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 . ${\displaystyle E}$ ${\displaystyle \Gamma \subset \operatorname {Sym} _{E}}$ ${\displaystyle E}$

Then:

${\displaystyle \Gamma }$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 . ${\displaystyle \Gamma }$ ${\displaystyle \Gamma }$ ${\displaystyle \Gamma }$

To the history of the sentence

Study of a Church.
Leonardo da Vinci (around 1488).
In Manuscript B of the Paris Manuscripts , Institut de France, Paris.

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

• Three-reflection set

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

1. ↑ George E. Martin: The Foundations of Geometry and the Non-Euclidean Plane. 1982, p. 386 ff
2. ↑ ^{a b} H. SM Coxeter: Immortal Geometry. 1963, p. 54
3. ↑ Martin, op. Cit., Pp. 386, 391-392
4. ^ Daniel Pedoe: Geometry and the Visual Arts. 1983, pp. 258-261
5. ^ David L. Johnson: Symmetries , Springer-Verlag, Berlin 2001, ISBN 1-85233-270-0 , Chapter 6.1: Leonardo's Theorem .
6. ↑ Martin, op.cit., P. 392
7. ↑ Pedoe, op.cit., P. 96
8. ^ Hermann Weyl: Symmetry. 1955, pp. 71,102