Leonardo theorem
The set of Leonardo ( English Leonardo's theorem ) is a theorem of absolute geometry , which the mathematician Hermann Weyl on According to Leonardo da Vinci is due. The sentence deals with the question of the structure of finite isometric groups of absolute planes .
Formulation of the sentence
The sentence can be stated in modern terms as follows:
- Given a level of absolute geometry and also a finite group of isometries on .
- Then:
- is either a cyclic group or is isomorphic to a dihedral group . The first case is when it consists only of rotations , while the second case is given when, in addition to rotations, at least one straight line reflection is included which is not the identical image .
To the history of the sentence
According to Hermann Weyl , Leonardo discovered the phrase when, in his architecture studies, he investigated the question of how to add chapels and niches to a building without destroying the symmetry of the building 's core.
See also
literature
- HSM Coxeter : Immortal Geometry . Translated into German by JJ Burckhardt (= Science and Culture . Volume 17 ). Birkhäuser Verlag, Basel / Stuttgart 1963, p. 54 ( MR0692941 ).
- George E. Martin: The Foundations of Geometry and the Non-Euclidean Plane (= Undergraduate Texts in Mathematics ). Springer Verlag, New York / Heidelberg / Berlin 1982, ISBN 0-387-90694-0 , p. 386 ff . ( MR0666074 - Reprint).
- Daniel Pedoe : Geometry and the Visual Arts . Reprint of the edition 1976. Dover Publications, New York 1983, ISBN 0-486-24458-X , pp. 95 ff., 258-261 .
- Hermann Weyl: symmetry . Birkhäuser Verlag, Basel / Stuttgart 1955, p. 71, 102 ( MR0079586 ).
Individual evidence
- ↑ George E. Martin: The Foundations of Geometry and the Non-Euclidean Plane. 1982, p. 386 ff
- ↑ a b H. SM Coxeter: Immortal Geometry. 1963, p. 54
- ↑ Martin, op. Cit., Pp. 386, 391-392
- ^ Daniel Pedoe: Geometry and the Visual Arts. 1983, pp. 258-261
- ^ David L. Johnson: Symmetries , Springer-Verlag, Berlin 2001, ISBN 1-85233-270-0 , Chapter 6.1: Leonardo's Theorem .
- ↑ Martin, op.cit., P. 392
- ↑ Pedoe, op.cit., P. 96
- ^ Hermann Weyl: Symmetry. 1955, pp. 71,102