Hi my name is Rob Burchett. I have been tutoring Math, Physics and Chemistry in York Region and Toronto for over 15 years. I can tutor in person or online. I usually find a combination of these two works best. Currently, I am tutoring in person at the Thornhill Community Centre Library in Thornhill, Ontario. Now, I have two students in grade 12 Calculus, a grade 10 Math and grade 5 English and Math student. You can find some information in regards to my tutoring at the left under the category: Tutoring.

I’ve had a lot of success tutoring many students over the years. In some cases I have been able to take students who are failing and raise their grades into the 90’s. I have tutored regular high school students, gifted students, students with learning disabilities and adult students. I can tutor a student as he is taking a course or plan ahead of time for a course he is going to take.

My usual fee is $20/hr. I would be happy to provide references. I can be reached by phone or text message at 647-218-1407 or by email at robburchett1@gmail.com.

I apologize for the current construction going on at my website and hope to have it back to normal shortly.

I am seeing three students of my own this semester. A grade 5 student for Math and English, with my wife Angela. Also, I am seeing his brother, a grade 11 student who is taking calculus ahead of time. I saw him last year for grade 11 and grade 12 Math also for grade 10 English, Civics and Religion. He is doing very well.

I also see in person a student in person and online who is taking grade 10 math. I helped prepare him for this semester, one semester back. He is getting 90% in Math now. The plan is to see him in the summer months for both Math and English.

I have two students through Brainiacs Online, a grade 9 student and a grade 12 student. I am preparing the grade 12 student for Advanced Functions and also Calculus and Vectors for next semester.

I am working with my own students, tutoring grade 10 and 11 Math in person and online. I also tutor online with Brainiacs Online, online tutoring grade 10, 11 Math and grade 11 Physics. I am looking for new students in grade 9,10 or 11 Math and Physics.

Abstract: Here I introduce concept sharing. In uncovering extended space, I develop new ways of understanding knots.

Overlapping shadows:

An interesting notion can come about from something observed in nature, the overlapping of shadows. Consider a cup of tea placed on a table and lit with two lights from above, one from the left and another from the right, as illustrated below:

Three areas of shadows are formed. The one in the center is the overlapping of two shadows.

The concept sharing of a number:

A number is a mathematical concept. In the above case, we can think of the number of shadows at the center. The shadows have exact boundaries and some way of showing we have two there or three or more there.

Numbers are further like the shadows in that they are ideas and not physical objects so they could exist as two or more, hidden as one, instead of the usual singular case as long as the numbers referred to two different mathematical objects.

Then borrowing from the notion of overlapping shadows we should be able to hide numbers together and they would be “two hidden as one” as well (concept sharing). Mathematical objects represented by the numbers have the same boundaries, like the shadows at the center.

Other than the further darkness of the overlapping shadows, we cannot see or imagine that there are two separate shadows there. Similarly with two objects hidden as one we can not see or imagine them together. Yet our logic tells us this can be so.

Let us create another number level, a level of number of numbers. Let the usual case be that the number of numbers is only 1. But now let us expand into the next level and allow the number of numbers to be 2.

So for example with the number 1, let us take away the original number 1 (since we have another underlying level, we can do this) and replace it with two new numbers 1’(1) and1’(2). These are together with the same position as 1 and are like the two shadows but do not form one number since they refer to two different mathematical objects.

The objects are somehow different from each other. We can extend concepts to form a concept of concept space. It must be possible for mathematical concepts to extend into such a space. The necessity of being able to remove a concept and replace it with any number of hidden concepts makes such a space necessary. The objects represented by the numbers are concepts too. Then since the objects allow this extension two or more concepts can move to two or more concept of concepts. We give the two hidden objects two new numbers 1’(1) and 1’(2).

In the case of the shadows, we can tell that there are two shadows there and give one the number 1 and the other the number two.

In the case of mathematical objects there is no external way of telling how many objects there are, previously it was assumed it was only one. We can state how many we wish at the onset thus fixing a certain mathematical system.

Thus we need the concept sharing of a number to indicate how many objects we wish to be there and that we are aware that it could have been a different number.

A new plane:

Points are also exact concepts. In the Euclidean plane they are places, with the notion of no extent, in the plane. We should be able to hide two together and number them using the two new numbers 0’(1) and 0’(2) identifying that we have two points. (0 is indicating an origin)

Two hidden objects of no extent would still have no extent- but there could be two objects here, under another mathematical system.

The two points 0’(1) and 0’(2) can be different by first uncovering a new place dimension, a place of places. This must already exist because there must be some way to have two points exist together and still be two points.

In a similar way as we uncovered the new number dimension (the number of numbers) we can uncover the new place dimension.

Take the original point out (we can do this since we have a new underlying dimension of place, a place of places) and replace it with the two new points. This can be done for the whole plane of points.

That is, there is nothing special about the origin, so each point of the usual plane can be removed and we can replace it with a “sharing” of two points. So that we have a whole plane of doubled points co-existing with a plane of places of places.

One of the new points can be fixed, while the other one is capable of “shifting” away in this new dimension of place. In this way these two can be different. Then all of the sharings in the new plane can become new origins-one point being fixed while the other point is capable of shifting away.

Then labeling the points 0’(1) and 0’(2) we can “shift” them apart, keeping 0’(1) fixed and moving 0’(2) into the plane of places and of places of places. This can be done by shifting space itself (the space of one point). See below:

We are shifting away from an origin in place of place space. This is not a movement as usual as these are not two distinct points in the usual plane indicated by a movement apart. This is the movement of one part of a sharing pair away from another by shifting it over a co-existing space of places of places.

In order for two different sharings to move away from one another, we have to bring 0’(1) and 0’(2) back together. Then we can treat the sharing as a single item, similar to the usual concept of a single point.

Then have another sharing a small distance away from the usual sharing. Then the sharings can move apart. The sharing takes up doubled locations as it moves away.

Then there is this combination between what we already know about space and this new knowledge for each individual sharing.

Then when we have two points at two places of places. We can have a unit of shift between them which is equal to the unit of distance in the usual plane.

Measure shift (another type of distance-this can be thought of as negative distance) between the center and 0’(1) and 0’(2) if we expand out in space. Also we can have the same value considering distance or shift (if we keep the unit the same for distance and shift). Then we can have a “mixed” space with both usual space and new space overlapping. (Some points doubled and separated, other points still together).

A connected and possibly knotted loop:

We can create a loop of these doubled points in places of places space in three dimensions of place and place of places space. Let the basic topology of this be the same as it is in the usual space. That is, we allow space to shift by isotopy, to expand or contract, to form crossings, etc. As we have another level of place, let it act basically as another dimension to space.

We can have a basic topology of space in places of places, mirroring the topology of points or closed curves, in the usual space. Additionally, there are other things we can now do.

I can shift the copy of the points away from the original loop. In this way I can compare two loops which might be knotted. Start with a knot in the space, form the doubles, then shift one

copy . This may then be manipulated to see if we can form a congruence between the moved copy and the inverse or mirror image knot. If these can be shown to be congruent then so are the original knots. Also we can seek other diagrams with a different order of crossings/joinings.

Creation of shifted diagrams:

Diagrams have crossings in R^3 or S^3 (usual space). Consider the trefoil.

Label these crossings Z(1), Z(2), Z(3),ect. The locations of Z-crossings/sharings (mixed space).

Place a, b at each Z. An a or a b is an extra moving point (which can be called a “part”-as in part of a whole) which keeps track of the crossing/sharing.These can come from the surrounding space..

A moving crossing is now a crossing/sharing as we are capable of having 2 parts at a vertex with a and b always moving along with the moving crossing/sharing.

There is a new type of Crossing/sharing possible. A Q-type. This is a crossing/sharing of two points which do not cross in the original diagram. Additionally r and s parts which make up the joining can also pass through each other. This is not allowed for a and b parts. So the knot keeps its knottedness.

Z(1), Z(2), ect. Are representing moving crossings/sharngs in P^3 where “placement space” is the name I give the larger space.(places of places). But in the shifting diagram they have some freedom. They can travel along as they are moving sharings/crossings, reachable through shifts of the diagram (that is, all shifts are reversible) or they can stop at a specific placement and the a, b pairs (joinings) example (a(1)(r(1)) can shift on forward by rotation. This is still reversible as I can get back to this Z, as I can reverse from forward shifting. At the end, after I go all the way back I come back to the same location.

Then all these diagrams are equivalent and complete as long as we don’t cut the diagram.

Then R0, R1,R2,R3, have their equivalents as well. (call them S0,S1,S2,S3) If I create a new sharing, not already present at the beginning as a crossing, I call it a Q-sharing. This is a different kind of crossing/ sharing from a Z-sharing. I have rotation of the locations, creation of labels, joinings, movements of joinings and labels through sharings.I can move a joining through a sharing too. And that’s all (this is a complete list of what’s possible).

Let there be another diagram D(2) in R^3 and we wish to compare this to the original diagram. Move it to mixed space. Concept share its connected parts and one part of it shifts off. If we can make a congruence between this and the other diagram in mixed space then the two diagrams in the usual space are also congruent.

So we need to look for a match of the labels, joinings when we simplify the diagrams.

If they are the same, then the information should be contained in one diagram. That is, shifts of one diagram should be able to produce the second one. So we need to look at one diagram to see if we can produce another(using all shifts available).

The new shifts are rotation by S0 (which isn’t usually considered) and movement of labels and joinings through sharings.

So for the diagram set for the Trefoil:

Start with the Trefoil positively oriented in R(3) or S(3) fig1

Realize the new space with concept sharing of a point (creation of parts, placement space)

Label the sharings/crossings with a(1),b(1), Z(1),…ect. fig.2

Double up the parts, move one copy away.

Move from positive space to mixed space (presence of placement space). Creation of Q sharings/crossings. Z(a,b) exists in both spaces Q(r,s) exists in only mixed space.

S0,S1,S2,S3 can be made leading back to a final diagram. Fig. 5. I can put back the joining pairs to reform the sharings only in a specific way

There is an invariant of the Trefoil shown. Fig. 6a. We can compare it to itself using S0 looking to see if we can have an inverse or a mirror image of the knot.

There may be a more direct way to compare two knots. Fig 6b.

We can look for a different shape as joinings can move past other joinings.

In the Trefoil, in particular, there are no other shapes, as well we cannot change any orientations, as shown.

Therefore the Trefoil is knotted and chiral.

In conclusion, we can use concept sharing to understand knots better. We can compare any two knots using new shifts of space in an uncovered dimension of place of places.

In concept sharing we can state that there can be any number of concepts sharing a concept of concept space. Yet this can be specified before hand or it can be allowed to be two different numbers. This is because there is no way of telling from the outside, how many concepts are actually present. Unless we are told or told that there are more than one number and told these numbers.

If there are two numbers of concepts of concepts we can have an equivalence of numbers. Given a number of number of numbers. (2).

In this sense the “false” equations of mathematics ie. 1=2, 3=5, etc. have a solution using concept sharing.

It is only an illusion that each day is basically the same, that we go through a series of routines to make our world. Each day is vastly different! Try to see the novelty and variety we are surrounded by. The true richness of the world. To function in all this try to find anchors which you can create and hold onto.

With the true richness of the world uncovered, what need is there for what is usually thought of as richness? Here, then one can live in true richness.