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Moulton Planes

Published online by Cambridge University Press:  20 November 2018

William A. Pierce
William A. Pierce*
Affiliation:
Syracuse University Syracuse 10, New York
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In 1902, F. R. Moulton (12) gave an early example of a non-Desarguesian plane. Its ‘points” are ordered pairs (x, y) of real numbers. Its “lines” coincide with lines of the real affine plane except that lines of negative slope are “bent” on the x-axis, line {y = b + mx}, for negative m, being replaced by {y = b + mx if y ≤ 0, y = [m/2]. [x + (b/m)] if y > 0}. A certain Desarguesian configuration in the classical plane is shifted just enough to vitiate Desargues’ Theorem for Moulton's geometry. The plane is neither a translation plane (“Veblen-Wedderburn” in the sense of Hall (7), p. 364) nor even the dual of one (Veblen and Wedderburn (17). It is natural to ask if the same construction is feasible when real numbers are replaced by elements from an arbitrary field.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 13 , 1961 , pp. 427 - 436
Copyright
Copyright © Canadian Mathematical Society 1961

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