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line as a dashed line in Euclidean Geometry may imply MaxwellEquations PostComment #2015 of 5th ed
Recently I was doing a correction of Euclidean Geometry, of their gigantic contradiction at the heart of Euclidean geometry where they had just two items, a point and a line, yet a line cannot exist with length if geometry has just point and line. What they missed was "empty space between successive points". That empty space is picked up by realizing the borderline between finite and infinity as a microinfinity of empty space between two successive finite points. So that in the 10 Grid where 10 is pretend to be the last largest finite number we have a microinfinity of 1/10 or .1, so that between 0 and .1 is empty space of just infinity numbers. A number like 0.005 is an infinite number, an infinite point in the 10 Grid.
So, in True Geometry, we can have a solid line, or solid line ray or solid line segment. But we can have some special also, due to empty space between successive finite points. We can have a dashed line segment or a dashed line ray or a dashed line. So for example the line from 0 to 10 on the xaxis in 10 Grid can be a solid line where it connects 0 to .1 to .2 to .3 on up to 9.8 to 9.9 to 10. But we can also have the dashed line where it connects 0 to .1 then leaves ..1 to .2 empty, then connects .2 to .3 then leaves .3 to .4 empty etc etc. Now I hope I have enough points so that the last solid dash is 9.9 to 10. Now in Old Fake Geometry with their contradiction of point and line of length, their lines were all solid lines. In True Geometry where you have three entities of point, empty space, line, then a line can be either dashed or solid, depending on the total number of points, for we cannot leave a point unconnected. For example the line segment 0 then .1 then .2 can only be a solid line segment for a dashed line would leave one point unconnected. Now it appears to me that such a new concept of dashed line in Geometry makes little difference to Geometry itself. However, the dashed line in Physics is likely to be a huge new understanding of how particles and waves travel in space. For example, the dashed line is very much like a Compression or longitudinal wave where the empty space is a rarefaction. And, the dashed line can serve as a transverse wave where the empty space is a sine wave. Now ever since 1930 with Quantum Mechanics well established, it is remarkable how little of physics rubbed off onto the mathematicians. Since QM was so successful in the 20th century that one would think the smart people of mathematics would wake up and say since physics is quantized, Space must be quantized and not continuous. So that math had such silly notions of continuum, countinuous, and always a third point between any two points. As if the mathematicians of the 20th century went on a Rip Van Winkle century long sleep, never learning anything from physics. For if they had learned from physics, they would have known that point and line need the third item of empty space (the quantum) between one point and its successor point. So, I bring up this dashed line for I sense that it will come handy with the Maxwell Equations, even though it be only a curiosity for math. AP 
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