Abstract
The first part of this chapter uses the Plane Separation Axiom to show that a line in a plane has two disjoint sides, and to prove the basic properties of segments, rays, and lines that are needed for a coherent geometry. The remainder of the chapter is a study of the basic interactions between lines, angles, triangles, and quadrilaterals, comprising Pasch geometry.
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Notes
- 1.
The process for doing this is somewhat complex and involves some subtleties. A reader desiring an overview of Pasch geometry (cf Definition PSH.7) without indulging in the details of a strict development may proceed as follows: first, peruse the statements below of the Postulate of Pasch and the Plane Separation Axiom; then accept Theorem PSH.12 (the Plane Separation Theorem) as an axiom, and go on from there. It must be noted, however, that Theorem PSH.8 is needed in the development following Theorem PSH.12.
- 2.
It could be interesting to construct a theory in which space is divided into two half-spaces by a plane, in a manner analogous to the theory developed here which treats division of the plane into two half-planes by a line. We have not pursued this, but it is said to have been carried out by B. L. van der Waerden, in De logische grondslagen der Euklidische meetkunde (Dutch), Chr. Huygens 13, 65–84, 257–274 (1934) [22]. Axiomatizations of Pasch-like statements for hyperplanes (i. e., statements that hyperplanes divide the space into two half-spaces) have been presented by E. Sperner in Die Ordnungsfunktionen einer Geometrie, Math. Ann. 121, 107–130, (1949) [19]. For the significance of Sperner’s work in ordered geometries, see H. Karzel, Emanuel Sperner: Begründer einer neuen Ordnungstheorie, Mitt. Math. Ges. Hamburg 25, 33–44 (2006) [12].
- 3.
Part (B) is sometimes called “Pasch’s Theorem.”
References
Karzel, H., Sperner, E.: Begründer einer neuen Ordnungstheorie. Mitteilungen der Mathematischen Gesellschaft in Hamburg, Hamburg 25, 33–44 (2006)
Pasch, M.: Vorlesungen Über Neuere Geometrie (Lectures Over More Recent Geometry), Ulan Press (2012, English reprint) ASIN: B009ZJXQ3A
Sperner, E.: Die Ordnungsfunktionen einer Geometrie. Mathematische Annalen (Springer) 121, 107–130 (1949)
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). Pasch Geometry (PSH). In: Euclidean Geometry and its Subgeometries. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-23775-6_5
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