The main idea is simple, that two ‘points’ can exist together and still be two ‘points’. Since ‘points’ have no extent, this is possible. We have the concept of the point being an exact location in the plane. Two new ‘points’ can be created together if they are different from the usual idea of points. One may say why two? Why not more? But we can state at the beginning that we have only two.

We have to have a new co-existing plane different plane so that we can take away one ‘point’ that already exists. This plane is a plane of ‘locations of locations’. Once this plane is in place, the usual points are altered only in that they can be then mobile. I need to take the one ‘point’ away otherwise I have both one and two ‘points’ there when I go to create the two ‘points’ together. Needing to be able to take this one ‘point’ away creates the need and existence of this new level.

I need to take away the concept of no extent being a point and replace it with a new concept of no extent. This can be an object which has no length, breadth or height but does have parts to it. In this special case the number of parts is two. (this is not like removing a point from a set of points, I am removing the concept of a point).

I can’t take a ‘point’ away without a new level. I need this space to be able to put two ‘points’ in. If If this new plane isn’t here then the two points are coincident and we have just one point.

I want to remove a ‘point’, leaving a ‘location of a location’ and then I can have the two new ‘points’ created together. Then this means a plane of location of locations exists. The usual locations are fixed in this plane. This new plane of ‘locations of locations’ is considered fixed, for now.

These new entities(‘points’) I call ‘parts’ (as in parts of the whole). Parts are different from points, as they are mobile in this new plane.

We may imagine the empty space in a three-dimensional container. We can extend the point concept stated above to spaces as I can have a set of points be the three-dimensional space. This empty space can be moved to take up the same “space of spaces” as the still spaces, if we move two empty spaces together. Then two empty spaces take up the same “space of spaces” left. . The spaces may be full of something such as the string which makes up a knot. Then the two full spaces can still be at one “space of spaces”. Two items are not together at the same space.

This may be the reality we live in. I am working on some physics in another section of this website. It may be for the far off future, but if we can move space itself then we may be able to travel to one of the exoplanets. Since the speed of light is limited by space but we may not be limited by this speed if we move space itself.

The new level can be what I call placement space. It is a space where a location may have another ‘location of a locations’. Or a small piece of space can have another “space of spaces”. Basically a location or section of space is mobile. At the beginning it is stated how many locations are at any given placement. For the work on knots this is two, and the resulting entity I call a “doublet”. Then two different locations may share the same placement (location of locations) and I call this a “sharing”. A special type of sharing may exist in the manipulation of diagrams which I call a “joining”.

I use the knottedness and chirality or achirality of knots to demonstrate the usefulness of these ideas.