w-Axis. The 3D Point is somehow Flat relative to 4D Space. This is difficult to understand for a 3D Observer but hopefully the analogy with the 2D case helps. Spheres are added to form two 4D Points. The 4D Points can now touch. The 4D Points touch at a single Point as indicated. The 3D Observer sees the component Spheres of the 4D Points displayed across the x-Axis. The 3D Observer imagines that each component Sphere is in a separate layer of 3D Space somewhere.

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:

• nD Line: Touching nD Points arranged across an Axis to form a Line.
• nD Square: Touching nD Lines arranged across an Axis to form a Square.
• nD Cube: Touching nD Squares arranged across an Axis to form a Cube.
• nD Hyper Cube: Touching nD Cubes arranged across the Hyper Space Axis to form a Hyper Cube.

A Line, a Square, or a Cube constructed from nD Points will be referred to as an nD Line, an nD Square, or an nD Cube. For simplicity we will usually use the minimum number of nD Points to construct nD objects. So an nD Line will be constructed with two nD Points, an nD Square with four nD Points, an nD Cube with eight nD Points, and an nD Hyper Cube with 16 nD Points. Note that the two Points in the above Animations can be considered to be an nD Line.

In previous sections the rotational behavior of Lines, Circles, Spheres and Hyper Spheres was explored. An nD Point can be a Line, a Circle, a Sphere, or a Hyper Sphere. We therefore have already explored the rotational behavior of nD Points. We will use nD Points instead of Points to construct compound Objects in up coming sections. Part of the difficulty of understanding 4D is that the connections between Points are not intuitive. If we consider Points to be 4D Points then the connections become easier to visualize as will be shown.

## Rotations On A Line And On A Plane Of Dimensional Points

In this section we will explore how a 2D Observer might understand the Rotation of nD Points arranged on a Line and how a 3D Observer might understand the Rotation of nD Points arranged on a Plane. A Line arrangement constructed using nD Points will be referred to as an nD Line. A Plane arrangement constructed using nD Points will be referred to as an nD Plane. Of course an nD Line must have at least two nD Points and an nD Plane must have at least three nD Points. We will see that there is an extra way to connect (n+1)D Points compared with nD Points, and this is the key to understanding Rotations on a Line for a 2D Observer and Rotations on a Plane for a 3D Observer. Empty nD Points will be used in these Animations because we are primarily interested in the surface behavior of these objects.

In this section we will describe a transition from 2D Space to 3D Space and use that knowledge to understand a transition from 3D Space to 4D Space. To construct 3D Space we can conceptually start with a 2D Space and then add component 2D Spaces with offsets on the positive and negative y-Axis. When we consider a Sphere in the newly created 3D Space we only have to consider a finite number of 2D Spaces because the Sphere will be made out of Circles that each exist in only one 2D Space. Similarly, to construct 4D Space we can conceptually start with a 3D Space and then add component 3D Spaces with offsets on the positive and negative w-Axis. When we consider a Hyper Sphere in the newly created 4D Space we only have to consider a finite number of 3D Spaces because the Hyper Sphere will be made out of Spheres that each exist in only one 3D Space.

In the 2D to 3D case we can further specify that the component 2D Spaces are parallel to the original 2D Space and therefore perpendicular to the y-Axis. The 2D Observer will have difficulty understanding this but a 3D Observer easily understands it. In the 3D to 4D case we can further specify that the component 3D Spaces are parallel to the original 3D Space and perpendicular to the w-Axis. But we have difficulty understanding how 3D Spaces can be parallel or perpendicular to something. This is because we are trapped in 3D Space with 3D Brains. From the 4D perspective 3D Space is Flat (but still 3D) and there are an infinite number of them. From our 3D perspective we can not imagine where to put even one other 3D Space. In 4D our 3D Space is the mathematical Hyper Plane. We are unable to imagine a 3D Hyper Plane, but theoretically knowing it exists will let us be able to talk about parallel 3D Spaces. Now we can also talk about a 3D Space being perpendicular to the w-Axis.

The Animation below shows a 2D Point World with a 2D Line and an xz-Rotation. Since a Line is defined by 2 Points we can limit the construction to only two nD Points. The 2D Points are Circles. Only the 2D Point at the origin can Rotate in place the other touching 2D Point must orbit around the first 2D Point.

The 2D Observer knows that a Line of Points can rotate in place in 3D Space but can never directly see this and can only understand it theoretically. In the following Animation the Line will be constructed on the y-Axis, which is the Hyper Axis for a 2D Point World. A 2D Observer can use Axis Sharing to begin to understand a Rotation in which two touching nD Points Rotate in place. But it is hopeless for the 2D Observer to use 2D Points while trying to understand this. We will show that 3D Points must be used for this to make sense.

In the Animation below a 2D Point is constructed at the origin and then another one 1 Diameter away in the positive direction on the y-Axis. An xz-Rotation is then started. The 3D Reference shows the actual placement and the Axis Shared area shows what a 2D Observer would see. The 2D Observer sees that the two 2D Points are rotating but not touching. In fact they can not touch no matter how close you bring them. They can only touch when they are on top of each other because a 2D Point has zero extension in the direction of the y-Axis. A 2D Point is Flat in 3D Space. Circles are added to form two 3D Points. The 3D Points can now touch and rotate In Place. The 3D Points touch at a single Point as indicated. The 2D Observer can see that there are two Points on the surface of a Sphere that Rotate In Place. The 2D Observer also sees that all other Points orbit in circles around a central Point in the particular 2D Space. The 2D Observer sees that the Spheres touch at a single Point that is Rotating In Place. This is confusing for the 2D Observer because he can not see the actual 3D Points in 3D Space but must see the Axis Sharing depiction in 2D Space. The Axis Sharing area depicts a kind of sideways rendition of the actual 3D Space.

A 3D Observer can easily see how the two 3D Points can touch and Rotate In Place. The 3D Observer can see that all the component Circles of the 3D Points are Rotating on the same Axis Line and completely understands how an Empty Sphere has two stationary Points that Rotate In Place. The 3D Observer can see that any Rotation in a component 2D Plane must be duplicated in all other 2D Planes. The Axis Point of Rotation is at the same location in any of the component 2D Planes that make up the 3D Space. It is easy to see how this forms a line in 3D Space.

The 2D Observer can not directly see all this but must interpret what is depicted in the Axis Sharing area. The 2D Observer sees the component Circles of the 3D Points displayed across the x-Axis. The 2D Observer imagines that each component Circle is in a separate parallel 2D Space somewhere and that they are all somehow rotating on the same Axis Point but in different Spaces. A 2D Observer is still not quite sure how something can Rotate on a Line. The 2D Observer will have a hard time seeing how the different Axis Points in the different 2D Spaces can form a Line. Since we as 3D Observers can easily understand this it should give us an advantage when we study 4D.

The Animation below shows three touching 3D Points with an xz-Rotation, an xy-Rotation, and then a zy-Rotation. The arrangement forms a corner shaped Object and will be called a 3D Corner, since it is made out of 3D Points. Only the 3D Point at the origin can Rotate In Place the other two touching 3D Points must orbit around the 3D Point at the origin depending on the type of Rotation.

A 3D Observer can completely understand the transition from 2D Space to 3D Space and we will try to use that inherent knowledge to help us understand 4D Space. First we must assume that 4D Space is constructed from layers of 3D Space in an analogous way to how 3D Space is constructed from layers of 2D Space. We will also use the observation from 2D Space that a Rotation in one 2D Plane at an Axis Point will be duplicated at the same location on all the component 2D Planes that make up a 3D Space. Extending that observation to 3D Space we can say that any Rotation on any Axis Line in a 3D Space component will be duplicated in all other 3D Space components that make up the 4D Space. The 3D Observer knows that three touching Points can rotate in place in 4D Space but can never directly see this and can only understand it theoretically. We will start by considering a 3D Line. The Line will be constructed on the w-Axis and the 3D Points are Spheres.

In the Animation below a 3D Point is constructed at the origin and then another one at 1 Diameter away on the w-Axis. An xz-Rotation, an xy-Rotation, and then a zy-Rotation is started. All three possible Rotations will cause the 3D Points to Rotate In Place in the same way. The 3D Reference can only show the 3D Point that is in our 3D Space. The 3D Observer sees that the two 3D Points are rotating but not touching. Assuming an analogous geometric situation to the 2D case we can say that they can not touch no matter how close you bring them. They can only touch when they are on top of each other because a 3D Point has no extension in the direction of the w-Axis. The 3D Point is somehow Flat relative to 4D Space. This is difficult to understand for a 3D Observer but hopefully the analogy with the 2D case helps. Spheres are added to form two 4D Points and the three Rotations are again started. The 4D Points can now touch and rotate in place. The 4D Points touch at a single Point as indicated. The 3D Observer sees the component Spheres of the 4D Points displayed across the x-Axis. The 3D Observer imagines that each component Sphere is in a separate 3D Space somewhere and that they are all somehow rotating on the same Axis Line but in different Spaces.

The next Animation shows the above Animation with the y-Axis Hyper Factored and with only an xz-Rotation. Since there are many Spheres Rotating In Place there will now be many Points that are Rotating In Place. These Points form an Empty Circle around each 4D Point and is the key to how 4D Points can connect in new ways.

The Animation below shows the Animation above with another 4D Point located at 1 Diameter up on the y-Axis to form a 4D Corner constructed in the yw-Plane with an xz-Rotation. There are now two Touching Points. The 4D Points are now on a Plane and the Rotation of these three 4D Points is on a Plane.

It is easy to see how another 4D Point could be added to form a 4D Square. The Animation below shows the above Animation with a fourth 4D Point located to form a Square in the yw-Plane with an xz-Rotation. There are now four Touching Points. They touch at Points that are Rotating In Place.

We can now make some general observations about Rotations in different dimensions. In 2D Space Rotation are on a 2D Point Axis, in 3D Space Rotations are on a 3D Line Axis , and in 4D Space Rotations are on a 4D Plane Axis. In each case the the Axis object has no extension in two of it's dimensions.

## Flatness of nD Space relative to (n+1)D Space

Whenever the next higher dimension, (n+1)D, is added to an nD Space any object in the nD Space will now have two new sides that did not exist in the original nD Space. One side will face in the positive direction and another side will face in the negative direction of the added Space. These two new sides have no extension into (n+1)D Space so the nD object will seem Flat in the (n+1)D Space. Also, each nD Point that makes up any nD Object will be Flat in the (n+1)D Space. A Square will seem Flat to a 3D Observer and a Cube will seem Flat to a 4D Observer. A 2D Observer will have difficulty understanding how a Square could ever be seen as Flat and a 3D Observer will have difficulty understanding how a Cube could ever be seen as Flat.

This difficulty is simply because a Square is really not Flat in 2D Space even though it is Flat in 3D Space, and a Cube is really not Flat in 3D Space even though it is Flat in 4D Space. So we are stipulating that, when using Dimensional Points, a Square is a fundamentally different kind of thing when it is in 2D Space than when it is in 3D Space. The Square is different because it is constructed from 2D Points in 2D Space and 3D Points in 3D Space. A Cube is also a different kind of thing when it is in 3D Space than when it is in 4D Space. Our 3D perceptual handicap will not allow us to ever see the Flatness of a Cube from the 4D point of view. We will try to show that not only is the Cube Flat but the whole 3D Space is Flat in 4D.

Let's first consider a Square in 2D Space constructed using four 2D Points. This is a 2D Square by previous definitions of an nD Square. From the point of view of a 3D Space this Square is Flat. The Square is Flat and the actual Points are Flat. The Square has identically zero thickness and we say it is Absolutely Flat. If we replace the 2D Points with 3D Points then the Square would obviously not be Absolutely Flat in the sense described. It is now a 3D Square and has some extension into 3D Space in any direction. The Square now has Differential thickness and we can say the Square is Differentially Flat. As long as we are using nD Points in nD Space there can not be any Absolutely Flat Objects only Differentially Flat Objects. We will almost always be working with nD Points in nD Space so the qualifiers will be dropped and we will just say that something is Flat with the assumption that we are intending differential Flatness.

The following Animation shows a Square constructed using four 3D Points on the zy-Plane. A Hyper Boundary circle is drown around each of the 3D Points. The yellow circles are around the two 3D Points at z = 0 and the green circles are around the two 3D Points that start at z = +1 diameter. The Square is given an xz-Rotation. It can be seen that the Square Rotates on the two 3D Points at z = 0. All the component Circles that are at z = 0 Rotate in place and all the component Circles a z = +1 diameter orbit around their counterparts that are Rotating In Place. The two green Hyper Boundary circles track their respective contained 3D Points.

In the following Animation the Square is given an xz-Rotation but stops at 90 degrees. The green Hyper Boundary circles will be in the same xy-Plane as the yellow Hyper Boundary circles after the 90 degree Rotation. The two yellow Hyper Boundary circles do not move. The Spheres that were at z = 1 diameter will now be located at x = -1 diameter. All 3D Points are now arranged so that the Square looks Flat from the point of view of the z-Axis. A 2D Observer is amazed that a Square can look Flat in 3D Space given the correct orientation.

We will describe in detail what a 2D Observer will see. Let a 2D Observer start at xyz coordinate (0, 0, 5d) where the d the diameter of a 3D Point. The 2D Observer would be located directly 5 diameters in front of the 3D Point at (0, 0, 0) and would be looking at the dark purple central component Circle of that 3D Point. A move of 1 diameter in the negative x direction would locate the Observer directly in front of the 3D Point at (-d, 0, 0). The Observer will find that he is 5 diameters in front of it also. Now the Observer moves +1 diameter in the y direction and is directly in front of the 3D Point at (-d, d, 0) and sees that he is also 5 diameters in front of it. Lastly, the Observer moves +1 diameter in the x direction and is directly in front of the 3D Point at (0, d, 0) and realizes that he is 5 diameters away from it also. He further realizes that since he was exactly the same distance in front of all four 3D Points that from the point of view of the z-Axis the Square is somehow Flat. The 2D Observer is also able to see how the upper pair of 3D Points is connected to the lower pair of 3D Points through 3D Space.

Similarly, the following Animation shows a Cube constructed using eight 4D Points in yzw-Space. A Hyper Boundary circle is drown around each of the 4D Points. The yellow circles are around the four 4D Points at z = 0 and the green circles are around the four 4D Points at z = +1 diameter. The Cube is given an xz-Rotation. It can be seen that the Cube Rotates on the four 4D Points that are at z = 0. All the component Spheres that are at z = 0 Rotate In Place and all the component Spheres a z = +1 diameter orbit around their counterparts that are Rotating In Place. The four green Hyper Boundary circles track their respective contained 4D Points.

In the following Animation the Cube is given an xz-Rotation but stops at 90 degrees. The green Hyper Boundary circles will be in the same xy-Plane as the yellow Hyper Boundary circles after the 90 degree Rotation. The four yellow Hyper Boundary circles do not move. All 4D Points are now arranged so that the Cube looks Flat from the point of view of the z-Axis. A 3D Observer is amazed that a Cube can look Flat in 4D Space even given the correct orientation.

We could describe what the 3D Observer would see for each of the eight 4D Points. But it is easy to see that a 3D Observer can now move in 1 diameter increments in the x direction, the y direction, and the w direction and still be 5 diameters away from all eight of the 4D Points. Specifically the Observer would be directly in front of the largest dark purple component Circle of each 4D Point. The 3D Observer realizes that since he is exactly the same distance in front of all eight 4D Points that from the point of view of the z-Axis the Cube is somehow Flat. It is Flat in 3 dimensions which would seem impossible for a 3D Observer.

The next thing we could do with the above Animation is construct a copy of the the resultant Flat Cube in a parallel plane at z = + 1 diameter. This will add eight more 4D Points that touch the original eight 4D Points. This forms a sixteen 4D Point Hyper Cube. Looking again at the Above Animation it is easy to see how all sixteen 4D Points could touch each other even without actually drawing the additional parallel plane. This will be explored in future TBD sections.