### new student from Wilfred Laurier

I am helping a new student from Wilfred Laurier University. She is taking an online course over the summer which is designed for her to go at her own pace.

I am helping a new student from Wilfred Laurier University. She is taking an online course over the summer which is designed for her to go at her own pace.

I have a new student through ‘ Tutor by Numbers’ he is in grade 12 and is trying to get into Durham College for Automotive Engineering. I have met with him 4 times and will likely be reviewing with him in the summer.

I am working at the Prestige private school located in Richmond Hill. I am tutoring six groups of four students from grade levels 2 to 11 in math.

I am starting a new branch of my business called “Tutor by Numbers”. I am seeking students from grade 4-12 who would be interested in meeting in-person in my local area of Markham/Vaughan. I am specifically looking for but I am not limiting myself to students with learning disabilities.

I am Currently having great success tutoring a student with a learning disability through the Learning Disabilities Association of York Region. He recently received a grade of 100% on two quizzes in math.

Anyone interested in this service can call me at 905-731-2145 or email me at robburchett1@gmail.com.

I am teaching two students, in person, at J.D.L. consulting in Richmond Hill. One student is in grade 9 and the other in grade 10. I also continue teaching a student in grade 8, through the learning disabilities association of York region.

I am still working for JDL consulting in Richmond Hill. I am teaching grade 4 and 5 math. I still continue to tutor a grade 8 student through the learning disabilities association of York Region.

The idea is that a location may have another place other than it usually occupies. Places where we might put a location somewhere else-relative to other locations. This can be done if locations were sharing the same points of the plane with other objects, locations of locations, or placements (l*l).

We know what we mean by a moving point. But here this is not the case, here we are moving the location. We can imagine that a location could have another l*l if the meaning of a location is kept. We can do this by changing back and forth from the usual plane to an alternative plane only when all locations are back in their usual places. To do this we will need to invent a new, alternative plane where a location could be in other places.

A physical object can’t be in two places at the same time. but a mathematical object could be if we consider the following:

- A point is an undefined object meant to express the concept of an entity with no extent.
- Since the object has no physical reality to it, that is it is not a concrete object we can go forward and suggest that the point may be duplicated.
- This can be expressed in this way p–>p+p

The concept for an isolate point is that although p is not extended in any concept of extended space, it is duplicated since we can’t tell if there are two or one there.

Our imagination fails us if we try to think of two points being there but there could be two there nevertheless as the duplication of two entities of no extent still possess no extent.

Once we duplicate a point at a placement we can leave an original copy where it is and move a copy to another placement.

Start with the usual plane with every point given a location (a,b) where a and b are numbers. Then form an alternative plane with l*l- a( )b a and b are numbers- at the left the a is the x-co-ordinate of the l*l and the b at the right is the y co-ordinate of the l*l- placed together with the locations we can form the plane: a((a,b))b. The l*l is written outside of the location only to indicate that the l*l are fixed and the locations can move between them. There may be other geometries where the l*l are not fixed but we do not consider these here.

The double bracket is meant to indicate that this is a one step ‘deeper’ level of location. ‘deeper’ in the sense that now locations have a location called l*l. For brevity the location of a location is called an l*l. Places where locations can be.

To give a location another location we have to be able to remove it from where it is and move it to another l*l. This means there has to be some other mathematical objects at the points where we have locations. This can be l*l, a( )b. Two mathematical objects, the location and the l*l exist at the same point. In the beginning then we have a((a,b))b for all a,b. There are higher levels such as a((a((a,b))b))b but we only consider two levels for now.

We can think of this as if we have two overlapping shadows in one plane, each shadow cast from a separate light source. See the teacup shadow diagram below. The two shadows are independent as when one light source is shut off we still have the other shadow from the other light source. One shadow can be thought of as consisting of locations and the other shadow can be thought of as made of placements.

a( )b and (a,b) are two different mathematical objects. Since both a( )b and (a,b) have no extension, they can be together yielding a result of no extension. This can be done with two different locations also ex. (c,d)*(e,f) and we call this a ‘joining’.

The notation: a((a,b))b is called a second-co-ordinate and is meant to indicate this combination of two objects. The outer part is the l*l. The inner part can take the form of a single location, a void, or a finite series of joined locatons ex. (c,d)*(e,f)*…..

We can start simply, with each location where it usually is but label these places. As an example suppose the location (1,1) starts at the l*l- 1( )1 in an alternative plane. In this double bracket alternate plane the locations can be placed anywhere. Then create a new set of axis centered at 1( )1 and use the notation 1((1,1))1 to indicate that at the placement 1( )1 we can place the location (1,1).

Similar to the Tangent Plane, a( )b acts a center of a((a,b))b. (a,b) can move off with an axis centered at a( )b.

In the example below, if we start at 4((4,4))4 and move the location (4,4) to the l*l 7( )7 the ‘vector’ grows as (4,4) is moved along.

So we can start with the plane (a,b) and then transform to the alternative plane a((a,b))b. Then we can move any location. As long as we move all locations back to there original spots we can transform back to (a,b).

** **

We say that a point has a location. But I want to say that that specific point or location can have another ‘location of location’ in relation to the plane of other locations.

To do this I put a plane of l*l a( )b in so that every location (a,b) has an l*l- a( )b. We stop at only two levels for now. So every location of the plane has a ‘location’ itself. These two together, the location of the location, and the location itself are called the second co-ordinates and take the form a((c,d)*(e,f)*…))b.

Each placement of a location initially acts as a center for its specific location. Any location can move away from its center. There can be different types of movement. Sometimes, in order for a location to move we have to create a ‘joining’ of two locations. Locations can ‘join’ with other locations. This is discussed a little later.

One thing we can do is move a circle of locations through a coincident circle of l*l. Underlined in red are locations, moving around 90 degrees clockwise through the coincident level of l*l. The plane can be divided into two types of locations. e-type parts and p-type parts. The circle of e-type parts can move through a circle of p-type parts defined in a plane of fixed p-type locations coincident to a plane of placements.

Also I can translate, for example the location (0,0) at the l*l-0( )0 to the l*l-1( )1. But (0,0) is not in isolation. It has to be in combination with other locations in order to get to and to be at 1( )1.

Let it be in ‘joining’ with other locations. Two locations can be in joining if they share the same l*l. Notate this with a *. Then when (0,0) reaches ((1,1)) we can notate this (0,0)*(1,1). This is an expanded notation. Both locations (0,0) and (1,1) are at the l*l- 1( )1. To express this, I cannot write them on top of one another, so I use this notation. This can be notated 1((0,0)*(1,1))1.

Both locations are in the same place, but there are two locations there and not one. Since l*l exist as a background level, we have a place to put (0,0)*(1,1). Whereas before there was no background level where I could put one location relative to another. Now we can put two in the same place, before we couldn’t identify two locations to the same place.

(0,0)*(1,1) can be thought of as sharing placement akin to two overlapping shadows appearing on a plane surface, the plane surface being a plane of l*l and the overlapping shadows thought of as overlapping locations-think of the shadows as locations themselves. As described elsewhere in this weblog, if the shadows come from two light sources, and I remove one light, one shadow remains.

The goal of these four pages is to find a way to define the most simple knot in its most simple form-but this is done in a different space than it is used to being defined in. This may open up new moves which we could apply to more complex knots.

In ordinary maps two points may be mapped to a single point, and then we have the inverse “mapping”: a single point may be mapped back to both different points.

But here we want to be able to work with knot diagrams defined in a space made out of “parts” and placements and not points. “Parts” and placements are discussed elsewhere on this website. By parts we mean locations.

The central idea is that instead of points which combine by coincidence (+) into a single point, parts can combine by joining(*). So that two parts brought together as in two lines of parts coming together to form a joining(*), may be written as for example (e(1,1)*e(1,2)). Meaning at one placement we have the first part e(1,1) and the second part e(1,2) sharing the placement.

In usual geometry a point has no extent and no parts or pieces. Since it has no parts or pieces it can’t be separated or divided into two or more parts or pieces. If we place two points together by moving two lines together until we have a coincident (+) point, we superimpose two wholes with no parts or pieces and the result is a single whole with no parts or pieces.

But in the geometry of placements and parts we have another entity “e” which has no extent and no parts or pieces and is also a whole. E is a group of moving locations in the space of placements and locations.

“e” is different from a point p, in the usual geometry in that we can place two e’s together by joining(*) and have the result not be a single whole-like with points-but two e’s in the same placement-similar to two overlapping shadows. “e” has no extent like a point p, but is not like a point p-we regard it as being a part of a combination as in e*e as well as a whole in itself.

The two e’s would have to be distinguished in some other way then their placement. “e’s” and points are different in this manner. P-type parts are also different from points, in the same manner and are usually fixed in the space of placements and locations.

There is a subtle difference between division and separation here. By division we mean the ability to form two or more entities like the original, from the original, smaller than the original. We therefore cannot divide a point or the entity (e(1,1)*e(1,2)) into pieces. But we can separate (e(1,1)*e(1,2)) into two parts. Then there is the difference between parts and pieces.

We let p denote usually fixed parts and e denote moving parts. e’s are not distinguished by their placement, but can be distinguished by some other rule. In the case of knot diagrams the first number indicates which location we are at and the second number indicates whether the e was originally below, 1 or above, 2.

So we start then with a three dimensional space which consists of placements and parts. We can first define a mapping of a trefoil knot in three-dimensional space of regular points to the one in the space of placements and parts. Then I can move the e’s of the trefoil until three sets of two are joining at the placements shown in the diagram close to the center of the page.

- I have to have all the elements back in place after a series of moves to go back to the second level from the third. Only joinings could be rearranged in that the order of separation could be altered (up or down at the joining)
- To get back to 3-space I need to move from z-joinings to normal crossings. So these have to be made up of 1+, 1-, 2+,2-,3+,3- but they don’t have to be in the same arrangement as they were formed. Then it is possible we could end up with the unknotted loop.

- I can duplicate a point in the plane.
- If I have a plane of locations, I can’t move a location to another place.
- Consider a plane of locations of locations co-existing with the plane of locations
- 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).
- Then each point of the plane has a location and the location of the location of it’s duplicated point
- Suppose we have a set of connected locations which form a loop in 3-space. We can move from the plane to 3-space.
- 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.

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)

It’s important to have a perspective on things. Try not to forget how far you’ve come.

You can help people in more ways than the more obvious. Just keeping someone company or keeping someone’s mood buoyed up can be a great help to them.

You have got to keep healthy and keep those you love healthy too.

It’s a hard thing to admit sometimes, but you have to realize you have limitations. But that is the same for everyone. Those around you who are close to you realize you have limitations. But you have to tell people when your limits are reached. They will understand and respect your limits.

It’s very important to stay organized especially with much more to do.

This entry talks about realizing how important it is to help all the people you are close to, how to keep healthy by remembering everything you know and how to keep positive, encouraging yourself to accomplish everything you can.

When being at a multiple number of levels, one has to relate one level at a time. As well as you go forward there is always temptation to do too much, temptation to do too little. It takes a long time to get into good habits. But once you are in them, you will be much stronger and safer. Sometimes you have to adjust your habits depending on your circumstances. This is okay, it’s okay to have dynamic habits.