# Interactive Online Tutoring Services

## December 21, 2017

Filed under: clarification of demonstration,knots — Rob @ 8:33 pm
1.  I can duplicate a point in the plane.
2.  If I have a plane of locations, I can’t move a location to another place.
3.  Consider a plane of locations of locations co-existing with the plane of locations
4.  Then I can move one copy of the location into the second plane where it joins with the location that is already there. (We can keep the rest of the locations still).
5.  Then each point of the plane has a location and the location of the location of it’s duplicated point
6.  Suppose we have a set of connected locations which form a loop in 3-space. We can move from the plane to 3-space.
7.  Then we can imagine transforming to the second level and twisting the loop so that the loop joins itself in the centre. We have the location from the still locations in 3-space and the two moving locations joining at the centre.

## December 16, 2017

### The third level of location

It seems a feature of this geometry is that we can have two different locations at one location of locations.

So if a circle is moved to ‘intersect’ itself, I can separate the ‘intersection’ as there are two different locations at this ‘intersection’.

If these two locations are labelled as the ‘intersection we can move these two labels out along the path of the circle and bring

two locations together which are not the intersection.

But this has to separate into a different space than would the two locations A and B separate into a space of locations of locations.

Let the distance between A and B be zero and some distance at a third level. A and B then move in a space of locations of locations of locations.

Then D(1) is a joining of C and D. A virtual joining.

Since we are at the third level we move the diagram to the circle again by separating C and D at D(1) like A and B are separated at Z(1)

## June 14, 2016

### diagram associated with e and p theory

Filed under: knots — Rob @ 2:30 pm

### e and p theory

Filed under: knots — Rob @ 2:22 pm

Suppose we have two overlapping shadows from one light source. An example would be two strings placed one above the other with a light source from above, slightly to the right. There is another background light source so that we can see the shadows from above, projecting onto a plane below (see the second drawing above).

Just at the four corners of the intersecting shadows we have four points for which two points from each string are projected together from the light source . For each set of these two points of the shadow intersection I have two shadow edges in coincidence. I can label one of these corners-p. A point p  from one shadow edge in coincidence (+) with a point p from the other shadow edge is the same shadow point p. We can say this because if I turn off the light source no shadow points remain. This can be an understanding of the coincidence of shadow edges. Briefly, this can be written p(+)p=p. One point p, contributed from each string are the same p, in coincidence.

We can also have two overlapping shadows from the same object from two light sources. An example would be the overlapping shadows that result when a cup of tea is placed on a table (see the first drawing above) with two light sources above, one from the left and one from the right and again some background light so that we can see the shadows.

At the edges of the overlapping shadows we have one point from the projection of the first light source e and one point from the second light source e placed together on the shadow plane. We can write e(*)e =e(*)e.  This is to indicate that the combination e(*)e does not become e. It is unlike the coincidence of shadow edges since I have to turn off both lights for both e’s to disappear. So in this case  we have a “part” “sharing” (*) with another “part”, one “part” from each shadow. If I turn off one light source, one set of shadow parts remains.

We can imagine one way to put two points together by what we usually think of as coincidence. Since the result must be an object which also has no extent and so far we only have one object like this, a point, two points of two coincident lines can lead to a single point of coincidence.

To come up with something different from coincidence, let e be a different entity than a point. E is not a point, but something else-a “part” for which we have sharing in which e(*)e does not equal e, but stays as e(*)e. This is an object which still has no extent-since I am placing together two items of no extent-but has two parts. Two different parts, one from each light source.

But the two e’s are not otherwise distinguishable so I label the location e(*)e. I can label them e(1) and e(2) but they are in the same location of the plane so that they are not otherwise distinguishable, I am not concerned which light source generated which e.

Then using this model, we can create a geometry where we have two different types of entities which have no parts and no extent- p and e. To do this we have used sharing-something different from coincidence.

Filed under: knots — Rob @ 1:44 pm

## June 7, 2016

Filed under: knots — Rob @ 10:46 am

I decided to add some notes below, on an extension of knot theory. I will add explanation to the notes a little bit at a time.

## June 4, 2016

### knot note 4

Filed under: knots — Rob @ 10:50 am

### knot note 5

Filed under: knots — Rob @ 10:48 am

### knot note 6

Filed under: knots — Rob @ 10:47 am

### knot note 7

Filed under: knots — Rob @ 10:46 am

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