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. There are no continuous objects in Point World only individual Points. Objects such as Lines, Circles, Spheres, and even Hyper Spheres will be constructed from individual Points. The goal was to find a way to see how each Point in a more complex object behaves during a 4D Rotation or Translation.

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 Solid Sphere. In Point World the small Solid Spheres represent actual Points. Each Solid Sphere is constructed using 72 flat polygons. The shading effect works on the individual polygons to create the overall shading of the Solid Sphere. Solid Spheres are just markers for where a Point is located. Solid Spheres can be drawn packed in more tightly or more loosely. Solid Spheres can be packed so tight that they overlap but they still represent Points. The actual diameter of a Solid Sphere is not important just the central Point location 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. A problem with Mathematical Points is that an n-dimensional Point Man (observer) will intuitively think of a Mathematical Point as something more like a tiny Sphere object for that dimension. We will call this tiny Sphere object a Physical Point. A Physical Point 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 Physical Point is a pair of Mathematical Points, a 2D Physical Point is a Circle of Mathematical Points, a 3D Physical Point is a Sphere of Mathematical Points, and a 4D Physical Point is a 4D Hyper Sphere of Mathematical Points. The surface of a Physical Point is constructed from as many Mathematical Points as you want. A Physical Point 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 Physical Point goes to identically zero then all the Mathematical Points on the surface collapse onto a central Mathematical Point resulting in a single Mathematical Point.

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

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

In the following Animation two 2D Physical Points will move toward each other and touch at the origin. The Solid Spheres representing the Points are varied in size to emphasize that the Solid Spheres are just markers for Points 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. Mathematical Points in a 2D Point World would ideally be drawn as 2D Circles from the Point of View of the 2D Point Man. But since we are 3D Observers the Mathematical Points in the 2D Point World will still be drawn as 3D Spheres. The 3D Spheres represent Mathematical Points in 2D, and the 3D look of a Mathematical Point does not mean there is any implied extension into 3D space. Mathematical Points are depicted as 3D objects because we are stuck in a 3D world and that's what we are familiar with.

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

In the following Animation Point Man travels around a 2D Physical Point and stops to look from various locations. The Mathematical Points are shown as 2D Circles and then as 3D Spheres. 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 2D Circles are used. Using 3D Spheres enables visibility of the Points even when the 2D Point World is viewed on edge. A 2D Point Man sees his 2D world by scanning left and right so that his visual window into the 2D world is 1-dimensional. The Width and the Depth information are both superimposed onto the 1-dimensional image that he sees. This is similar to how our 3D Depth is superimposed onto our 2D visual image.

(Animation3) Edge View Of 2D Point World

 

 

Using Axis Sharing To Represent Hyper Points

A Hyper Point is a Mathematical Point 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 Physical Point until he saw the topmost Mathematical Point of the 3D Physical Point. Similarly if he could move down into negative 3D Hyper Space he would see lower and lower parts of the 3D Physical Point until he saw the lowest Mathematical Point. The following Animation shows the 3D Physical Point 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 Physical Point from Animation4 using Axis Sharing. A 2D Point Man runs an algorithm that draws the various y-Axis components of the 3D Physical Point 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 Physical Point, such as that as he goes further and further into 3D Hyper Space he sees smaller and smaller 2D Physical Points until there is only a single Mathematical Point. So he realizes that a 3D Physical Point is made out of smaller and smaller 2D Physical Points locateded further and further into 3D Hyper Space. He also realizes that he would be able to move all around any particular 2D Physical Point and look at all of it's Mathematical Points. Note that the 2D Point Man can not see inside a 2D Physical Point 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 Physical POint using Axis Sharing. A 3D Point Man runs an algorithm that draws the various w-Axis components of the 4D Physical Point 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 Physical Point, such as that as Point Man goes further and further into 4D Hyper Space he sees smaller and smaller 3D Physical Points until there is only a single Mathematical Point. So we realize that a 4D Physical Point is made out of 3D Physical Points locateded further and further into 4D Hyper Space. We can also see that we can go all around any particular 3D Physical Point and look at all of it's Mathematical Points. Note that we can not see inside a 3D Physical Point 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 Physical Point into smaller (n-1)-dimensional Physical Points might seem to violate the definition of a Physical Point as being the smallest thing. How can there be smaller and larger Physical Points? It's because the definition is imposed on the n-dimensional Physical Point. The (n-1)-dimensional Physical Points are just artifacts of the n-dimensional Physical Point. Each Mathematical Point of any (n-1)-dimensional Physical Point is still dR from the center of the n-dimensional Physical Point regardless of how small the radius of the (n-1)-dimensional Physical Point might be. If this doesn't help then just think of a Physical Point 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 Mathematical Point 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 Physical Point 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 Physical Point 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 Mathematical Points in the yz-Plane. When the z-Axis is Hyper Factored a more circular looking arrangement of the Mathematical 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 Point Man probably already suspects that the arrangement of Mathematical Points for a 3D Physical Point in 3D Hyper Space is more complicated than a 2D Physical Point 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 Physical Point.

(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 Mathematical Points in the yw-Plane or zw-Plane. The 3D Point Man is not physically able to see a 4D Physical Point 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 Mathematical Points for a 4D Physical Point in 4D Hyper Space is more complicated than a 3D Physical Point 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 Physical Point. 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 Mathematical 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 he x value of all Mathematical Points is set to zero. The result of Zero Factoring the x-Axis is that Mathematical Point arrangements that have x-Axis components collapse to the Hyper Factored y-Axis values. The following Animation shows that when a 3D Physical Point is shown using Axis Sharing and the x-Axis is Zero Factored that the resulting arrangement collapses to lines of Mathematical Points parallel to the z-Axis. This shows that each 2D Physical Point contains Mathematical Points that all have the same y-Axis component. Mathematical Points 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 Mathematical 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 Mathematical Point arrangements that have x-Axis components collapse to Hyper Factored w-Axis values. The following Animation shows that when a 4D Physical Point is shown using Axis Sharing and the x-Axis is Zero Factored that the resulting arrangement collapses to disks of Mathematical Points parallel to the yz-Plane. This shows that each 3D Physical Point contains Mathematical Points 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

In a 1D Point World the whole world exists on a single line (the x-Axis) and the surface of the 1D Physical Point is a Mathematical Point Pair at -dR and dR. In a 2D Point World the whole world exists on a single plane (the xz-Plane) and the surface of a 2D Physical Point is a 2D Circle around the origin with radius dR. In a 3D Point World the y-Axis is added to the xz-Plane to form a 3D space and a 3D Physical Point is the usual 3D Sphere around the origin with radius dR. In a 4D Point World a fourth axis (the w-Axis) must be added to the graph, but since the Point World representation of space is limited to 3 dimensions the w-Axis will have to be handled using Axis Sharing. A Physical Point in a 4D world is a 4D Hyper Sphere around the origin with radius dR.

Point World uses a Physical Point construction technique that proves to be useful when analyzing how Mathematical Points move through Hyper Space. Starting with the trivial 1D case on the x-Axis it is seen that there is only one way to build the 1D Physical Point which is at x=-dR and x=dR and which defines the extent of the 1D Physical Point on the x-Axis. Even though there are only two Mathematical Points we can still say that the radius is dR. For the 2D case the z-Axis is added a 2D Physical Point is built by adding 1D Physical Points to the graph at an Offset decreasing by equal -Delta increments in the -z direction and at an Offset increasing by equal +Delta increments in the z direction up to z=-dR and z=dR. The radius of the added 1D Physical Point will become smaller as the Offset increases in a mathematical relationship that forms the 2D Physical Point. At the largest Offsets of z=-dR and z=dR the 1D Physical Point becomes a single Mathematical Point that defines the extent of the 2D Physical Point on the z-Axis. This method of building the 2D Physical Point is called the Equal Offset method. This method produces a Circle that has Points that are closer together at small Offsets and further apart at large Offsets. Another method that produces a Circle with Points evenly spaced around the Circle is called the Equal Angle method. Both these methods produce an arrangement of Points that all have the same Radius to a central Point. Either of these methods will be used in the Animations depending on the purpose of the Animation. Also, 3D Spheres can be created from 2D Circles and 4D Hyper Spheres can be created from 3D Spheres using these methods. The following Animation shows how 2D Circles can be constructed using Point Pairs and how the Circles look for the two different methods.

(Animation14) Constructing 2D Empty Physical PointsUsing 1D Point Pairs

The following Animation shows how 2D Full Circles can be constructed using Point Lines and how the Circles look for the two different methods.

(Animation15) Constructing 2D Full Physical PointsUsing 1D Point Pairs

The following Animation shows how 3D Spheres can be constructed using 2D Circles and how the Spheres look for the two different methods.

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

The following Animation shows how 4D Hyper Spheres can be constructed using 3D Spheres and how the Hyper Spheres look for the two different methods. For 4D Hyper Spheres the Equal Angles cannot properly be shown when using Axis Sharing so both methods show the Offsets. Note how the Offsets are not equal for the Equal Angle method.

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

 

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

Equations For Equal Angle Method:
Angle = n * DeltaAngle
Offset = ObjRadius * sin(Angle)
Radius = ObjRadius * cos(Angle)

Where:
-N <= n <= N.
DeltaOffset is the Equal Offset increment.
DeltaAngle is the Equal Angle increment.
Offset is the Offset of the Pair, Circle, or Sphere component of the object.
Angle is the Angle of the Pair, Circle, or Sphere component of the object.
Radius is the Radius of the Pair, Circle, or Sphere component of the object.
ObjRadius is the Radius of the object (Circle, Sphere, or Hyper Sphere).

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

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

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

 

 

Rotations In Point World

The

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