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May 31, 2016

knot note 12

Filed under: knots — Rob @ 3:51 pm

knots12

knot note 13

Filed under: knots — Rob @ 3:50 pm

knots13

May 30, 2016

knot note 14

Filed under: knots — Rob @ 2:56 pm

knots14

knot note 15

Filed under: knots — Rob @ 2:54 pm

knots15

May 29, 2016

knot note 16

Filed under: knots — Rob @ 3:31 pm

knots16

knot note 17

Filed under: knots — Rob @ 3:29 pm

knots17

May 25, 2016

Introduction to the geometry of points and parts

Filed under: the geometry of points and parts — Rob @ 8:15 pm

Suppose we have two overlapping shadows from one light source. An example would be two strings placed one above the other with a light source from above, slightly to the right. There is another background light source so that we can see the shadows from above, projecting onto a plane below (see the second drawing above).

Just at the four corners of the intersecting shadows we have four points for which two points from each string are projected together from the light source . For each set of these two points of the shadow intersection I have two shadow edges in coincidence. I can label one of these corners-p. A point p  from one shadow edge in coincidence (+) with a point p from the other shadow edge is the same shadow point p. We can say this because if I turn off the light source no shadow points remain. This can be an understanding of the coincidence of shadow edges. Briefly, this can be written p(+)p=p. One point p, contributed from each string are the same p, in coincidence.

 

We can also have two overlapping shadows from the same object from two light sources. An example would be the overlapping shadows that result when a cup of tea is placed on a table (see the first drawing above) with two light sources above, one from the left and one from the right and again some background light so that we can see the shadows.

At the edges of the overlapping shadows we have one point from the projection of the first light source e and one point from the second light source e placed together on the shadow plane. We can write e(*)e =e(*)e.  This is to indicate that the combination e(*)e does not become e. It is unlike the coincidence of shadow edges since I have to turn off both lights for both e’s to disappear. So in this case  we have a “part” “sharing” (*) with another “part”, one “part” from each shadow. If I turn off one light source, one set of shadow parts remains.

 

We can imagine one way to put two points together by what we usually think of as coincidence. Since the result must be an object which also has no extent and so far we only have one object like this, a point, two points of two coincident lines can lead to a single point of coincidence.

To come up with something different from coincidence, let e be a different entity than a point. E is not a point, but something else-a “part” for which we have sharing in which e(*)e does not equal e, but stays as e(*)e. This is an object which still has no extent-since I am placing together two items of no extent-but has two parts. Two different parts, one from each light source.

But the two e’s are not otherwise distinguishable so I label the location e(*)e. I can label them e(1) and e(2) but they are in the same location of the plane so that they are not otherwise distinguishable, I am not concerned which light source generated which e.

Then using this model, we can create a geometry where we have two different types of entities which have no parts and no extent- p and e. To do this we have used sharing-something different from coincidence.

May 1, 2016

Deciding to add some notes

Filed under: the mathematics of position — Rob @ 1:04 pm

Below I have entered four pages of notes. I will add explanation to the diagrams over the next little while. Firstly, I will explain the notes using text and then I will add in the symbolism after that.

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