Abstract
A belineation is a bijection of a plane that preserves betweenness. This chapter shows that every belineation on a Pasch plane is a collineation, and explores the interactions between belineations and segments, rays, lines, sides of a line, angles, and triangles.
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Notes
- 1.
We hope the reader is not offended by this rather odd name; believe it or not, we bandied about some other names that were even stranger—such as betweeneation.
References
Bachmann, F.: Aufbau der Geometrie aus dem Spiegelungsbegriff, 2nd edn. Grundlehren der mathematischen Wissenschaften. Springer, Berlin (1973). ISBN 978-3642655388
Behnke, H., Bachmann, F., Fladt, K., Kunle, H. (eds.): Fundamentals of Mathematics, vol. 2. MIT, Cambridge (1974). ISBN 978-0262020695. Chapter 3, Affine and Projective Planes, R. Lingenberg and A. Bauer; Chapter 4, Euclidean Planes, J. Diller and J. Boczec; also Chapter 5, Absolute Geometry, F. Bachmann, W. Pejas, H. Wolff, and A. Bauer; tr. by S. Gould
Birkhoff, G.D., Beatley, R.: Basic Geometry, 2nd edn. AMS Chelsea Publishing Co., New York (2000, Reprint). ISBN 978-0821821015
Burton, D.M.: The History of Mathematics: An Introduction, 7th edn. McGraw Hill, New York (2010). ISBN 978-0073383156
Doneddu, A.: Étude de géométries planes ordonnées. Rend. Circ. Mat. Palermo Ser. II 19(1), 27–67 (1970). ISSN 0009-725X
Euclid of Alexandria: Elements. The Thirteen Books of Euclid’s Elements (3 vols., tr. by Thomas L. Heath, Cambridge University Press, 1925 edn.) (facsimile). Dover, New York (1956). ISBN 978-0486600888 (vol. 1), ISBN 978-0486600895 (vol. 2), ISBN 978-0486600901 (vol. 3); also available in a single volume from Digireads.com. ISBN 978-1420934762
Ewald, G.: Geometry: An Introduction. Ishi Press (2013). ISBN 978-4871877183
Greenberg, M.J.: Euclidean and Non-Euclidean Geometries, Development and History, 4th edn. W.H. Freeman and Co., New York (2008). ISBN 978-0716799481
Halmos, P.R.: Finite Dimensional Vector Spaces, 2nd edn. Springer, New York (1974, 1987). ISBN 978-0387900933
Hilbert, D.: Foundations of Geometry, 2nd edn. (tr. by P. Bernays). Open Court, Chicago (1971). Also available via Project Gutenberg at http://www.gutenberg.org/ebooks/17384 in a translation by E.J. Townsend, Open Court (1950). ISBN 978-0875481647
Holme, A.: Geometry, Our Cultural Heritage, 2nd edn. Springer, Heidelberg (2010). ISBN 978-3642144400
Karzel, H., Sperner, E.: Begründer einer neuen Ordnungstheorie. Mitteilungen der Mathematischen Gesellschaft in Hamburg, Hamburg 25, 33–44 (2006)
Lumiste, U.: Foundations of Geometry. Estonian Mathematical Society, Tartu (2009)
Martin, G.E.: Transformation Geometry: An Introduction to Symmetry, Chapter 14. Springer, New York (1982). ISBN 978-0387906362
Millman, R.S., Parker, G.D.: Geometry, a Metric Approach with Models, 2nd edn, Chapter 10. Springer, New York (1993). ISBN 978-0387974125
Moulton, F.R.: A simple non-desarguesian plane geometry. Trans. Am. Math. Soc. 3(2), 192-195 (1902). American Mathematical Society, Providence. ISSN 0002-9947, JSTOR 1986419
Pasch, M.: Vorlesungen Über Neuere Geometrie (Lectures Over More Recent Geometry), Ulan Press (2012, English reprint) ASIN: B009ZJXQ3A
Rosen, K.: Discrete Mathematics and Its Applications, 7th edn. McGraw Hill, Boston (2011). ISBN 978-0073383095
Sperner, E.: Die Ordnungsfunktionen einer Geometrie. Mathematische Annalen (Springer) 121, 107–130 (1949)
Tarski, A.: What is elementary geometry? In: Henkin, L., Suppes, P., Tarski, A. (eds.) The Axiomatic Method, with Special Reference to Geometry and Physics. Proceedings of an International Symposium at the Univ. of Calif., Berkeley, 26 December 1957–4 January 1958. Studies in Logic and the Foundations of Mathematics, pp. 16-29. North-Holland, Amsterdam. Brouwer Press (2007, Reprint). ISBN 978-1443728126. MR0106185
Tarski, A.: In: Tarski, J. (ed.) Introduction to Logic: And to the Methodology of Deductive Sciences, 4th edn. Dover, New York (1995). ISBN 978-0486284620
van der Waerden, B.L.: De logische grondslagen der Euklidische meetkunde (Dutch). Chr. Huygens 13, 65–84, 257–274 (1934)
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Specht, E.J., Jones, H.T., Calkins, K.G., Rhoads, D.H. (2015). Collineations Preserving Betweenness (COBE). In: Euclidean Geometry and its Subgeometries. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-23775-6_7
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