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  • Projective Geometry has no Axiom of Equality.

  • Size is a Sum of Equal Sub-divisions of some Interval.

  • So Projective Geometry Cannot Measure.

 

Projectively Proscribed Items

transmission

                           tyre                          compasses

roller

Projective Geometry is Exclusive Geometry

It preserves Incidence, but not Absolute Measure.
In particular, Equality and Isotropy are not preserved,

and there is no Axiom of Curvature,

In consequence,
large and small are without meaning,
there are no units, equations, algebras, functions, or limits
(because these use equality),

and the staples of the Euclideo-Newtonian Dogma, namely,

Parallelity, Rectangularity and Circularity,

are not “self-evident” givens. They are meaningless, and in fact undetectable.

Scalars, Vectors and Tensors do not work.

Solid, Rigid Rotation is not available,
so neither are
Surfaces of Revolution.

Slip-free, Measure-Preserving, Rolling Operations, such as the rolling of a truck wheel on the ground, or printing from a cylindrical plate on to a plane, or belt-and-pulley transmission, cannot be had from strictly axiomatic geometric construction.

Forms cut by tools such as the twist drill depicted above also cannot be had from strictly axiomatic geometric construction.

Figures made by compasses cannot be shown to be circles.

The hyperboloid adumbrated above cannot be made by rotation around a line of a plane containing a hyperbola, because that hyperbola is not rigidly preserved during rotation - in other words, the hyperbola is not “imprinted”, or “mounted” like a painting on a wall, on the rotating plane, so cannot be transported by it.

 

 

Nineteenth-century mathematicians
saw a need to heal this breach
between Projective and Euclidean Geometries.
litbox

cumberland_hotel

It is unlikely that just three vanishing points can accommodate all the features of a real building like this one in London, although those indicated above seem to do it, if in a rough, ‘broad brush’ fashion. Perhaps the full set would cluster about these three.

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