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Saccheri quadrilaterals. A 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 1733 book Euclides ab omni naevo vindicatus (Euclid freed of every flaw), an attempt to prove the parallel postulate using the method reductio ad absurdum.
Saccheri is primarily known today for his last publication, in 1733 shortly before his death. Now considered an early exploration of non-Euclidean geometry, Euclides ab omni naevo vindicatus (Euclid Freed of Every Flaw) languished in obscurity until it was rediscovered by Eugenio Beltrami, in the mid-19th century.
English: Diagram of Saccheri quadrilaterals (right, obtuse, acute) Italiano: Quadrilatero di Saccheri (retto, ottuso, acuto) Français : Quadrilatère de Saccheri (droit, obtus, aigu)
Furthermore, rectangles can exist (a statement equivalent to the parallel postulate) only in Euclidean geometry. A Saccheri quadrilateral is a quadrilateral which has two sides of equal length, both perpendicular to a side called the base. The other two angles of a Saccheri quadrilateral are called the summit angles and they have equal measure ...
In geometry, many uniform tilings on sphere, euclidean plane, and hyperbolic plane can be made by Wythoff construction within a fundamental triangle, (p q r), defined by internal angles as π/p, π/q, and π/r. Special cases are right triangles (p q 2).
In hyperbolic geometry the fourth angle is acute, in Euclidean geometry it is a right angle and in elliptic geometry it is an obtuse angle. A Lambert quadrilateral can be constructed from a Saccheri quadrilateral by joining the midpoints of the base and summit of the Saccheri quadrilateral. This line segment is perpendicular to both the base ...
Hero of Alexandria (c. AD 10–70) – Euclidean geometry; Pappus of Alexandria (c. AD 290–c. 350) – Euclidean geometry, projective geometry; Hypatia of Alexandria (c. AD 370–c. 415) – Euclidean geometry; Brahmagupta (597–668) – Euclidean geometry, cyclic quadrilaterals
The polytopes of rank 2 (2-polytopes) are called polygons.Regular polygons are equilateral and cyclic.A p-gonal regular polygon is represented by Schläfli symbol {p}.. Many sources only consider convex polygons, but star polygons, like the pentagram, when considered, can also be regular.