I am seeing five students so far in the Fall. One online and the rest in person at the Thornhill Community Centre Library. I have one grade 9, one grade 10, one grade 11 and two grade 12 Math students.

I am also working on two articles for two different mathematics educator’s magazines. One for Vector, based in B.C and one for the AMTNYS based in New York.

I have five summer students whom I am reviewing the previous grade and teaching ahead the next grade level for. One student is online and the rest are in person at the Thornhill Community Centre Library. I am giving homework to 3 of the five students.

Also I am working on an article for Vector Magazine, based in B.C. The article uses some of the math I created, which is shown at left under the category: Mathematics.

Hi my name is Rob Burchett. I have been tutoring Math, Physics and Chemistry in York Region and Toronto for over 18 years. I can tutor in person or online. I usually find a combination of these two works best. Currently, I am tutoring math for all grade levels and science up to grade 10 in person at the Thornhill Community Centre Library in Thornhill, Ontario.

Now, I have six students; a grade 11 Math student, two grade 10 Math students, a student whom I’m reviewing grade 7,8 and 9 Math for, an online student who is going into grade 10 and a grade 10 Math and Science 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.

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.

It seems the axiomatic foundation of Mathematics is made of concepts and thus subject to concept sharing. Then the axioms must have other levels. Then there is no irreducible axiomatic system.

A better foundation of Mathematics is seen when we look at the truths that concept sharing allows. Since we must have concept sharing we need to look at the Mathematics that comes from this.

As an entry into Concept Sharing it’s best to start with a specific example. Then let’s start with the concept of a point. In math this is the notion of an entity with no extent, or in Cartesian geometry the notion of something with position only.

Consider a teacup placed on a table with two lights from above. One from the left and one from the right. See below:

Now as seen in the overlap, two points of the shadows can take up the space of one point.

We have the familiar idea of two items just touching or resting upon one another as we see in everyday life. For example a book resting on a table, or books packed tightly together, on a shelf.

But there is something else that is hinted at here. That is that points could be multiple. A point could be in combination with multiple copies of itself. There could be multiple copies of the one point present within the space of one point.

Look at the turtle diagram below. Points are tangent to one another, if we represent points with circles. This represents points which are together, which could be separated.

But we could also have multiplicity of the individual circles, representing multiplicity of points.

The trouble is that this must fit into our current structure of mathematics as I am not adding any new notion in, merely clarifying. We already have this notion of a point as being solitary, we have to extend this.

The solution is that since this other case must be possible, it must be possible to take out the point that is there and replace it with the possibly multiple point. I say possibly because we allow the case where the point could be single too.

This necessitates a “lower” level of places which contain original places, as points are thought of as places in Cartesian geometry.

So we must have a plane or a space in which the ordinary places of Cartesian geometry take on other places of places.

A picture of this would look like below:

This is a movement of one piece of a doubled origin.

Not only the origin but each identified point of the new space can act as its own centre, moving away from its copy.

We could have a closed loop of these points all moving together. As well, this loop could be knotted.

Then this is also the entry into Concept Sharing as math concepts such as number, set, group, ect. Can all be thought of as point-like. That is to say they are all ideas which could have multiple expressions.

They are all exact and have no physical reality, they are just ideas.

Since they can all be multiple, there must exist lower concept spaces.

The concept sharing of a number:

Numbers are exact concepts. In the above case, we can think of them as the number of overlapping shadows at the center. They have exact boundaries and some way of showing we have two overlapping shadows there or three there, etc.

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) if the mathematical objects represented by the numbers had 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 numbers hidden as one we can not see or imagine them together. Yet our logic tells us this can be so.

Then to this end let us create another number dimension, a dimension 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 dimension 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 dimension, we can do this) and replace it with two new numbers 1(1) and1(2). These are together like the two shadows but do not form one number. The number then has a one-dimensionality to it as it is at least possible to separate it linearly.

Keep in mind that these numbers are different. They do not represent two obviously separate objects, but represent two mathematical objects hidden as one. Or you can think of the two number 1’s as fitting into each other.

Since numbers are exact we can consider the objects to be like points. Since something with no extent placed onto something with no extent would still have no extent but there could be two objects here. Also we could think of a combined single object of no extent( at the start) created form two other objects of no extent.

When we are numbering two mathematical objects which fit into each other, the objects are somehow different from each other and we give the two hidden objects two new numbers 1(1) and 1(2).

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. Then we need the concept sharing of a number to indicate how many concepts we wish to be present.

The twin prime conjecture:

There is a lower level of prime numbers using the concept sharing of a number. For examples: (1(1), (2(2), (3(3), (5(5),…I drop the (1), (2) notation for clarity and brevity.

Yet there is now a plane of numbers as shown below. With entries such as (3(5), (5(7), (5(11). These are partially shared prime numbers. We can make a squares as shown, with a gap of two on all sides, centered at (4(4). This connects the prime shared numbers.

We might also make copies of these prime’s associated with this square. Yet there is a countable infinity of these possible.

The shared prime pairs can’t be anywhere else in the plane but on the midline. Since we need to recover the ordinary prime’s, there can only be one copy of these. The rest of the copies are moved into the infinity that is available. Then in this way we are able to recover the prime numbers. Take away all the shared points except for one at each step. Then (1(1),(2(2),(3(3),(5,5),(7(7) can become 1,2,3,5,7…

Since I can make an indefinite amount of copies, the same gap must be possible infinitely as we go further up. Therefore the twin prime conjecture is seen. Also, we see how other gaps must be repeated too.

Starting with the diagram of the Culprit knot, I am trying to find a way of showing that I can form the unknot without increasing the crossing number.

I do so by placing the knot in placement space. The the locations are free to move in a space of locations of locations. We keep the locations connected in the knot as they were originally connected.

Specifically, they can move around in a loop along the path of the knot.

Once we decompose two crossings into a joining (alpha-beta’s), we can double up the knot diagram again. One diagram is still and the other I can move the locations around again along the path of the new knot diagram. Then I can decompose again, etc. This means I can move one alpha-beta past another.

I then decompose completely and look for another way to put the knot back together. This new diagram is obtainable from the original diagram by the usual Reidmeister moves.

Hide and seek is a fun game. But suppose instead of children, numbers would like to hide.

Suppose you are the number 1. Where could you hide? Numbers are so obvious. They go 1,2,3,4…etc. How could 1 hide?

But suppose I have two number 1’s. I could hide both together, by putting them together so that the two of them could not be seen, they just look like 1 number 1.

LIke so 1…..>1(x2)<……1.

Now wait you say, there aren’t two number 1’s!

But why not? We only have to take the usual 1 that’s there out of the way and two number 1’s can be where that 1 was.

To do this we need another level of numbers like a one story house has a basement. Then I could remove the number 1, like removing the first story, and still have the basement. Then build a two story house on top of the basement that’s left over. For some clarity on this see the notes below.

Now since two number 1’ s are playing hide and seek, how do we tell that it is not the case that we have the usual case of 1 number 1?

We can show that there are really 2, 1’s there by calling it (1(1) like so:

(1)(x2) = (1<….(1) =(1(1).

The left 1 is really together with the right 1, yet I need to work with both of these so both of these are shown.

So not only would 1 like to play this game but the other numbers would like to join in. 2,3,4,5,6,…etc would also like to hide.

So we can have (2(2), (3(3), (4(4)…

Then also we can have the other games that numbers like to play like addition (+) and multiplication (x).

So (1(1)+(1(1)=(2(2) and (2(2)x(3(3)=(6(6) for examples.

These numbers can all be on a line. The hidden number line:

But now 1 says I would like to hide with 2. Since part of 2 is 1 I could hide with part of 2. Then this could be labeled (1(2) and it would be equal to (2(1). We just have a smaller number hiding with a larger number. 1 and 2 are in the exact same place.

But where are these hidden numbers on the hidden number line?

The answer is that these numbers are on a number plane. A plane of numbers that looks like the diagram below:

On the midline we have the usual hidden numbers.

Then we can ask, how can these hidden numbers play the usual games of numbers? (addition and multiplication)

Let’s investigate!

(1(2) + (1(2) =(2(4)

But also,

(1(2) + (2(1)=(3(3). So it appears that in order for (1(2) to play (2(4)=(3(3). But there’s nothing to say this isn’t okay, since there are no rules for these partially hidden numbers-yet!

Goldbach’s Conjecture:

On June 7th, 1742 Christian Goldbach wrote a letter to Leonard Euler In which Goldbach guessed or “Conjectured” that every even number is the sum of two prime numbers. So for example: 16=3+13.

Let’s look at the guess in terms of the hidden numbers!

We can think about new types of numbers (1(1)(1(2)), (2(1)(2(2)), (3(1)(3(2)),..These are lower numbers than the naturals, the idea of concept sharing being deeper than the usual idea of concepts. The notation is meant to show that the numbers are together in the number space, fitting into one another or hidden as one. I show a revelation of one number “peeking out” from concept sharing with another number. Shown more clearly here: Like so (a)x2….(a<……(a) leading to (a(a). For clarity and brevity I drop the (1),(2) extra designation from now on.

There are other numbers such as (2(3), (3(4), (3(5),… We can think of this as partial sharing. For (2(3) think of three dots colored red. We can have two blue dots overlapping with two red dots forming two purple dots. I say that to give the idea, numbers are not colored dots. Numbers are the exact concept of amount or position only. So that (2(3) can be thought of as two sharing with three and (3(2) can be thought of as three sharing with two. In the first case 1 is left over and in the second case 1 is an extra number. These are the same concept though so (2(3)=(3(2).

Consider the even shared numbers (4(4), (6(6),…

We can break them down into a sum of two prime shared pairs such as (4(4)=(2(2)+(2(2) if we define the other types of shared numbers with dissimilar basic numbers to fill in the rest of the possibilities of a number plane. They can form products too.

Some numbers have more than one decomposition. Such as (16(16)=(13(3)+(3(13) and (16(16)=(11(5)+(5(11).

Now think of the even shared numbers as being created from the primitives. I designate them (4(4) … but these numbers are not created from the usual numbers on the number lines, as these are lower numbers, these lower numbers; (4(4)…. must come from somewhere else. These even shared numbers must go on indefinitely, as to lead to the numbers 1,2,3,…at the higher level . Then the natural numbers are all at a higher level from these numbers.

For multiplication, in the usual number system we have prime numbers. We can look for the lower level of prime numbers. Suppose I build a lower level of primes from the usual prime numbers by concept sharing two prime numbers. For example (7(3)=(3(7).

Then starting at (4(4), we can ask, how is (4(4) created? (4(4)=(2(2)(2(2). (4(4) is an even shared number so there is a division by (2(2) possible. This can be the definition of an even shared number. When I divide both sides by (2(2) I have a shared number on both sides.

Then also (6(6)=(2(2)(3(3). Also (6(6) =(2(2)(5(1) also as (5(1)+(5(1)=(5(1)+(1(5)=(6(6).

But we wish a prime factorization. For example (2(2) is not equal to (2(1)(1(2) as this is the shared number counterpart to not allowing a prime to be expressed as a product of 1 and itself. (this is a decomposition but not a prime factorization).

So in the shared number system we need a new definition of prime factorization. Let’s look at some more examples.

(6(6)=(2(2)(4(2) but (4(2)=(4(1)(1(2)=(2(1)(2(1)(1(2) which is does not work as I can break it down further into (6(6)=(2(2)(2(1)(2(1)(1(2)= (2(1)(2(1)(1(2) +(1(2)(1(2)(1(2)=(4(2)+(1(8)=(5(10). 5+10=15 not 12.

Also (8(8) is not equal to (2(2)(4(4) as (2(2)(4(4)=(2(2)(4(2)(1(2)=(2(2)(2(2)(2(1)(1(2). This can be further broken down to (2(2)(2(1)(1(2)+(2(2)(1(2)(1(2)=(4(4)+(2(8)=(6(12). This is not allowed. We find (8(8)=(5(3)+(3(5)=(2(2)*(3(5)=(2(2)*(5(3)=(10(6), and 10+6=16=8+8.

It turns out we do not allow for example, (12(12)=(2(2)(11(1) as this cannot appear as a prime factorization.

And so on. Each even shared number must be a multiple of (2(2) and another prime shared number.

We can prove this. Suppose I have an even shared number (2n(2n) then this divides into (2(2)(n(n) and (n(n) is our starting number. Then we can see if we reduce the left number by one as we increase the right number by one we can eventually reach, (2n(2n)=(2(2)(1(2n-1) This is not allowed since 1 is seen as a factor. When we look at other possible factors we see there are three possibilities. Either we have (2n(2n)=(2(2)(p(1)(p(2)) where p(1) and p(2) are prime numbers-this is allowable or we reach (2n(2n)=(2(2)(m(p)=(2(2)(p(m) where p is prime and m is composite, or (2n(2n)=(2(2)(a(b)=(2(2)(b(a) where a and b are both composite. In the second case we can break the product down to (2(2)(p(c)(1(d) where cXd=m. In the second case we can always break the product down to (2(2)(a(e)(1(f) where eXf=b, then further break a down to (2(2)(p(1)(e)(p(2)(1)(p(3)(1)….(p(l)(1)(1(f). Where p(1),p(2),p(3)….p(l) is some finite sequence of primes. This factorization is not allowed since I can use switching to create a contradiction.

Every even shared number can be divided by (2(2). The other factor must be a prime shared number as we need this to work in the shared number system. For example (12(12)=(2(2)(6(6). (6(6) must separate into two prime numbers, the left number moving up by one and the right number moving down by one. Otherwise (12(12) can’t appear (we assumed it was there). It must exist since I need to have 12. I can now take (12(12) out and replace it by 12. So we must have a decomposition with (2(2)(a(b) with a, b both being ordinary prime numbers.

You see as we now have (4(4), (6(6), (8(8),…. we can now go on to create the usual natural numbers. Divide (4(4) by (2(2) to get (2(2) and divide (2(2) again by (2(2) to get (1(1). The we can go forward by dividing (6(6) by (2(2) to get (3(3) and (8(8) by (2(2) to get (4(4) etc. Then all this sequence can lead to the natural numbers. I can replace (1(1) by 1, (2(2) by 2, (3(3) by 3 etc.

This also demonstrates the equality of every primitive decomposition as well (which the lower level was hinting at) since for any even shared number there may be more than one decomposition. We go back to form (16(16) and then can separate once again to find another decomposition. Here we find two, (2(2)(13(3) and (2(2)(11(5)

Then if we look at one half of these binary decompositions, that is, for example, look at 16(16)=(13(3)+(3(13) and see 16 =13+3 we can see Goldbach’s conjecture is true. It was just a part of a deeper understanding of numbers.

Notes:

As an entry into Concept Sharing it’s best to start with a specific example. Then let’s start with the concept of a point. In math this is the notion of an entity with no extent, or in Cartesian geometry the notion of something with position only.

Consider a teacup placed on a table with two lights from above. One from the left and one from the right. See below:

Now as seen in the overlap, two points of the shadows can take up the space of one point.

We have the familiar idea of two items just touching or resting upon one another as we see in everyday life. For example a book resting on a table, or books packed tightly together, on a shelf.

But there is something else that is hinted at here. That is that points could be multiple. A point could be in combination with multiple copies of itself. There could be multiple copies of the one point present within the space of one point.

Look at the turtle diagram below. Points are tangent to one another, if we represent points with circles.

But we could also have multiplicity of the individual circles, representing multiplicity of points.

The trouble is that this must fit into our current structure of mathematics as I am not adding any new notion in, merely clarifying. We already have this notion of a point as being solitary, we have to extend this.

The solution is that since this other case must be possible, it must be possible to take out the point that is there and replace it with the possibly multiple point.

This necessitates a “lower” level of places which contain original places, as points are thought of as places in Cartesian geometry.

So we must have a plane or a space in which the ordinary places of Cartesian geometry take on other places of places.

A picture of this would look like below:

This is a movement of one piece of a doubled origin.

Not only the origin but each identified point of the new space can act as its own centre, moving away from its copy.

We could have a closed loop of these points all moving together. As well, this loop could be knotted.

Then this is also the entry into Concept Sharing as math concepts such as number, set, group, ect. Can all be thought of as point-like. That is to say they are all ideas which could have multiple expressions.

They are all exact and have no physical reality, they are just ideas.

Since they can all be multiple, there must exist lower concept spaces.

The concept sharing of a number:

Numbers are exact concepts. In the above case, we can think of them as the number of overlapping shadows at the center. They have exact boundaries and some way of showing we have two overlapping shadows there or three there, etc.

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) if the mathematical objects represented by the numbers had 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 numbers hidden as one we can not see or imagine them together. Yet our logic tells us this can be so.

Then to this end let us create another number dimension, a dimension 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 dimension 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 dimension, we can do this) and replace it with two new numbers 1(1) and1(2). These are together like the two shadows but do not form one number. The number then has a one-dimensionality to it as it is at least possible to separate it linearly.

Keep in mind that these numbers are different. They do not represent two obviously separate objects, but represent two mathematical objects hidden as one. Or you can think of the two number 1’s as fitting into each other.

Since numbers are exact we can consider the objects to be like points. Since something with no extent placed onto something with no extent would still have no extent but there could be two objects here. Also we could think of a combined single object of no extent( at the start) created form two other objects of no extent.

When we are numbering two mathematical objects which fit into each other, the objects are somehow different from each other and we give the two hidden objects two new numbers 1(1) and 1(2).

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. Then we need the concept sharing of a number to indicate how many concepts we wish to be present.

For Concept Sharing we needed to start with sharing numbers. The idea is to form sharing concepts. For this we need these new numbers, to be specific so that we can start with certain mathematical systems.

In mathematics, concepts are mental constructions. They are ideas like shadows with boundaries. They can be thought of like points. The foundation of mathematics is based on concepts. Here then we need to find a new, extended foundation.

We can concept share a point since two items of no extent will still have no extent, but there could be two items here.

So similarly numbers, as we have seen, points, sets, groups, ect. Can all concept share.

In order to do this we must remove the concept which is present initially and replace it with 2 or more sharing concepts. Since it is possible to share concepts there must exist a more underlying concept space.

The two sharing concepts must be different in some way which we can specify based on the nature of the concept itself.

Then an infinite level concept space can form as I can continue into the next level of the concept and so on. One may utilize as many levels as is necessary.

Additionally there can be a finite number, an infinite countable number of an infinite uncountable number of sharing concepts, as this is the understanding of numbers.

Abstract: Here I introduce concept sharing. In uncovering extended space, I show a new way of understanding knots.

As an entry into Concept Sharing it’s best to start with a specific example. Then let’s start with the concept of a point. In math this is the notion of an entity with no extent, or in Cartesian geometry the notion of something with position only.

Consider a teacup placed on a table with two lights from above. One from the left and one from the right. See below:

Now as seen in the overlap, two points of the shadows can take up the space of one point.

We have the familiar idea of two items just touching or resting upon one another as we see in everyday life. For example a book resting on a table, or books packed tightly together, on a shelf.

But there is something else that is hinted at here. That is that points could be multiple. A point could be in combination with multiple copies of itself. There could be multiple copies of the one point present within the space of one point.

Look at the turtle diagram below. Points are tangent to one another, if we represent points with circles.

But we could also have multiplicity of the individual circles, representing multiplicity of points.

The trouble is that this must fit into our current structure of mathematics as I am not adding any new notion in, merely clarifying. We already have this notion of a point as being solitary, we have to extend this.

The solution is that since this other case must be possible, it must be possible to take out the point that is there and replace it with the possibly multiple point.

This necessitates a “lower” level of places which contain original places, as points are thought of as places in Cartesian geometry.

So we must have a plane or a space in which the ordinary places of Cartesian geometry take on other places of places.

A picture of this would look like below:

This is a movement of one piece of a doubled origin.

Not only the origin but each identified point of the new space can act as its own centre, moving away from its copy.

We could have a closed loop of these points all moving together. As well, this loop could be knotted.

Then this is also the entry into Concept Sharing as math concepts such as number, set, group, ect. Can all be thought of as point-like. That is to say they are all ideas which could have multiple expressions.

They are all exact and have no physical reality, they are just ideas.

Since they can all be multiple, there must exist lower concept spaces.

The Concept Sharing of a Number:

Numbers are exact concepts. In the above case, we can think of them as the number of shadows at the center. They have exact boundaries and some way of showing we have two there or three there, ect.

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) if the mathematical objects represented by the numbers had 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. Two or more numbers can be hidden as one since natural numbers represent exact positions. Similarly with two numbers hidden as one we can not see or imagine them together. Yet our logic tells us this can be so.

Then to this end let us create another number dimension, a dimension of number of numbers. This must already exist since it should be possible to put two or more numbers together at a beginning. Let the usual case be that the number of numbers is only 1. But now let us expand into the next dimension 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 dimension, we can do this) and replace it with two new numbers 1’(1) and1’(2). These are together like the two shadows but do not form one number.

The two numbers together can be notated ((1’(1)((1’(2)). 1’(1) is “peeking out” from behind 1’(2). Shown by the use of a half parentheses. Seen more clearly here: (a(a).

Keep in mind that these numbers are different. They do not represent two obviously separate objects, but represent two mathematical objects, also concept sharing, hidden as one.

The objects are concept sharing as well and are somehow different from each other. We give the two hidden objects two new numbers 1’(1) and 1’(2).

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. Then we need the concept sharing of a number to indicate how many objects we wish to be there.

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 place two together using two new numbers 0’(1) and 0’(2) identifying that we have two points. (0 is indicating an origin)

An object of no extent created together with another object 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 interest of the usual plane can be removed and we can replace it with a “sharing” of two points. So that we have a subset of sharing 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.