A Saccheriquadrilateral is a quadrilateral with two equal sides perpendicular to the base. It is named after Giovanni Gerolamo Saccheri, who used it extensively in his book Euclides ab omni naevo vindicatus first published in 1733, an attempt to prove the parallel postulate using the methodReductio ad absurdum. The first known consideration of the Saccheri quadrilateral was by Omar Khayyam in the late 11th century, and it may occasionally be referred to as the Khayyam-Saccheri quadrilateral. For a Saccheri quadrilateral ABCD, the sides AD and BC are equal in length, and also perpendicular to the base AB. The top CD is the summit or upper base and the angles at C and D are called the summit angles. The advantage of using Saccheri quadrilaterals when considering the parallel postulate is that they place the mutually exclusive options in very clear terms: As it turns out:
Saccheri himself, however, thought that both the obtuse and acute cases could be shown to be contradictory. He did show that the obtuse case was contradictory, but failed to properly handle the acute case.
History
Saccheri quadrilaterals were first considered by Omar Khayyam in the late 11th century in Book I of Explanations of the Difficulties in the Postulates of Euclid. Unlike many commentators on Euclid before and after him, Khayyam was not trying to prove the parallel postulate as such but to derive it from an equivalent postulate he formulated from "the principles of the Philosopher" : Khayyam then considered the three cases right, obtuse, and acute that the summit angles of a Saccheri quadrilateral can take and after proving a number of theorems about them, he refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid. It was not until 600 years later that Giordano Vitale made an advance on Khayyam in his book Euclide restituo, when he used the quadrilateral to prove that if three points are equidistant on the base AB and the summit CD, then AB and CD are everywhere equidistant. Saccheri himself based the whole of his long and ultimately flawed proof of the parallel postulate around the quadrilateral and its three cases, proving many theorems about its properties along the way.
Saccheri quadrilaterals in hyperbolic geometry
Let ABCD be a Saccheri quadrilateral having AB as base, CD as summit and CA and DB as the equal sides that are perpendicular to the base. The following properties are valid in any Saccheri quadrilateral in hyperbolic geometry:
The summit angles are equal and acute.
The summit is longer than the base.
Two Saccheri quadrilaterals are congruent if:
* the base segments and summit angles are congruent
* the summit segments and summit angles are congruent.
The line segment joining the midpoint of the base and the midpoint of the summit:
Tilings of the Poincaré diskmodel of the Hyperbolic plane exist having Saccheri quadrilaterals as fundamental domains. Besides the 2 right angles, these quadrilaterals have acute summit angles. The tilings exhibit a *nn22 symmetry, and include:
