Mathematical Points and Dimensional Points
A Mathematical 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.
https://www.theintermind.com/ExploringT ... nYAxis.gif
