1 Answers
In Euclidean geometry, the bundle theorem is a statement about six circles and eight points in the Euclidean plane. In general incidence geometry, it is a similar property that a Möbius plane may or may not satisfy. According to Kahn's Theorem, it is fulfilled by "ovoidal" Möbius planes only; thus, it is the analog for Möbius planes of Desargues' Theorem for projective planes.
Bundle theorem. If for eight different points A 1 , A 2 , A 3 , A 4 , B 1 , B 2 , B 3 , B 4 {\displaystyle A_{1},A_{2},A_{3},A_{4},B_{1},B_{2},B_{3},B_{4}} five of the six quadruples Q i j := { A i , B i , A j , B j } , i < j , {\displaystyle Q_{ij}:=\{A_{i},B_{i},A_{j},B_{j}\},\ i The bundle theorem should not be confused with Miquel's theorem. An ovoidal Möbius plane in real Euclidean space may be considered as the geometry of the plane sections of an egglike surface, like a sphere, or an ellipsoid, or half a sphere glued to a suitable half of an ellipsoid, or the surface with equation x 4 + y 4 + z 4 = 1 {\displaystyle x^{4}+y^{4}+z^{4}=1} , etc. If the egglike surface is a sphere one gets the space model of the classical real Möbius plane, which is the "circle geometry" on the sphere.