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This summer I am having the pleasure of working for JDL Consulting in Richmond Hill. I am teaching grade 10 and 11 mathematics and science to students who have recently come to Canada from China.

Below I have entered four pages of notes. I will add explanation to the diagrams over the next little while. Firstly, I will explain the notes using text and then I will add in the symbolism after that.

It has been seen from the idea of concept sharing that a place can have a place of places. So two places can have two places of places too. Then it should be possible to switch the two places and move in space. We need to be able to “move” as places.

Then instead of us actually being places we need to create a space we can climb into which acts like a point from the outside. Thus we need to find a way of turning a sphere into a point, on the outside. The first step then could be to find a way to do this two-dimensionally, that is find a way to map a circle to a point.

This note starts with the two locations A and B. But the space we are in is like two overlapping shadows, since we also have the further background of placements. That is, there is another coexisting plane where locations can be in other places, different from where they usually are relative to each other.

In order for a location to itself have another ‘location’ we need to have a coexisting plane which gives another ‘placement’ to any location. We give a new name to a ‘location’ of a location, since we can’t use the name ‘location’ again.

Let a plane of placements be created and coexist with a plane of locations. At the beginning a set of second co-ordinates is created. ((a,b),((a,b))). a and b are any real numbers. Each location (a,b) is at its usual place ((a,b)). The further explanation of this can be seen here, in the article on the twin prime conjecture:

Online Tutoring Services Ontario Canada » the twin prime conjecture

We can move locations in a closed loop or we can have a geometry where two or more locations share the same placement. This is called a joining. The parts of a joining are indistinguishable.

Then with A and B we can move A and B through other locations (creating a line of joinings) to a center where we have three locations and one placement.

This second note starts with a picture of a circle with radius r with a direction in the space modelled after the overlapping shadows.

The indicated locations can move through the three-dimensional space of placements and locations to eventually create a self-intersecting loop of locations. In order to do this a continuous series of joinings are gone through and a circle of voids, or just placements is left.

But we can also have the usual motion of a loop through space to give a self-intersecting loop. We can call this a move of the diagram, while in the other case we can call this a shift of the diagram. So the extra background gives us two different ways of altering the diagram.

What we can do then is move the diagram to create a self-intersecting loop in the usual sense. But I can reach this self-intersecting loop in two possible ways either by twisting through space one way or the other.

What I can do is create one diagram which covers both cases. If I place the center, as created in the first note at the self-intersection point of the self-intersecting  loop and then let the loop unloop itself but this time by shifting, not by moving. Then the multiple point b(1) has components e(1) and e(2) and e(1) and e(2) can switch positions. Then this models both twist cases.

See a detail here using D and Q labels from knot diagrams:

Online Tutoring Services Ontario Canada » Diagram 6