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Utterly Fundamental Stuff (2)       — On Intervals —

Projective geometry and Euclid both measure intervals by intervals, but Euclid
  • assumes that an interval has an absolute size,
  • assumes that two or more intervals can be of equal size [video],
  • and, using sums of equal ‘units’, tacitly conflates number with geometry.

Projective Geometry and Euclid part ways here, because there is no exclusively geometric way to validate these assumptions, meaning that,

for Projective Geometry, size and equality do not exist,
and, as a direct consequence,
Projective Geometry can compare intervals only through incidence.

In Addition :-

  1. An interval is always the incidence of no fewer than three geometric elements (see sketch on the right).

  2. Because an element is not three things, but just one (namely, itself),
    • and an interval is three things (elements)
    • an element cannot be an interval.

  3. So we have that geometric elements are sizeless, because
    • all sizes are intervals,
    • and no geometric element is an interval.
Points haven't size (not even zero):
Lines haven't length (not even ∞):
Planes haven't area (not even ∞2).

As there are three elements – point, line and plane – there are three fundamental types of interval ...

  1. Interpunctual - between two points incident with a line
  2. Interlinear     - between two lines incident in a point and a plane
  3. Interplanar    - between two planes incident in a line

We note that, for every type, whenever one interval is formed, two are in fact formed.  For example, we may trace out one interval going right from A to B on the line l, shown on the left above, and may also trace out a second interval going left from A to B.  (The latter tracing works because a line has no ends.)

Elements have no ends, but can, by ‘two-of-the-same’ incident with one ‘not-the-same’, serve as ends – of intervals.

Intervals have ends, Elements do not.

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