Horizon - Projective Geometry

Projective Geometry

The horizon was identified as a central line in graphical perspective by Girard Desargues in his Perspective of 1636. He wrote

The words on the level, level, parallel to the horizon are ... used to mean the same thing.
The words vertically, perpendicular to the horizon, and square to the horizon are also used to mean the same thing.

After introducing the terms "basis" and "subject", he continued:

When the basis of the subject is Level, the face which is turned upward towards the Sky is called the Upper Surface of the Basis of the Subject, and the other face which is turned down towards the earth is called the Lower Surface of the Basis of the Subject.

Standing on a floor plane where parallel lines converge toward a point on the horizon, one sees that the point of convergence on the horizon is a vanishing point, which geometers call a point at infinity. Since each point on the horizon corresponds to a convergence point for its own set of parallel lines, the horizon is a line at infinity that represents the various sets of parallel lines. The science of perspective led eventually to projective geometry. John Stillwell describes these developments in the chapter titled "Horizon" in his book Yearning for the Impossible (2006). After introducing parallelism through traditional axioms, he introduces coordinates, which are "a natural consequence of the parallel axiom", and the slope of a line. Then moving to visual perspective of a a tiled floor, he reviews the construzione legittima in 1505 by Jean Pèlerin . Projective configurations provide a context to discuss "incidence" as a geometric primitive. Stillwell makes a review of other pioneers in the field besides Desargues: Etienne Pascal, Abraham Bosse, and Phillipe de la Hire. He illustrates the projective configurations associated with Desargues and Pappus of Alexandria. The role of these configurations as both theorems and axioms is discussed. Stillwell also ventures into foundations of mathematics in a section titled "What are the Laws of Algebra ?" The "algebra of points", originally given by Karl von Staudt deriving the axioms of a field, is considered along with supporting work by David Hilbert (1899) and Ruth Moufang (1932). Concluding the chapter with four axioms for the projective plane that determine Euclidean geometry as well as the laws of algebra, Stillwell writes

This discovery from 100 years ago seems capable of turning mathematics upside down, though it has not yet been fully absorbed by the mathematical community. Not only does it defy the trend of turning geometry into algebra, it suggests that both geometry and algebra have a simpler foundation than previously thought.

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