### working in the spring term

I am working at the Prestige private school located in Richmond Hill. I am tutoring five groups of four students from grade levels 3 to 11 in math.

I am working at the Prestige private school located in Richmond Hill. I am tutoring five groups of four students from grade levels 3 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’.

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 placement 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.

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

The double bracket is meant to indicate that this is a one step ‘higher’ level of location. ‘higher’ in the sense that now locations have a location called placement. For brevity the location of a location is called a placement. 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 location. This means there has to be some other mathematical objects at the points where we have locations. This can be the placement of a location, a( )b. Two mathematical objects, the location and the placement 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 here.

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 placement. 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 placement 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 placement 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’ in relation to the plane of other locations.

To do this I put a plane of ‘placements’ a( )b in so that every location (a,b) has a placement a( )b. We stop at only two levels for now. So every location of the plane has a ‘location’ itself. These two together, the placement 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 placements. Underlined in red are locations, moving around 90 degrees clockwise through the coincident level of placements. 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 placement 0( )0 to the placement 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 placement. 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 placement 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 placements 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 placements 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.

In the self-joining loop-‘joining’ is where two parts are together at a placement on one on each of two intersecting strands-the markers of the original crossing a and b can separate and move around the loop. There is a flow of parts which can exist. They can flow around the loop and through and past a and b.

A Z-joining type is then created when a and b are separated.

If we wish to change the loop to the form of a circle, we have to mark where we are going to do this with a 1+ and a 1- and take two parts from the space alpha and beta and place them at Z.

Then when Z is separated into D, alpha and beta move along with D and eventually come out when the loop disappears and this place is marked.

Its a problem that some people have that the speed that they are going, either too fast or two slow can get them in trouble. Its important to build character traits other than strength. Kindness, honesty, friendliness, reliability, a hard working attitude can all help you when your strength fails you, which is inevitable from time to time.

A high mood can hinder you, if you let it. If you concentrate on what you can do to help you, rather than just feeling good-this can help.

Here I can define a ‘constant part’. I can place a location on a line of locations. In a space of placements and locations. I move location a, to the ‘part-line’-a line of locations in the space of placements and locations.

The idea is to let the constant part keep it’s placement while other parts on the part-line are in motion. Two parts or locations, in the space of placements and locations can occupy the same placement.

Here, in pictures at the top is the idea of starting with a loop crossing itself in regular space to a conversion to placement and location space. a and b, two constant parts and then the idea that I can separate a and b.

Stress is the push that the world puts on you, in the many different ways it can. If you push back instead of just allowing yourself to be pushed, you won’t be pushed down, for one thing. This is easy to see as a cause of depression. Pushing back until you feel you are pushing too much is the way to go. This is not aggressive behavior. Then the only other thing that can happen is unexpected things-but good unexpected things, if they are in excess will balance off bad unexpected things.

These are my notes from 2004. I am editing them with a pencil to put in the missing ideas. Here the setting is a space of placements and locations at the same points of a plane. I did not originally make this clear. Calling a ‘part’ a location which is free to move in a space of placements or locations of locations.

It is a fact that some people have a difficulty with a connection to reality, due to problems with thinking, feeling or mood. This does not mean that they can’t still do well. It is one way to function to think of life as a sort of game. Decision making can be a difficult matter and also being able to push back when the world pushes at you. This is only natural, not aggressive behavior.

That strength has its limits is not usually talked about. But since anyone can become sick at anytime or an unforeseen accident can occur, it is best use strength when you have it. It is only a temporary thing. Still it can be built over time, so that it can be used now and then, it is very useful to build it.