Saccheri quadrilateral

While the quadrilaterals are named for Saccheri, they were considered in the works of earlier mathematicians. Saccheri's first proposition states that if two equal lines AD 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. Abner of Burgos's (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. 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 mathematician 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. ==Saccheri quadrilaterals in hyperbolic geometry==

Let ABCD be a Saccheri quadrilateral having base AB, summit CD, and legs CA and DB. The following properties are valid in any Saccheri quadrilateral in hyperbolic geometry: • The summit angles C and D 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: • Is perpendicular to the base and the summit, • is the only line of symmetry of the quadrilateral, • is the shortest segment connecting base and summit, • is perpendicular to the line joining the midpoints of the sides, • divides the Saccheri quadrilateral into two Lambert quadrilaterals. • The line segment joining the midpoints of the sides is not perpendicular to either side. Equations In the hyperbolic plane of constant curvature -1, the summit s of a Saccheri quadrilateral can be calculated from the leg l and the base b using the formulas :\begin{align} \cosh s &= \cosh b \cdot \cosh^2 l - \sinh^2 l \\[5mu] \sinh \tfrac12 s &= \cosh l\, \sinh\tfrac12b \end{align} A proof is in Tilings in the Poincaré disk model Tilings of the Poincaré disk model of the hyperbolic plane exist having Saccheri quadrilaterals as fundamental domains. Besides the two right angles, these quadrilaterals have acute summit angles. The tilings exhibit a symmetry (orbifold notation), and include: ==See also==