Saccheri quadrilateral

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Saccheri quadrilaterals

A Saccheri quadrilateral (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 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.[1]

For a Saccheri quadrilateral ABCD, the sides AD and BC (also called legs) are equal in length and perpendicular to the base AB. The top CD is called the summit or upper base and the angles at C and D are called the summit angles.

The advantage of using Saccheri quardrilaterals when considering the parallel postulate is that they place the mutually exclusive options in very clear terms:

Are the summit angles right angles, obtuse angles, or acute angles?

As it turns out, when the summit angles are right angles, the existence of this quadrilateral is equivalent to the statement expounded by Euclid's fifth postulate. When they are acute, this quadrilateral leads to hyperbolic geometry, and when they are obtuse, the quadrilateral leads to elliptical geometry (provided that other modifications are made to the postulates[2]). Saccheri himself, however, thought that both the obtuse and acute cases could be shown to be contradictory. He did show this in the obtuse case, but failed to properly handle the acute case.[3]

History[edit]

Saccheri quadrilaterals were first considered by Omar Khayyam (1048-1131) in the late 11th century in Book I of Explanations of the Difficulties in the Postulates of Euclid.[1] 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):

Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge.[4]

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.

It was not until 600 years later that Giordano Vitale made an advance on Khayyam in his book Euclide restituo (1680, 1686), 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, heroic, and ultimately flawed proof of the parallel postulate around the quadrilateral and its three cases, proving many theorems about its properties along the way.

Properties[edit]

Let ABCD be a Saccheri quadrilateral having AB as base, CA and DB the equal sides that are perpendicular to the base and CD the summit. The following properties are valid in any Saccheri quadrilateral in hyperbolic geometry.[5]

  • The summit angles (at C and D) are equal and acute.
  • The summit is longer than the base.
  • The line segment joining the midpoint of the base and the midpoint of the summit is mutually perpendicular to the base and summit.
  • The line segment joining the midpoints of the sides is not perpendicular to either side.
  • The above two line segments are perpendicular to each other.
  • The line segment joining the midpoint of the base and the midpoint of the summit divides the Saccheri quadrilateral into two Lambert quadrilaterals.
  • Two Saccheri quadrilaterals with congruent bases and congruent summit angles are congruent (i.e., the remaining pairs of corresponding parts are congruent).
  • Two Saccheri quadrilaterals with congruent summits and congruent summit angles are congruent.

A formula[edit]

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 formula

\cosh s = \cosh b \cdot \cosh^2 l - \sinh^2 l.[6]

Examples[edit]

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:

Hyperbolic domains 2233.png
*3322 symmetry
Hyperbolic domains ii22.png
*∞∞22 symmetry

See also[edit]

Notes[edit]

  1. ^ a b Boris Abramovich Rozenfelʹd (1988). A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space (Abe Shenitzer translation ed.). Springer. p. 65. ISBN 0-387-96458-4. 
  2. ^ Coxeter 1998, pg. 11
  3. ^ Faber 1983, pg. 145
  4. ^ Boris A Rosenfeld and Adolf P Youschkevitch (1996), Geometry, p.467 in Roshdi Rashed, Régis Morelon (1996), Encyclopedia of the history of Arabic science, Routledge, ISBN 0-415-12411-5.
  5. ^ Faber 1983, pp. 146 - 147
  6. ^ P. Buser and H. Karcher. Gromov's almost flat manifolds. Asterisque 81 (1981), page 104.

References[edit]

  • Coxeter, H.S.M. (1998), Non-Euclidean Geometry (6th ed.), Washington, D.C.: Mathematical Association of America, ISBN 0-88385-522-4 
  • Faber, Richard L. (1983), Foundations of Euclidean and Non-Euclidean Geometry, New York: Marcel Dekker, ISBN 0-8247-1748-1 
  • M. J. Greenberg, Euclidean and Non-Euclidean Geometries: Development and History, 4th edition, W. H. Freeman, 2008.
  • George E. Martin, The Foundations of Geometry and the Non-Euclidean Plane, Springer-Verlag, 1975