TY - JOUR

T1 - The Ubiquitous Axiom

AU - Pambuccian, Victor

AU - Schacht, Celia

N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

PY - 2021/8

Y1 - 2021/8

N2 - This paper starts with a survey of what is known regarding an axiom, referred to as the Lotschnittaxiom, stating that the perpendiculars to the sides of a right angle intersect. Several statements are presented that turn out to rather unexpectedly be equivalent, with plane absolute geometry without the Archimedean axiom as a background, to the Lotschnittaxiom. One natural statement is shown to be strictly weaker than the Lotschnittaxiom, creating a chain of four statements, starting with the Euclidean parallel postulate, each weaker than the previous one. It then moves on to provide surprising equivalents, expressed as pure incidence statements, for both the Lotschnittaxiom and Aristotle’s axiom, whose conjunction is equivalent to the Euclidean parallel postulate. The new incidence-geometric axioms are shown to be syntactically simplest.

AB - This paper starts with a survey of what is known regarding an axiom, referred to as the Lotschnittaxiom, stating that the perpendiculars to the sides of a right angle intersect. Several statements are presented that turn out to rather unexpectedly be equivalent, with plane absolute geometry without the Archimedean axiom as a background, to the Lotschnittaxiom. One natural statement is shown to be strictly weaker than the Lotschnittaxiom, creating a chain of four statements, starting with the Euclidean parallel postulate, each weaker than the previous one. It then moves on to provide surprising equivalents, expressed as pure incidence statements, for both the Lotschnittaxiom and Aristotle’s axiom, whose conjunction is equivalent to the Euclidean parallel postulate. The new incidence-geometric axioms are shown to be syntactically simplest.

KW - Aristotle’s axiom

KW - Euclidean parallel postulate

KW - incidence geometry

KW - Lotschnittaxiom

KW - plane absolute geometry

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U2 - 10.1007/s00025-021-01424-3

DO - 10.1007/s00025-021-01424-3

M3 - Article

AN - SCOPUS:85105864229

VL - 76

JO - Results in Mathematics

JF - Results in Mathematics

SN - 1422-6383

IS - 3

M1 - 114

ER -