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 make a one-to-one invertible map from an immeasurable diamond to a measurable one but with a finer ruler we can measure the immeasurable diamond. Then this itself can come from a map below to another level of immeasurablilty- a further immeasurable diamond, and so on.

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.