the following axioms hold: C1: For any pair of non parallel points there is exactly one point with . C2: For any point and any cycle there are exactly two points with . C3: For any three points , pairwise non parallel, there is exactly one cycle which contains . C4: For any cycle , any point and any point and there exists exactly one cycle such that , i.e., touches at point . C5: Any cycle contains at least 3 points. There is at least one cycle and a point not in . For investigations the following statements on parallel classes (equivalent to C1, C2 respectively) are advantageous. C1′: For any two points we have C2′: For any point and any cycle we have: First consequences of the axioms are Analogously to Möbius and Laguerre planes we get the connection to the linear geometry via the residues. For a Minkowski plane and we define the local structure and call it the residue at point P. For the classical Minkowski plane is the real affine plane . An immediate consequence of axioms C1 to C4 and C1′, C2′ are the following two theorems. Minimal model The minimal model of a Minkowski plane can be established over the set of three elements: Parallel points: if and only if if and only if . Hence and . Finite Minkowski-planes For finite Minkowski-planes we get from C1′, C2′: This gives rise of the definition: For a finite Minkowski plane and a cycle of we call the integer the order of . Simple combinatorial considerations yield Miquelian Minkowski planes We get the most important examples of Minkowski planes by generalizing the classical real model: Just replace by an arbitrary field then we get in any case a Minkowski plane . Analogously to Möbius and Laguerre planes the Theorem of Miquel is a characteristic property of a Minkowski plane . Theorem (Miquel): For the Minkowski plane the following is true: If for any 8 pairwise not parallel points which can be assigned to the vertices of a cube such that the points in 5 faces correspond to concyclical quadruples, then the sixth quadruple of points is concyclical, too. (For a better overview in the figure there are circles drawn instead of hyperbolas.) Theorem (Chen): Only a Minkowski plane satisfies the theorem of Miquel. Because of the last theorem is called a miquelian Minkowski plane. Remark: