Saccheri square
A Saccheri quadrangle is a quadrangle in absolute geometry with the properties that two adjacent interior angles are right angles and two opposite sides, on which these angles are applied, are of equal length. Such squares were defined and examined by the Italian mathematician Giovanni Girolamo Saccheri in the first third of the 18th century, after whom they are named today. His original aim was to derive Euclid's 5th postulate, the axiom of parallels , from the other axioms with a proof of contradiction .
A square of this type was first studied by the Persian mathematician Omar Chayyam in the late 11th century, so the square is also (more correctly) called the Chayyam-Saccheri square . Whether Saccheri knew of Khayyam's writings is unknown.
History and characteristics
- In plane Euclidean geometry, of course, every Saccheri quadrangle is a rectangle .
- In plane absolute geometry, the following theorem applies:
- If the corner points of a quadrilateral are consecutively labeled A, B, C and D and the interior angles at A and B are right, then the side DA is longer, the same length or shorter than the side CB, depending on whether the interior angle at D is smaller than is equal to or greater than the angle at C.
- From this theorem it follows that the two interior angles of a Saccheri quadrilateral, about which nothing is stated in its definition, must always be the same in absolute geometry.
- Essentially correctly, Saccheri showed that in the axiom system of absolute geometry he used, which was by and large equivalent to axiom groups I-III and V in the axiom system of Euclidean geometry defined much later by David Hilbert , the angles in question were not obtuse could be. Hilbert formulated his system of axioms in such a way that his axioms without the axiom of parallels allow both Euclidean (there is a unique parallel through a point) and hyperbolic (there are several parallels through a point) models of geometry. In order to also be able to axiomatically grasp elliptical geometry, in which there is no parallel due to a point outside a straight line , Hilbert's axioms of arrangement (group II) and congruence (group III) are often replaced by weaker axioms of motion in absolute geometry . A more recent axiomatic of absolute geometry, which is based entirely on the concept of motion, is metric absolute geometry .
- In contrast, Saccheri's proof that these angles cannot be acute was flawed.
The axiom of parallels expressed by Saccheri quadrilaterals
Each of the following statements is based on the axioms of absolute geometry (according to Hilbert) equivalent to the parallels axiom (Axiom IV according to Hilbert):
- There is a rectangle.
- In one and therefore in every Saccheri quadrilateral, all interior angles are right.
literature
- Girolamo Saccheri: Euclides from omni naevo vindicatus . sive conatus geometricus quo stabiliuntur prima ipsa universae geometriae principia / Hieronymus Saccherius. 1733, doi : 10.3931 / e-rara-10433 .
- HSM Coxeter: Non-Euclidean Geometry . 6th edition. Mathematical Association of America, Washington DC 1998, ISBN 0-88385-522-4 .
- Richard L. Faber: Foundations of Euclidean and Non-Euclidean Geometry . Marcel Dekker, New York 1983, ISBN 0-8247-1748-1 .
- MJ Greenberg: Euclidean and Non-Euclidean Geometries: Development and History . 4th edition. WH Freeman, 2008.
- George E. Martin: The Foundations of Geometry and the Non-Euclidean Plane . Springer-Verlag, 1975.
- Richard Trudeau: The Geometric Revolution . Birkhäuser Verlag, Basel / Boston / Berlin 1998, ISBN 3-7643-5914-5 .
- Benno Klotzek: Euclidean and non-Euclidean elementary geometries . 1st edition. Harri Deutsch, Frankfurt am Main 2001, ISBN 3-8171-1583-0 .