### Placements

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).

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