### Immeasurable, indivisible length

I am adding notes in on the idea of immeasurable, indivisible length. I will add explanations to make it clear.

I am adding notes in on the idea of immeasurable, indivisible length. I will add explanations to make it clear.

Shadows have area, but no height. The easiest way to give them extendable area is to let any dot of a shadow actually be a tiny, unseen square. This way it can be imagined that a shadow could be built by placing one square onto another. The smaller the dot, the less jagged are the edges of the shadow. We can have the small dot at the center of a diagram, and the dot can be as small as we like. It can always be reversibly mapped to a large square at the outside of the diagram. Expanded, so that we can see it is a square.

Shrink the dot until we reach an immeasurable state and at this time the dot become indivisible as well, as explained later. This state can be temporary until we find new rulers which are finer and allow further division. On the page with the ruler at 45 degrees the squares are shown at 0 and 1. I can’t measure anything below a*( root 2), where a is the side length of the square, for a division on a ruler is marked with a number and a number is considered exact, so anything within the division is considered not measurable by the ruler and not divisible also. Let us also use a ruler that is just the right size for the dot.

Change the ruler on an angle of 45 degrees to one which is horizontal by altering the square to be a diamond with diagonal length 2a, so as to make the diagram easier to imagine. Anything less than 2a I cannot measure with the coarser ruler, as the division has width 2a and we mark this with an exact number, so also 2a is not divisible.

But the finer ruler shows us that if we measure from the center we have thickness 2a if we place two divisions together. Any length equal to or greater than this I can measure with the coarser ruler’s divisions and I can make a unit length.

We can represent a diamond of immeasurable, indivisible length by the diagram with 4 e’s , one at each corner of the square. Let p-type dots be anywhere on lines from any e to any e.

Then anywhere I try to divide e—e I would be separating into e–p and p–e . These would be a different type of entity from e—e. Division makes sense if I can divide into two like items. So indivisibility is represented. e is an entity where I can overlap by sharing (shown in the theory of points and parts). 4e’s can be at one location e*e*e*e.

With the appearance of a finer ruler, the diamond becomes measurable and changes to the diagram with p(1),p(2),p(3),p(4). But there then will be again another immeasurable, indivisible diamond below.

Suppose I want to represent a diamond that is immeasurable and indivisible for all time. Then I can regard the last diagram on the page. We have a diamond with e’s at the corners as before but now the lines between the e’s are allowed to become ones with variable length. Then I can always make this smaller than any ruler.

The idea is that locations can have other placements. That is, I can have a location in the plane such as (0,0) and it is where it is.

But lets invent another plane where locations can be in other places relative to other locations. A location of a location or placement, for example 0( )0 is where (0,0) is usually placed relative to other locations.(use the special notation to indicate a placement)

A location with a placement may be written 0((0,0))0. The inner pair of numbers is the location and the outer pair of numbers is the placement of the location. Call this a second-coordinate.

We can start with all locations at their usual placements to give a plane of second coordinates a((a,b))b a,b are numbers.

The two overlapping shadows can be modeling two planes. In this case, the two different planes can be 1. the plane of locations( a plane of points with locations as subsets) 2. the plane of locations of locations or placements. Let Placements be fixed relative to each other so that they do not move with the usual plane as background. (for this reason they are given the outside positions)

Since locations can move relative to each other (we have another background since we have another plane) we can have something called a joining. It is two or more locations at the same placement.

We can have a geometry where selected pairs of locations switch by moving through the already existing locations to form joinings.