Lotschnittaxiom
The Lotschnittaxiom (German for "axiom of the intersecting perpendiculars") is an axiom in the foundations of geometry, introduced and studied by Friedrich Bachmann.[1] It states:
Perpendiculars raised on each side of a right angle intersect.
Bachmann showed that, in the absence of the Archimedean axiom, it is strictly weaker than the rectangle axiom, which states that there is a rectangle, which in turn is strictly weaker than the Parallel Postulate, as shown by Max Dehn. [2] In the presence of the Archimedean axiom, the Lotschnittaxiom is equivalent with the Parallel Postulate.
Equivalent formulations[edit]
As shown by Bachmann, the Lotschnittaxiom is equivalent to the statement
Through any point inside a right angle there passes a line that intersects both sides of the angle.
It was shown in[3] that it is also equivalent to the statement
The altitude in an isosceles triangle with base angles of 45° is less than the base.
and in [4] that it is equivalent to the following axiom proposed by Lagrange:[5]
If the lines a and b are two intersecting lines that are parallel to a line g, then the reflection of a in b is also parallel to g.
As shown in,[6] the Lotschnittaxiom is also equivalent to the following statements, the first one due to A. Lippman, the second one due to Henri Lebesgue [7]
Given any circle, there exists a triangle containing that circle in its interior.
Given any convex quadrilateral, there exists a triangle containing that convex quadrilateral in its interior.
Three more equivalent formulations, all purely incidence-geometric, were proved in:[8]
Given three parallel lines, there is a line that intersects all three of them.
There exist lines a and b, such that any line intersects a or b.
If the lines a_1, a_2, and a_3 are pairwise parallel, then there is a permutation (i,j,k) of (1,2,3) such that any line g which intersects a_i and a_j also intersects a_k.
In Bachmann's geometry of line-reflections[edit]
Its role in Friedrich Bachmann's absolute geometry based on line-reflections, in the absence of order or free mobility (the theory of metric planes) was studied in [9] and in.[10]
Connection with the Parallel Postulate[edit]
As shown in,[3] the conjunction of the Lotschnittaxiom and of Aristotle's axiom is equivalent to the Parallel Postulate.
References[edit]
- ^ Bachmann, Friedrich (1964), "Zur Parallelenfrage", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 27 (3–4): 173–192, doi:10.1007/BF02993215, S2CID 186240918.
- ^ Dehn, Max (1900), "Die Legendre'schen Sätze über die Winkelsumme im Dreieck", Mathematische Annalen, 53 (3): 404–439, doi:10.1007/BF01448980, S2CID 122651688
- ^ a b Pambuccian, Victor (1994), "Zum Stufenaufbau des Parallelenaxioms", Journal of Geometry, 51 (1–2): 79–88, doi:10.1007/BF01226859, hdl:2027.42/43033, S2CID 28056805
- ^ Pambuccian, Victor (2009), "On the equivalence of Lagrange's axiom to the Lotschnittaxiom", Journal of Geometry, 95 (1–2): 165–171, doi:10.1007/s00022-009-0018-2, S2CID 121123017
- ^ Grabiner, Judith V. (2009), "Why did Lagrange "prove" the parallel postulate?" (PDF), American Mathematical Monthly, 116: 3–18
- ^ Pambuccian, Victor; Schacht, Celia (2019), "Lippmann's axiom and Lebesgue's axiom are equivalent to the Lotschnittaxiom", Beiträge zur Algebra und Geometrie, 60 (4): 733–748, doi:10.1007/s13366-019-00445-y, S2CID 149747562
- ^ Lebesgue, Henri (1936), "Sur le postulatum d'Euclide", Bulletin International de l'Académie Yougoslave des Sciences et des Beaux-Arts, 29/30: 42–43
- ^ Pambuccian, Victor; Schacht, Celia (2021), "The ubiquitous axiom", Results in Mathematics, 76 (3): 1–39, doi:10.1007/s00025-021-01424-3, S2CID 236236967
- ^ Dress, Andreas (1966), "Lotschnittebenen. Ein Beitrag zum Problem der algebraischen Beschreibung metrischer Ebenen", Journal für die Reine und Angewandte Mathematik, 1966 (224): 90–112, doi:10.1515/crll.1966.224.90, S2CID 118080739
- ^ Dress, Andreas (1965), "Lotschnittebenen mit halbierbarem rechtem Winkel", Archiv für Mathematik, 16: 388–392, doi:10.1007/BF01220047, S2CID 122588823
Sources[edit]
- Bachmann, Friedrich (1964), "Zur Parallelenfrage", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 27 (3–4): 173–192, doi:10.1007/BF02993215, S2CID 186240918
- Pambuccian, Victor (1994), "Zum Stufenaufbau des Parallelenaxioms", Journal of Geometry, 51 (1–2): 79–88, doi:10.1007/BF01226859, hdl:2027.42/43033, S2CID 28056805
- Pambuccian, Victor; Schacht, Celia (2019), "Lippmann's axiom and Lebesgue's axiom are equivalent to the Lotschnittaxiom", Beiträge zur Algebra und Geometrie, 60 (4): 733–748, doi:10.1007/s13366-019-00445-y, S2CID 149747562
- Pambuccian, Victor; Schacht, Celia (2021), "The ubiquitous axiom", Results in Mathematics, 76 (3): 1–39, doi:10.1007/s00025-021-01424-3, S2CID 236236967