Abstract
This chapter discusses the various aspects of projectivities. Functions with domain and range consisting of points or lines play a fundamental role in many studies of geometry. The chapter discusses some additional functions of this type and some properties to be used in the subsequent development. In a desarguesian projective plane, a projectivity between distinct lines given by a composite of three perspectivities can be realized as a composite of two perspectivities. In a desarguesian projective plane, a projectivity between lines and can be realized as a composite of no more than two or three perspectivities. It is found that if an example of a desarguesian projective plane exists, which is not pappian, then necessarily it is not finite. In particular, if the example is an algebraic incidence basis with elements from a division ring where multiplication is noncommutative, the division ring is not finite. An example is constructed using an infinite division ring with a noncommutative multiplication. This will be the division ring of quaternions.
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