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Elements

 
 

The three irreducible elements of geometry are Point, Line and Plane.

They are unary qualities.

Unary”, here, signifies that geometric elements have exactly one quale each.
These quales differ, each from every other.
Each can stand alone.
Elements of a kind are identical, but not unique.

 
 
  • A Point is just a place—without size of any value (such as, say, zero) whatsoever
  • A Line is just extension—without ends, or place, or sides, or thickness of any value (such as zero) whatsoever, or length of any value at all (such as ∞)
  • A Plane is just spread—without edges, or thickness of any value (such as zero), or area of any value (such as ∞²)
 
 

Elements are unquantifiable, and sizeless,
because
none is, of and by itself, a calibrated interval. Thus -

A point cannot be made more - or less - of a place than it is;
a line cannot be made more - or less - of a line than it is;
a plane cannot be made more - or less - of a plane than it is.

Each instance of an element is a single, whole, unbroken thing,
Projective Geometry is for this reason sometimes called, “Synthetic” Geometry,
to bring out the (stark) contrast with so-called, Analytic” Geometry,
which (because to “analyse” is to “break into bits”) is very much concerned with broken elements,
and with counting the bits.

Thus, also, the common intuition that lines and planes are composed of points is mistaken,
since it implies that some elements are sums of other elements.
However, while sums are quantities,
elements are not
.
It is simply that there is nothing available to sum.

For example, since points are sizeless places,
they cannot be made to extend as a line
by being serially abutted (added).

Accordingly, neither measurement, nor its concomitant, metrication, is native to geometry.

Because units have (equal) size, and geometric elements do not, there are no units.

This means that Projective Geometry cannot specify absolute sizes—of anything.

 
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