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

The Sphere in Animation1 shows how a 3D Sphere can 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.

 

 

Mathematical Points And Physical Points

When the term Point is used in Point World it is assumed that the reference is to a Mathematical Point (MP). A problem with MPs is that an n-dimensional Point Man (observer) will intuitively think of a MP as something more like a tiny Sphere object for that dimension. We will call this tiny Sphere object a PP. A PP for a dimension is just the set of all points that are equidistant from the origin using a differential Radius (dR), where dR-->0. A 1D PP is a pair of MPs, a 2D PP is a Circle of MPs, a 3D PP is a Sphere of MPs, and a 4D PP is a Hyper Sphere of MPs. The surface of a PP is constructed from as many MPs as you want. A PP can be thought of as the smallest thing that can exist while retaining the dimensionality of the particular dimension under study. Note that dR-->0 but can not equal zero because if the radius of a PP goes to identically zero then all the MPs on the surface collapse onto a central MP resulting in a single Mathematical Point.

An MP has identically zero diameter and has no dimensional properties. If you let dx be a differential distance between the MPs on a line then you can make dx as small as you like and the MPs will still never touch. Only when dx is identically zero do the MPs touch each other, but they also all collapse onto a single MP. This means you can not arrange MPs next to each other in a line configuration where they touch each other. Two MPs can only touch each other when they are on top of each other as a single MP.

This inability to depict Mathematical Points as touching each other means they are not very useful in this analysis. The real utility of PPs is that two or more PPs can touch. Since the surfaces of PPs are made out of MPs, we can define two PPs as touching when an MP from one is at the same location in space as an MP from the other one. PPs can be arranged like MPs to form Lines and Planes, but unlike MPs the PPs can touch. Also, a PP has surface structure that can visually be seen to rotate, whereas an MP cannot visually rotate.

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

(Animation2) Touching Physical Points

 

 

Special Considerations For A 2D Point World

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. MPs 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 MPs in the 2D Point World will still be drawn as Balls. Balls represent MPs in 2D, and the 3D look of an MP Ball does not mean there is any implied extension into 3D space. MPs 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 2D PP and stops to look from various locations. The MPs 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 MPs 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.

(Animation3) Edge View Of 2D Point World

 

 

Using Axis Sharing To Represent Hyper Points

A Hyper Point is an MP that has a component on the 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.

(Animation4) 2D Point Man looking at 3D Physical Point

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 3D PP until he saw the topmost MP of the 3D PP. Similarly if he could move down into negative 3D Hyper Space he would see lower and lower parts of the 3D PP until he saw the lowest MP. The following Animation shows the 3D PP from Animation4 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.

(Animation5) 2D Point Man Moving In 3D Looking At 3D Physical Point

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 PP from Animation4 using Axis Sharing. A 2D Point Man runs an algorithm that draws the various y-Axis components of the 3D PP 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 3D PP, such as that as he goes further and further into 3D Hyper Space he sees smaller and smaller 2D PPs until there is only a single MP. So he realizes that a 3D PP is made out of smaller and smaller 2D PPs locateded further and further into 3D Hyper Space. He also realizes that he would be able to move all around any particular 2D PP and look at all of it's MPs. Note that the 2D Point Man can not see inside a 2D PP like we can from our 3D perspective. A 2D Point Man sees everything on a 1D edge using his 1-dimensional vision capabilities.

(Animation6) 2D Point Man Looking At 3D Physical Point Using Axis Sharing

The following Animation shows a 4D PP using Axis Sharing. A 3D Point Man runs an algorithm that draws the various w-Axis components of the 4D PP 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 4D PP, such as that as Point Man goes further and further into 4D Hyper Space he sees smaller and smaller 3D PPs until there is only a single MP. So we realize that a 4D PP is made out of 3D PPs locateded further and further into 4D Hyper Space. We can also see that we can go all around any particular 3D PP and look at all of it's MPs. Note that we can not see inside a 3D PP like a 4D Point Man could. We see everything on a 2D edge using our 2-dimensional vision capabilities.

(Animation7) 3D Point Man Looking At 4D Physical Point Using Axis Sharing

The deconstruction of an n-dimensional PP into smaller (n-1)-dimensional PPs might seem to violate the definition of a PP as being the smallest thing. How can there be smaller and larger PPs? It's because the definition is imposed on the n-dimensional PP. The (n-1)-dimensional PPs are just artifacts of the n-dimensional PP. Each MP of any (n-1)-dimensional PP is still dR from the center of the n-dimensional PP regardless of how small the radius of the (n-1)-dimensional PP might be. If this doesn't help then just think of a PP as a Circle, Sphere, or Hyper Sphere with non differential radius. The analysis is conceptually the same if we use these non differential objects. A differential Circle is after all a Circle. But a differential Circle is also a MP like thing and a Circle is not.

 

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 3D PP using Axis Sharing and varying Hyper Factors.

(Animation8) 2D Point World With 3D Physical Point 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 4D PP using Axis Sharing and varying Hyper Factors.

(Animation9) 3D Point World With 4D Physical Point 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 MPs in the yz-Plane. When the z-Axis is Hyper Factored a more circular looking arrangement of the MPs 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 Point Man probably already suspects that the arrangement of MPs for a 3D PP in 3D Hyper Space is more complicated than a 2D PP and is somehow circular in nature. The following Animation shows how Hyper Factoring the z-Axis will let the 2D Point Man begin to get a feel for the shape of a 3D PP.

(Animation10) 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 MPs in the yw-Plane or zw-Plane. The 3D Point Man is not physically able to see a 4D PP 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 Point Man (and we) already suspect that the arrangement of MPs for a 4D PP in 4D Hyper Space is more complicated than a 3D PP and is somehow circular in nature. The following Animation shows how Hyper Factoring the y-Axis will let the 3D Point Man begin to get a feel for the shape of a 4D PP. 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.

(Animation11) 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 MP 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 he x value of all MPs is set to zero. The result of Zero Factoring the x-Axis is that MP arrangements that have x-Axis components collapse to the Hyper Factored y-Axis values. The following Animation shows that when a 3D PP is shown using Axis Sharing and the x-Axis is Zero Factored that the resulting arrangement collapses to lines of MPs parallel to the z-Axis. This shows that each 2D PP contains MPs that all have the same y-Axis component. MPs that originally had a zero y component would collapse to the origin leaving only Hyper Factored y components on the x-Axis.

(Animation12) 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 MP 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 MP arrangements that have x-Axis components collapse to Hyper Factored w-Axis values. The following Animation shows that when a 4D PP is shown using Axis Sharing and the x-Axis is Zero Factored that the resulting arrangement collapses to disks of MPs parallel to the yz-Plane. This shows that each 3D PP contains MPs that all have the same w-Axis component. Mathemtical Points that originally had a zero w component would collapse to the origin leaving only Hyper Factored w components on the x-Axis.

(Animation13) 3D Point World With Zero Factored Axes

 

 

Constructing Physical Points In Point World

We start with the assumption that a 0D PP is a single MP at the origin. There cannot be an actual PP in a 0D Point World because there are no dimensions to extend into. This Point World consists of nothing because a single MP is really nothing and is a degenerate PP . Next we add the x-Axis to make a 1D Point World and put an MP at x=-dR and at x=+dR to make a 1D PP at the origin. We have used 2 0D PPs to make the 1D PP. The 1D PP consists of only two MPs on its surface and has zero surface area which also makes it a degenerate PP.

In a 2D Point World the z-Axis is added perpendicular to the x-Axis of the 1D Point World to form a 2D Point World (xz-Plane) and the surface of a 2D PP is a 2D Circle of MPs located around the origin with radius dR. We cann use two different methods to assign the MPs that are located on the 2D Circle. The first method will be called the Equal Offset method. We decide how many 1D PPs we want to construct on each side of the x-Axis and use the Equations for the Equal Offset Method below. This method produces a 2D PP defined by a Circle that has MPs that are closer together at small Offsets and further apart at large Offsets. This method constructs a 2D PP that doesn't look good but behaves very well in the animations. The second method is called the Equal Angle method. Again we first decide how many 1D PPs we want to construct on each side of the x-Axis and use the Equations For Equal Angle Method below. This method produces a 2D PP defined by a Circle that has MPs that are evenly spaced around the Circle. This method constructs a 2D PP that looks good but does not behave as well as the Equal Offset method in the Animations.

The PPs in n-dimensional Point World will be constructed from the PPs of (n-1)-dimensional Point World. This construction process will produce PPs with well behaved MPs as seen in later Animations.

Note that the equations are included for completeness and for those who need to know every detail. It is not necessary to understand the equations in order to understand the Animations.

So far we have been describing the construction of Empty PPs but there can also be Full PPs. A 1D Full PP is a line of MPs from the MP at -dR to the MP at dR. The following Animation shows the construction of 2D Empty PPs using 1D Empty PPs.

(Animation14) Constructing 2D Empty Physical Points Using 1D Empty Physical Points

The following Animation shows how 2D Full PPs can be constructed using 1D Full PPs.

(Animation15) Constructing 2D Full Physical Points Using 1D Full Physical Points

In a 3D Point World the y-Axis is added perpendicular to the xz-Plane of the 2D Point World to form a 3D space (xyz-Space) and the surface of a 3D PP is a 3D Sphere of MPs located around the origin with radius dR. We can use the Equal Offset method or the Equal Angle method to assign the MPs that are located on the 3D Sphere. We first decide how many 2D PPs we want to construct on each side of the xz-Plane and use the Equations For Equal Offset Method or Equal Angle method below. The following Animation shows how 3D Empty PPs can be constructed using 2D Empty PPs. 3D Full PPs are constructed similarly and are not shown in an Animation.

(Animation16) Constructing 3D Empty Physical Points Using 2D Empty Physical Points

In a 4D Point World a fourth axis (the w-Axis) must be added perpendicular to xyz-Space, but since the Point World representation of space is limited to 3 dimensions the w-Axis will have to be handled using Axis Sharing. The surface of a 4D PP is a 4D Hyper Sphere of MPs located around the origin with radius dR. We can use the Equal Offset method or the Equal Angle method to assign the MPs that are located on the 4D Hyper Sphere. We first decide how many 3D PPs we want to construct on each side of the xyz-Space and use the Equations For Equal Offset Method or Equal Angle method below. For 4D Hyper Spheres the Equal Angles cannot properly be shown when using Axis Sharing so both methods show the Offsets. Note how the Offsets are not equal for the Equal Angle method. The following Animation shows how 4D Empty Hyper Spheres can be constructed using 3D PPs. 4D Full PPs are constructed similarly and are not shown in an Animation.

(Animation17) Constructing 4D Empty Physical Points Using 3D Empty Physical Points

 

Equations For Equal Offset Method:
DeltaOffset = dRadius / (2 * N)
Offset = n * DeltaOffset
Angle = asin(Offset / ObjRadius)
Offset = dRadius * sin(Angle)
Radius = dRadius * cos(Angle)

Equations For Equal Angle Method:
DeltaAngle = (Pi/2) / (2 * N)
Angle = n * DeltaAngle
Offset = dRadius * sin(Angle)
Radius = dRadius * cos(Angle)

Where:
N is the number of (n-1)-dimensional PPs on each side of the PP under construction.
-N <= n <= N.
DeltaOffset is the Equal Offset increment.
DeltaAngle is the Equal Angle increment.
Offset is the Offset of the (n-1)-dimensional PP of the n-dimensional PP.
Angle is the Angle of the (n-1)-dimensional PP of the n-dimensional PP.
Radius is the Radius of the (n-1)-dimensional PP of the n-dimensional PP.
dRadius is the differential Radius of the n-dimensional PP.

Note that these equations will produce MPs that have dRadius. This can be seen by recognizing that the Offset is on the Hyper Axis (y or w) and the Radius is perpendicular to the Hyper Axis regardless of the particular MP. It is easy in 3D space to see how all the MPs of a component 2D PP are at a Radius that is perpendicular to the y-axis. In 4D space it is more difficult to see how all the MPs on a component 3D PP can be perpendicular to the w-Axis. It must be realized that relative to 4D space our 3D space is flat and the 3D PP is also flat. The whole 3D space is perpendicular to the 4D w-Axis so the MPs of the 3D PP in 4D are all on the same 3D Hyper Plane and therefore the Radius lines are all perpendicular to the w-Axis. So the Radius line and the Offset line define a right triangle in the n-dimensional space and it is easy to see that the line from the MP to the origin completes the triangle and is the distance of the MP from the origin which is dRadius. So:

dRadius = sqrt(Offset^2 + Radius^2)

It is easy to see how 2D PPs are made out of 1D PPs and how 3D PPs are made out of 2D PPs but it is more difficult to see how 4D PPs are made out of 3D PPs because we can not ever really see a 4D PP. We can only theorize about it and see it using Axis Sharing or some other method. We actually need a different kind of Brain. We need a 4D Brain and Visual System. Similarly a 2D Point Man can not ever really see a 3D PP. He needs a 3D Brain and Visual System. Even as 3D beings we can not show a 2D Point Man how to see a 3D PP. A higher dimensional space is not just more of the lower dimensional space but rather it is a completely different thing. However it's not hopeless, because more can be learned about 4D space especially when it comes to rotations as will be shown next.

 

Rotations In Point World

The

(Animation18) 2D Th ere are 936 possible ways of looking at Rotations In Point World.