A Saccheri quadrilateral (also known as a Khayyam–Saccheri quadrilateral) 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 (literally Euclid Freed of Every Flaw) first published in 1733, an attempt to prove the parallel postulate using the method Reductio ad absurdum. The Saccheri quadrilateral may occasionally be referred to as the Khayyam–Saccheri quadrilateral, in reference to the 11th century Persian scholar Omar Khayyam.
For a Saccheri quadrilateral ABCD, the sides AD and BC (also called the legs) 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.
While the quadrilaterals are named for Giovanni Gerolamo Saccheri, they were considered in the works of earlier mathematicians. Saccheri's first proposition states that if two equal lines AC and BC form equal angles with the line AB, the angles at CD will equal each other; a version of this statement appears in the works of the ninth century scholar Thabit ibn Qurra. Rectifying the Curved, a 14th century treatise written in Castile, builds off the work of Thabit ibn Qurra and also contains descriptions of Saccheri quadrilaterals. Omar Khayyam (1048-1131) described them in the late 11th century in Book I of his Explanations of the Difficulties in the Postulates of Euclid. Unlike many commentators on Euclid before and after him (including of course Saccheri), 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" (Aristotle):
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 (correctly) refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid.
The 17th century Italian matematician Giordano Vitale used the quadrilateral in his Euclide restituo (1680, 1686) 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.
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:
In the hyperbolic plane of constant curvature , the summit of a Saccheri quadrilateral can be calculated from the leg and the base using the formula
Tilings of the Poincaré disk model 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 (orbifold notation), and include: